SOLUTIONS MANUAL for C TRIMBLE & ASSOCIATES PREALGEBRA NINTH EDITION Elayn Martin-Gay by StudyGuide

Contents

Chapter 1

Chapter 2

Chapter 3

Chapter 4

101

Chapter 5

164

Chapter 6

209

Chapter 7

246

Chapter 8

284

Chapter 9

330

Chapter 10

370

Appendices

389

Practice Final Exam

392

Chapter 1 Section 1.2 Practice Exercises 1. The place value of the 8 in 38,760,005 is millions. 2. The place value of the 8 in 67,890 is hundreds. 3. The place value of the 8 in 481,922 is tenthousands. 4. 54 is written as fifty-four. 5. 678 is written as six hundred seventy-eight. 6. 93,205 is written as ninety-three thousand, two hundred five.

5. In a whole number, each group of 3 digits is called a period. 6. The place value of the digit 4 in the whole number 264 is ones. 7. hundreds 8. To read (or write) a number, read from left to right. 9. 80,000 10. Dachsund Exercise Set 1.2

7. 679,430,105 is written as six hundred seventynine million, four hundred thirty thousand, one hundred five.

2. The place value of the 5 in 905 is ones.

8. Thirty-seven in standard form is 37.

6. The place value of the 5 in 79,050,000 is tenthousands.

9. Two hundred twelve in standard form is 212. 10. Eight thousand, two hundred seventy-four in standard form is 8,274 or 8274. 11. Five million, fifty-seven thousand, twenty-six in standard form is 5,057,026.

4. The place value of the 5 in 6527 is hundreds.

8. The place value of the 5 in 51,682,700 is tenmillions. 10. 316 is written as three hundred sixteen. 12. 5445 is written as five thousand, four hundred forty-five.

12. 4,026,301 = 4,000,000 + 20,000 + 6000 + 300 + 1

14. 42,009 is written as forty-two thousand, nine.

13. a.

16. 3,204,000 is written as three million, two hundred four thousand.

Find Australia in the “Country” column. Read from left to right until the “bronze” column is reached. Australia won 22 bronze medals.

b. Find the countries for which the entry in the “Total” column is greater than 60. The United States, China, ROC, and Great Britain won more than 60 medals. Vocabulary, Readiness & Video Check 1.2 1. The numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... are called whole numbers. 2. The number 1286 is written in standard form. 3. The number “twenty-one” is written in words. 4. The number 900 + 60 + 5 is written in expanded form.

18. 47,033,107 is written as forty-seven million, thirty-three thousand, one hundred seven. 20. 254 is written as two hundred fifty-four. 22. 105,447 is written as one hundred five thousand, four hundred forty-seven. 24. 1,110,000,000 is written as one billion, one hundred ten million. 26. 11,239 is written as eleven thousand, two hundred thirty-nine. 28. 202,700 is written as two hundred two thousand, seven hundred. 30. Four thousand, four hundred sixty-eight in standard form is 4468.

Chapter 1: The Whole Numbers

ISM: Prealgebra

32. Seventy-three thousand, two in standard form is 73,002. 34. Sixteen million, four hundred five thousand, sixteen in standard form is 16,405,016. 36. Two million, twelve in standard form is 2,000,012.

74. answers may vary 76. A quadrillion in standard form is 1,000,000,000,000,000. Section 1.3 Practice Exercises 1.

4135 + 252 4387

47,364 + 135,898 183, 262

38. Six hundred forty thousand, eight hundred eighty-one in standard form is 640,881. 40. Two hundred thirty-four thousand in standard form is 234,000.

1 1 11

42. One thousand, eight hundred fifteen in standard form is 1815. 44. Two hundred sixty million, one hundred thirtyeight thousand, five hundred sixty-nine dollars in standard form is $260,138,569. 46. Seven hundred fifteen in standard form is 715. 48. 789 = 700 + 80 + 9 50. 6040 = 6000 + 40 52. 20,215 = 20,000 + 200 + 10 + 5

3. Notice 12 + 8 = 20 and 4 + 6 = 10. 12 + 4 + 8 + 6 + 5 = 20 + 10 + 5 = 35 12 2

6432 789 54 + 28 7303

5. a.

54. 99,032 = 90,000 + 9000 + 30 + 2

b. 20 − 8 = 12 because 12 + 8 = 20

56. 47,703,029 = 40,000,000 + 7,000,000 + 700,000 + 3000 + 20 + 9

58. Mount Baker erupted in 1792, which is in standard form.

62. Mount St. Helens has an eruption listed in 1980. All other eruptions listed in the table occurred before this one. 64. More Golden retrievers are registered than German shepherds.

6. a.

9143 − 122 9021

Check: 9021 + 122 9143

978 − 851 127

Check:

7. a.

69 7 −49 64 8

70. The largest number is 77,753. 72. Yes

127 + 851 978

8 17

66. French bulldogs are second in popularity. 24 is written as twenty-four. 68. The maximum height of an average-size standard dachsund is 9 inches.

93 − 93 = 0 because 0 + 93 = 93.

d. 42 − 0 = 42 because 42 + 0 = 42.

60. Mount Shasta and Mount St. Helens have each had two eruptions listed.

14 − 6 = 8 because 8 + 6 = 14.

Check: 648 + 49 697

2 12

326 − 245 81

Check:

81 + 245 326

ISM: Prealgebra

1234 − 822 412

Chapter 1: The Whole Numbers

Check:

412 + 822 1234

40 0 −1 6 4 2 3 6

Check:

236 + 164 400

Check:

238 + 762 1000

9. 2 cm + 8 cm + 15 cm + 5 cm = 30 cm The perimeter is 30 centimeters. 10. 647 + 647 + 647 = 1941 The perimeter is 1941 feet. 11.

15, 759 − 458 15, 301 The radius of Neptune is 15,301 miles.

12. a.

11. 9625 − 647 = 8978

Vocabulary, Readiness & Video Check 1.3

91010

10 0 0 −762 238

10. 366 − 87 = 279

12. 10,711 − 8925 = 1786

8. 76 − 27 = 49 9. 147 − 38 = 109

9 3 1010

8. a.

7. 865 − 95 = 770

The country with the fewest threatened amphibians corresponds to the shortest bar, which is Peru.

b. To find the total number of threatened amphibians for Brazil, Peru, and Mexico, we add. 110 78 + 191 379 The total number of threatened amphibians for Brazil, Peru, and Mexico is 379. Calculator Explorations

1. The sum of 0 and any number is the same number. 2. In 35 + 20 = 55, the number 55 is called the sum and 35 and 20 are each called an addend. 3. The difference of any number and that same number is 0. 4. The difference of any number and 0 is the same number. 5. In 37 − 19 = 18, the number 37 is the minuend, the 19 is the subtrahend, and the 18 is the difference. 6. The distance around a polygon is called its perimeter. 7. Since 7 + 10 = 10 + 7, we say that changing the order in addition does not change the sum. This property is called the commutative property of addition. 8. Since (3 + 1) + 20 = 3 + (1 + 20), we say that changing the grouping in addition does not change the sum. This property is called the associative property of addition. 9. To add whole numbers, we line up place values and add from right to left.

2. 76 + 97 = 173

10. We cannot take 7 from 2 in the ones place, so we borrow one ten from the tens place and move it over to the ones place to give us 10 + 2, or 12.

3. 285 + 55 = 340

11. triangle; 3

4. 8773 + 652 = 9425

12. To find the additional money needed to purchase, subtract the savings account amount from the purchase price.

1. 89 + 45 = 134

5. 985 + 1210 + 562 + 77 = 2834 6. 465 + 9888 + 620 + 1550 = 12,523

Chapter 1: The Whole Numbers

ISM: Prealgebra

Exercise Set 1.3 2.

27 + 31 58

37 + 542 579

23 45 + 30 98

236 + 6243 6479

111

20.

1 11 2 1 2

10.

17, 427 + 821, 059 838, 486

12.

3 5 8 5 +7 28

22.

504, 218 321,920 38,507 + 594, 687 1, 459,332

24.

957 − 257 700

Check:

700 + 257 957

26.

55 − 29 26

Check:

26 + 29 55

28.

674 − 299 375

Check:

375 + 299 674

30.

300 − 149 151

Check:

32.

5349 − 720 4629

Check: 4629 + 720 5349

34.

724 − 16 708

Check: 708 + 16 724

36.

1983 − 1914 69

Check:

14.

64 28 56 25 + 32 205 11 2

16.

16 1056 748 + 7770 9590 1 111

18.

6789 4321 + 5555 16, 665

26 582 4 763 + 62,511 67,882

151 + 149 300 1

69 + 1914 1983

ISM: Prealgebra

Chapter 1: The Whole Numbers

11 11

38.

40, 000 − 23,582 16, 418

Check:

6050 − 1878 4172

Check:

16, 418 + 23,582 40, 000 111

40.

4172 + 1878 6050 11 11

42.

62, 222 − 39,898 22,324

44.

986 − 48 938

Check:

22,324 + 39,898 62, 222

80 93 17 9 +2 201

802 + 6487 7289 The sum of 802 and 6487 is 7289.

62. “Find the total” indicates addition. 12

89 45 2 19 + 341 496 The total of 89, 45, 2, 19, and 341 is 496.

66. “Increased by” indicates addition. 712 + 38 750 712 increased by 38 is 750.

48. 10, 000 − 1786 8214

68. “Less” indicates subtraction. 25 − 12 13 25 less 12 is 13.

1 11

50.

64. “Find the difference” indicates subtraction. 16 − 5 11 The difference of 16 and 5 is 11.

46.

60. “Find the sum” indicates addition.

12, 468 3 211 + 1 988 17, 667

52. 3 + 4 + 5 = 12 The perimeter is 12 centimeters. 54. Opposite sides of a rectangle have the same length. 9 + 3 + 9 + 3 = 12 + 12 = 24 The perimeter is 24 miles.

70. “Subtracted from” indicates subtraction. 90 − 86 4 86 subtracted from 90 is 4.

56. 6 + 5 + 7 + 3 + 4 + 7 + 5 = 37 The perimeter is 37 inches.

72. Subtract 7953 million from 8687 million. 8687 − 7953 734 The world’s projected population increase is 734 million.

58. The unknown vertical side has length 3 + 5 = 8 feet. The unknown horizontal side has length 8 + 4 = 12 feet. 8 + 3 + 4 + 5 + 12 + 8 = 40 The perimeter is 40 feet.

74. Subtract the discount from the regular price. 276 − 69 207 The first-year price is $207.

Chapter 1: The Whole Numbers 76.

78.

ISM: Prealgebra

164, 000 + 40, 000 204, 000 The total U.S. land area drained by the Ohio and Tennessee sub-basins is 204,000 square miles.

94. North Carolina has the fewest Aldi locations.

189, 000 − 75, 000 114, 000 The Upper Mississippi sub-basin drains 114,000 square miles more than the Lower Mississippi sub-basin.

98. The total number of Aldi locations in the states listed is 1335. 1335 + 923 2258 There are 2258 Aldi locations in the United States.

80. Opposite sides of a rectangle have the same length. 60 + 45 + 60 + 45 = 210 The perimeter is 210 feet. 82.

84.

59,320 − 55, 492 3 828 They traveled 3828 miles on their trip. 299,345 + 259,516 558,861 The total number of F-Series trucks and Silverados sold during the first six months of 2022 was 558,861.

96. 211 + 92 + 203 + 98 + 92 + 123 + 91 + 152 + 146 + 127 = 1335 The total number of Aldi locations in the ten states listed is 1335.

11 1

100.

102. The minuend is 2863 and the subtrahend is 1904. 104. The minuend is 86 and the subtrahend is 25. 106. answers may vary 21

108.

86. The shortest bar corresponds to the quietest reading. Leaves rustling is the quietest. 88.

90.

100 − 70 30 The difference in sound intensity between live rock music and loud television is 30 dB. 119 − 99 20 The difference in volume between the mid-size and a sub-compact car is 20 cubic feet.

92. Opposite sides of a rectangle have the same length. 18 + 12 + 18 + 12 = 60 The perimeter of the puzzle is 60 inches.

605 + 9779 10,384 The total highway mileage in Kansas is 10,384 miles.

773 659 + 481 1913 The given sum is correct. 1 2

110.

19 214 49 + 651 933 The given sum is incorrect, the correct sum is 933.

112.

389 + 89 478 The given difference is correct.

ISM: Prealgebra

Chapter 1: The Whole Numbers 3. a.

114.

116.

7168 + 547 7715 The given difference is incorrect. 7615 − 547 7068

b. To round 76,243 to the nearest hundred, observe that the digit in the tens place is 4. Since this digit is less than 5, we do not add 1 to the digit in the hundreds place. The number 76,243 rounded to the nearest hundred is 76,200.

10, 244 − 8 534 1 710

118. answers may vary Section 1.4 Practice Exercises 1. a.

To round 57 to the nearest ten, observe that the digit in the ones place is 7. Since the digit is at least 5, we add 1 to the digit in the tens place. The number 57 rounded to the nearest ten is 60.

b. To round 641 to the nearest ten, observe that the digit in the ones place is 1. Since the digit is less than 5, we do not add 1 to the digit in the tens place. The number 641 rounded to the nearest ten is 640. c.

2. a.

To round 325 to the nearest ten observe that the digit in the ones place is 5. Since the digit is at least 5, we add 1 to the digit in the tens place. The number 325 rounded to the nearest ten is 330. To round 72,304 to the nearest thousand, observe that the digit in the hundreds place is 3. Since the digit is less than 5, we do not add 1 to the digit in the thousands place. The number 72,304 rounded to the nearest thousand is 72,000.

b. To round 9222 to the nearest thousand, observe that the digit in the hundreds place is 2. Since the digit is less than 5, we do not add 1 to the digit in the thousands place. The number 9222 rounded to the nearest thousand is 9000. c.

To round 671,800 to the nearest thousand, observe that the digit in the hundreds place is 8. Since this digit is at least 5, we add 1 to the digit in the thousands place. The number 671,800 rounded to the nearest thousand is 672,000.

To round 3474 to the nearest hundred, observe that the digit in the tens place is 7. Since this digit is at least 5, we add 1 to the digit in the hundreds place. The number 3474 rounded to the nearest hundred is 3500.

To round 978,965 to the nearest hundred, observe that the digit in the tens place is 6. Since this digit is at least 5, we add 1 to the digit in the hundreds place. The number 978,865 rounded to the nearest hundred is 979,000.

4. 49 25 32 51 98

rounds to rounds to rounds to rounds to rounds to

50 30 30 50 + 100 260

3785 − 2479

rounds to rounds to

4000 − 2000 2000

11 16 19 + 31

10 20 20 + 30 80 The total distance is approximately 80 miles.

18, 617 − 1 607

rounds to rounds to rounds to rounds to

rounds to rounds to

18,600 − 1 600 17, 000 The difference in the reported number of pertussis cases between 2019 and 2021 was 17,000.

Vocabulary, Readiness & Video Check 1.4 1. To graph a number on a number line, darken the point representing the location of the number. 2. Another word for approximating a whole number is rounding.

Chapter 1: The Whole Numbers

ISM: Prealgebra

3. The number 65 rounded to the nearest ten is 70, but the number 61 rounded to the nearest ten is 60. 4. An exact number of products is 1265, but an estimate is 1000. 5. 3 is in the place value we’re rounding to (tens), and the digit to the right of this place value is 5 or greater, so we need to add 1 to the 3. 6. On a number line, 22 is closer to 20 than to 30. Thus, 22 rounded to the nearest ten is 20. 7. Each circled digit is to the right of the place value being rounded to and is used to determine whether or not we add 1 to the digit in the place value being rounded to. Exercise Set 1.4 2. To round 273 to the nearest ten, observe that the digit in the ones place is 3. Since this digit is less than 5, we do not add 1 to the digit in the tens place. The number 273 rounded to the nearest ten is 270. 4. To round 846 to the nearest ten, observe that the digit in the ones place is 6. Since this digit is at least 5, we add 1 to the digit in the tens place. The number 846 rounded to the nearest ten is 850. 6. To round 8494 to the nearest hundred, observe that the digit in the tens place is 9. Since this digit is at least 5, we add 1 to the digit in the hundreds place. The number 8494 rounded to the nearest hundred is 8500. 8. To round 898 to the nearest ten, observe that the digit in the ones place is 8. Since this digit is at least 5, we add 1 to the digit in the tens place. The number 898 rounded to the nearest ten is 900. 10. To round 82,198 to the nearest thousand, observe that the digit in the hundreds place is 1. Since this digit is less than 5, we do not add 1 to the digit in the thousands place. The number 82,198 rounded to the nearest thousand is 82,000. 12. To round 42,682 to the nearest ten-thousand, observe that the digit in the thousands place is 2. Since this digit is less than 5, we do not add 1 to the digit in the ten-thousands place. The number 42,682 rounded to the nearest ten-thousand is 40,000.

14. To round 179,406 to the nearest hundred, observe that the digit in the tens place is 0. Since this digit is less than 5, we do not add 1 to the digit in the hundreds place. The number 179,406 rounded to the nearest hundred is 179,400. 16. To round 96,501 to the nearest thousand, observe that the digit in the hundreds place is 5. Since this digit is at least 5, we add 1 to the digit in the thousands place. The number 96,501 rounded to the nearest thousand is 97,000. 18. To round 99,995 to the nearest ten, observe that the digit in the ones place is 5. Since this digit is at least 5, we add 1 to the digit in the tens place. The number 99,995 rounded to the nearest ten is 100,000. 20. To round 39,523,698 to the nearest million, observe that the digit in the hundred-thousands place is 5. Since this digit is at least 5, we add 1 to the digit in the millions place. The number 39,523,698 rounded to the nearest million is 40,000,000. 22. Estimate 7619 to a given place value by rounding it to that place value. 7619 rounded to the tens place is 7620, to the hundreds place is 7600, and to the thousands place is 8000. 24. Estimate 7777 to a given place value by rounding it to that place value. 7777 rounded to the tens place is 7780, to the hundreds place is 7800, and to the thousands place is 8000. 26. Estimate 85,049 to a given place value by rounding it to that place value. 85,049 rounded to the tens place is 85,050, to the hundreds place is 85,000, and to the thousands place is 85,000. 28. To round 41,529 to the nearest thousand, observe that the digit in the hundreds place is 5. Since this digit is at least 5, we add 1 to the digit in the thousands place. Therefore 41,529 miles rounded to the nearest thousand is 42,000 miles. 30. To round 60,149 to the nearest hundred, observe that the digit in the tens place is 4. Since this digit is less than 5, we do not add 1 to the digit in the hundreds place. Therefore, 60,149 days rounded to the nearest hundred is 60,100 days.

ISM: Prealgebra

Chapter 1: The Whole Numbers

32. To round 334,696,947 to the nearest million, observe that the digit in the hundred-thousands place is 6. Since this digit is at least 5, we add 1 to the digit in the millions place. Therefore, 334,696,947 rounded to the nearest million is 335,000,000. 34. To round 74,686 to the nearest hundred, observe that the digit in the tens place is 8. Since this digit is at least 5, we add 1 to the digit in the hundreds place. Therefore, $74,686 rounded to the nearest hundred is $74,700. 36. 2022: To round 13,700,000,000 to the nearest billion, observe that the digit in the hundredmillions place is 7. Since this digit is at least 5, we add 1 to the digit in the billions place. The number 13,700,000,000 rounded to the nearest billion is 14,000,000,000. 2021: To round 15,100,000,000 to the nearest billion, observe that the digit in the hundredmillions place is 1. Since this digit is less than 5, we do not add 1 to the digit in the billions place. The number 15,100,000,000 rounded to the nearest billion is 15,000,000,000. 38.

50. 542 + 789 + 198 is approximately 540 + 790 + 200 = 1530. The answer of 2139 is incorrect. 52. 5233 + 4988 is approximately 5200 + 5000 = 10,200. The answer of 9011 is incorrect. 54.

89 97 100 79 75 + 82

rounds to rounds to rounds to rounds to rounds to rounds to

90 100 100 80 80 + 80 530 The total score is approximately 530.

56.

588 689 277 143 59 + 802

rounds to rounds to rounds to rounds to rounds to rounds to

600 700 300 100 100 + 800 2600 The total distance is approximately 2600 miles.

52 33 15 + 29

rounds to rounds to rounds to rounds to

50 30 20 + 30 130

58.

1360 − 1240

rounds to rounds to

555 − 235

rounds to rounds to

560 − 240 320

1400 − 1200 200 The difference in price is approximately $200.

60.

64 41 + 133

rounds to rounds to rounds to

4050 3133 + 1220

rounds to rounds to rounds to

4100 3100 + 1200 8400

44.

1989 − 1870

rounds to rounds to

2000 − 1900 100

46.

799 1655 + 271

rounds to rounds to rounds to

40.

42.

60 40 + 130 230

The total distance is approximately 230 miles. 62.

6615 + 1737

rounds to rounds to

6600 + 1700 8300 The total enrollment is approximately 8300 students.

64. 588 hundred-thousands is 58,800,000 in standard form. 58,800,000 rounded to the nearest million is 59,000,000. 58,800,000 rounded to the nearest ten-million is 60,000,000.

800 1700 + 300 2800

48. 522 + 785 is approximately 520 + 790 = 1310. The answer of 1307 is correct.

66. 433 hundred-thousands is 43,300,000 in standard form. 43,300,000 rounded to the nearest million is 43,000,000. 43,300,000 rounded to the nearest ten-million is 40,000,000.

Chapter 1: The Whole Numbers

ISM: Prealgebra

68. 5698, for example, rounded to the nearest ten is 5700. 70. The largest possible number that rounds to 1,500,000 when rounded to the nearest hundredthousand is 1,549,999.

6. Area = length ⋅ width = (360 miles)(280 miles) = 100,800 square miles The area of Wyoming is 100,800 square miles. 7.

72. answers may vary 74.

5950 7693 + 8203

rounds to rounds to rounds to

6 000 7 700 + 8 200 21,900

The perimeter is approximately 21,900 miles.

16 × 45 80 640 720 The printer can print 720 pages in 45 minutes.

8. 5 × 84 = 420 12 × 3 = 36

Section 1.5 Practice Exercises 1. a.

(50)(0) = 0

d. 75 ⋅ 1 = 75

rounds to rounds to

200 × 400 80,000 There are approximately 80,000 words on 391 pages.

b. 30(2 + 3) = 30 ⋅ 2 + 30 ⋅ 3

1. 72 × 48 = 3456

7(2 + 8) = 7 ⋅ 2 + 7 ⋅ 8

2. 81 × 92 = 7452

3. a.

163 × 391

Calculator Explorations

6(4 + 5) = 6 ⋅ 4 + 6 ⋅ 5

2. a.

The total cost is $456.

6×0=0

b. (1)8 = 8

29 × 6 174 44

648 × 5 3240

306 × 81 306 24 480 24, 786

726 × 142 1 452 29 040 72 600 103, 092

420 + 36 456

3. 163 ⋅ 94 = 15,322 4. 285 ⋅ 144 = 41,040 5. 983(277) = 272,291 6. 1562(843) = 1,316,766 Vocabulary, Readiness & Video Check 1.5 1. The product of 0 and any number is 0. 2. The product of 1 and any number is the number. 3. In 8 ⋅ 12 = 96, the 96 is called the product and 8 and 12 are each called a factor. 4. Since 9 ⋅ 10 = 10 ⋅ 9, we say that changing the order in multiplication does not change the product. This property is called the commutative property of multiplication.

ISM: Prealgebra

Chapter 1: The Whole Numbers

5. Since (3 ⋅ 4) ⋅ 6 = 3 ⋅ (4 ⋅ 6), we say that changing the grouping in multiplication does not change the product. This property is called the associative property of multiplication. 6. Area measures the amount of surface of a region.

22.

9021 × 3 27, 063

24.

91 × 72 182 6370 6552

26.

526 23 1 578 10 520 12, 098

7. Area of a rectangle = length ⋅ width. 8. We know 9(10 + 8) = 9 ⋅ 10 + 9 ⋅ 8 by the distributive property. 9. distributive

10. to show that 8649 is actually multiplied by 70 and not by just 7 11. Area is measured in square units, and here we have meters by meters, or square meters; the answer is 63 square meters, or the correct units are square meters.

28.

708 × 21 708 14 160 14,868

30.

720 × 80 57, 600

12. Multiplication is also an application of addition since it is addition of the same addend. Exercise Set 1.5 2. 55 ⋅ 1 = 55 4. 27 ⋅ 0 = 0 6. 7 ⋅ 6 ⋅ 0 = 0 8. 1 ⋅ 41 = 41

32. (593)(47)(0) = 0 34. (240)(1)(20) = (240)(20) = 4800 36.

10. 5(8 + 2) = 5 ⋅ 8 + 5 ⋅ 2 12. 6(1 + 4) = 6 ⋅ 1 + 6 ⋅ 4 14. 12(12 + 3) = 12 ⋅ 12 + 12 ⋅ 3 16.

79 × 3 237

18.

638 × 5 3190

20.

882 × 2 1764

38.

1357 × 79 12 213 94 990 107, 203

807 127 5 649 16 140 80 700 102, 489

40.

1234 × 567 8 638 74 040 617 000 699, 678

Chapter 1: The Whole Numbers 42.

44.

ISM: Prealgebra

426 × 110 4 260 42 600 46,860

1876 × 1407 13 132 750 400 1 876 000 2, 639,532

62.

3310 × 3 9930

64.

14 × 8 112 There are 112 grams of fat in 8 ounces of hulled sunflower seeds.

66.

34 × 14 136 340 476 There are 476 seats in the room.

46. Area = (length)(width) = (13 inches)(3 inches) = 39 square inches

Perimeter = length + width + length + width = 13 + 3 + 13 + 3 = 32 inches

68. a. b.

48. Area = (length)(width) = (25 centimeters)(20 centimeters) = 500 square centimeters

Perimeter = length + width + length + width = 25 + 20 + 25 + 20 = 90 centimeters 50.

52.

982 × 650

rounds to rounds to

1000 × 700 700, 000

111 × 999

rounds to rounds to

100 × 1000 100, 000

20 × 3 60 There are 60 apartments in the building.

70. Area = (length)(width) = (60 feet)(45 feet) = 2700 square feet The area is 2700 square feet. 72. Area = (length)(width) = (776 meters)(639 meters) = 495,864 square meters The area is 495,864 square meters. 74.

64 × 17 448 640 1088 The 17 flash drives hold 1088 GB.

76.

365 × 3 1095 A cow eats 1095 pounds of grain each year.

54. 2872 × 12 is approximately 2872 × 10, which is 28,720. The best estimate is b. 56. 706 × 409 is approximately 700 × 400, which is 280,000. The best estimate is d. 58. 70 × 12 = (7 × 10) × 12 = 7 × (10 × 12) = 7 × 120 = 840

5 × 4 = 20 There are 20 apartments on one floor.

60. 9 × 900 = 8100

ISM: Prealgebra 78.

Chapter 1: The Whole Numbers

13 × 16 78 130 208 There are 208 grams of fat in 16 ounces.

98. 57 × 3 = 171 57 × 6 = 342 The problem is

57 × 63

100. answers may vary

Person

Number of persons

Cost per person

Student

$120

102. 3 × 161 = 483 2 × 479 = 958 483 + 958 + 254 = 1695 LeBron James scored 1695 points during the 2021−2022 regular season.

Nonstudent

$28

Section 1.6 Practice Exercises

Children under 12

$10

80.

Total Cost

Cost per Category

1. a.

8 9 72 because 8 ⋅ 9 = 72.

$158 b. 40 ÷ 5 = 8 because 8 ⋅ 5 = 40.

82. 3 × 26 = 78 There were 78 million Americans age 65+ in 2022. 84. −

126 8 118

24 = 4 because 4 ⋅ 6 = 24. 6

2. a.

7 = 1 because 1 ⋅ 7 = 7. 7

b. 5 ÷ 1 = 5 because 5 ⋅ 1 = 5.

86. 47 + 26 + 10 + 231 + 50 = 364 88.

90.

14 + 9 23 The total of 14 and 9 is 23.

10 = 10 because 10 ⋅ 1 = 10. 1

21 ÷ 21 = 1 because 1 ⋅ 21 = 21.

d. 4 ÷ 1 = 4 because 4 ⋅ 1 = 4.

92. 11 + 11 + 11 + 11 + 11 + 11 = 6 ⋅ 11 or 11 ⋅ 6 94. a.

3. a.

4 ⋅ 5 = 5 + 5 + 5 + 5 or 4 + 4 + 4 + 4 + 4

b. answers may vary 96.

11 1 11 because 11 ⋅ 1 = 11.

19 × 4 76 The product of 19 and 4 is 76.

31 × 50 1550

b. c.

0 = 0 because 0 ⋅ 7 = 0. 7

0 8 0 because 0 ⋅ 8 = 0. 7 ÷ 0 is undefined because if 7 ÷ 0 is a number, then the number times 0 would be 7.

d. 0 ÷ 14 = 0 because 0 ⋅ 14 = 0.

Chapter 1: The Whole Numbers

4. a.

5. a.

818 6 4908 −48 10 −6 48 −48 0 Check: 818 × 6 4908 553 4 2212 −20 21 −20 12 −12 0 Check: 553 × 4 2212 251 753 −6 15 −15 03 −3 0 Check: 251 × 3 753

ISM: Prealgebra

Check: 5100 × 9 = 45,900 6. a.

304 2128 −21 02 −0 28 −28 0

234 R 3 939 −8 13 −12 19 −16 3

Check: 234 ⋅ 4 + 3 = 939 b.

5 100 9 45,900 −45 09 −9 000

657 R 2 5 3287 −30 28 −25 37 −35 2

Check: 657 ⋅ 5 + 2 = 3287 7. a.

9067 R 2 9 81, 605 −81 06 −0 60 −54 65 −63 2

Check: 9067 ⋅ 9 + 2 = 81,605

Check: 304 × 7 = 2128

ISM: Prealgebra

Chapter 1: The Whole Numbers

5827 R 2 23,310 −20 33 −3 2 11 −8 30 −28 2

Check: 5827 ⋅ 4 + 2 = 23,310 524 R 12 8. 17 8920 −85 42 −34 80 −68 12

Calculator Explorations 1. 848 ÷ 16 = 53 2. 564 ÷ 12 = 47 3. 5890 ÷ 95 = 62 4. 1053 ÷ 27 = 39

49 R 60 9. 678 33, 282 −27 12 6 162 −6 102 60

32,886 = 261 126

143, 088 = 542 264

7. 0 ÷ 315 = 0

57 10. 3 171 −15 21 −21 0 Each student got 57 printer cartridges. 44 532 −48 52 −48 4 There will be 44 full boxes and 4 printers left over.

11. 12

12. Find the sum and divide by 7. 18 4 7 126 7 −7 35 56 16 −56 9 0 3 + 52 126 The average time is 18 minutes.

8. 315 ÷ 0 is an error. Vocabulary, Readiness & Video Check 1.6 1. In 90 ÷ 2 = 45, the answer 45 is called the quotient, 90 is called the dividend, and 2 is called the divisor. 2. The quotient of any number and 1 is the same number. 3. The quotient of any number (except 0) and the same number is 1. 4. The quotient of 0 and any number (except 0) is 0. 5. The quotient of any number and 0 is undefined. 6. The average of a list of numbers is the sum of the numbers divided by the number of numbers. 7. 0 8. zero; this zero becomes a placeholder in the quotient.

Chapter 1: The Whole Numbers

ISM: Prealgebra

9. 202 ⋅ 102 + 15 = 20,619 10. This tells us we have a division problem since division may be used to separate a quantity into equal parts. 11. addition and division Exercise Set 1.6

526 26. 4 2104 −20 10 −8 24 −24 0

Check: 526 ⋅ 4 = 2104

2. 72 ÷ 9 = 8 4. 24 ÷ 6 = 4 6. 0 ÷ 4 = 0 8. 38 ÷ 1 = 38 10.

49 =1 49

12.

45 =5 9

14.

12 is undefined 0

16. 6 ÷ 6 = 1 18. 7 ÷ 0 is undefined 20. 18 ÷ 3 = 6 22. 5

17 85 −5 35 −35 0

Check: 17 ⋅ 5 = 85 24. 8

80 640 −64 00

Check: 80 ⋅ 8 = 640

28.

0 =0 30 Check: 0 ⋅ 30 = 0

7 30. 8 56 −56 0

Check: 7 ⋅ 8 = 56 11 32. 11 121 −11 11 −11 0

Check: 11 ⋅ 11 = 121 34. 7

60 R 6 426 −42 06

Check: 60 ⋅ 7 + 6 = 426 413 R 1 36. 3 1240 −12 04 −3 10 −9 1

Check: 413 ⋅ 3 + 1 = 1240

ISM: Prealgebra

55 R 2 38. 3 167 −15 17 −15 2

Check: 55 ⋅ 3 + 2 = 167 833 R 1 40. 4 3333 −32 13 −12 13 −12 1

Check: 833 ⋅ 4 + 1 = 3333 32 42. 23 736 −69 46 −46 0

Check: 32 ⋅ 23 = 736 44. 42

48 2016 −168 336 −336 0

Check: 48 ⋅ 42 = 2016 46. 44

44 R 2 1938 −176 178 −176 2

Check: 44 ⋅ 44 + 2 = 1938

Chapter 1: The Whole Numbers

612 R 10 48. 12 7354 −72 15 −12 34 −24 10

Check: 612 ⋅ 12 + 10 = 7354 405 50. 14 5670 −56 07 −0 70 −70 0

Check: 405 ⋅ 14 = 5670 39 R 9 52. 64 2505 −192 585 −576 9

Check: 39 ⋅ 64 + 9 = 2505 47 54. 123 5781 −492 861 −861 0

Check: 47 ⋅ 123 = 5781 96 R 52 56. 240 23, 092 −21 60 1 492 −1 440 52

Check: 96 ⋅ 240 + 52 = 23,092

Chapter 1: The Whole Numbers

ISM: Prealgebra

3 040 68. 214 650,560 −642 85 −0 8 56 −8 56 00 −0 0

201 R 50 58. 203 40,853 −40 6 25 −0 253 −203 50

Check: 201 ⋅ 203 + 50 = 40,853 303 R 63 60. 543 164,592 −162 9 1 69 − 0 1 692 −1 629 63

13 R 3 70. 7 94 −7 24 −21 3 The quotient is 13 R 3.

Check: 303 ⋅ 543 + 63 = 164,592 13 62. 8 104 −8 24 −24 0 603 R 2 64. 5 3017 −30 01 −0 17 −15 2 1714 R 47 66. 50 85, 747 −50 35 7 −35 0 74 −50 247 −200 47

3 R 20 72. 32 116 −96 20 116 divided by 32 is 3 R 20.

15 R 3 78 −5 28 −25 3 The quotient is 15 R 3.

74. 5

58 4930 −425 680 −680 0 There are 58 students in the group.

76. 85

ISM: Prealgebra

Chapter 1: The Whole Numbers

252000 78. 21 5292000 −42 109 −105 42 −42 0 Each person received $252,000.

16 88. 320 5280 −320 2080 −1920 160 There are 16 whole feet in 1 rod.

412 80. 14 5768 −56 16 −14 28 −28 0 The truck hauls 412 bushels on each trip.

180 = 30 6

92.

23 R 1 84. 8 185 −16 25 −24 1 Yes, there is enough for a 22-student class. There is one 8-foot length and 1 additional foot of rope left over. That is, she has 9 feet of extra rope.

94.

16 96 −6 36 −36 0 The players each scored 16 touchdowns.

37 26 15 29 51 + 22 180 Average =

82. Lane divider = 25 + 25 = 50 105 50 5280 −50 28 −0 280 −250 30 There are 105 whole lane dividers.

86. 6

30 6 180 −18 00

90.

169 5 845 −5 34 −30 45 −45 0

121 200 185 176 + 163 845

Average =

845 = 169 5

92 96 90 85 92 + 79 534 Average =

96.

53 40 + 30 123

89 534 −48 54 −54 0

534 = 89 6

41 3 123 −12 03 −3 0

Chapter 1: The Whole Numbers

ISM: Prealgebra 120. answers may vary Possible answer: 2 and 2

98.

23 407 92 + 7011 7533

100.

712 × 54 2 848 35 600 38, 448

102.

712 − 54 658

122.

86 46 − 10 − 10 76 36 − 10 − 10 66 26 − 10 − 10 56 16 − 10 − 10 46 6 Therefore, 86 ÷ 10 = 8 R 6.

Mid-Chapter Review 1

9 R 25 106. 31 304 −279 25

42 63 + 89 194

7006 − 451 6555

108. The quotient of 200 and 20 is 200 ÷ 20, which is choice b.

87 × 52 174 4350 4524

104.

0 = 0 because 0 ⋅ 23 = 0 23

110. 40 divided by 8 is 40 ÷ 8, which is choice c. 112.

1,955, 000, 000 7,820, 000, 000 −4 38 −36 22 − 20 20 −2 0 0 000 000 The advertisers shown spent an average of $1,955,000,000.

2, 700, 000, 000 1,980, 000, 000 1,540, 000, 000 + 1, 600, 000, 000 7,820, 000, 000

562 4. 8 4496 −40 49 −48 16 −16 0

5. 1 ⋅ 67 = 67 36 is undefined. 0

114. The average will decrease; answers may vary.

116. No; answers may vary Possible answer: The average cannot be less than each of the four numbers.

7. 16 ÷ 16 = 1

118. 84 ÷ 21 = 4 The width is 4 inches. 20

ISM: Prealgebra

8. 5 ÷ 1 = 5 9. 0 ⋅ 21 = 0 10. 7 ⋅ 0 ⋅ 8 = 0 11. 0 ÷ 7 = 0 12. 12 ÷ 4 = 3 13. 9 ⋅ 7 = 63 14. 45 ÷ 5 = 9 15.

207 − 69 138

16.

207 + 69 276

17.

3718 − 2549 1169

18.

1861 + 7965 9826

182 R 4 19. 7 1278 −7 57 −56 18 −14 4

20.

1259 63 3 777 75 540 79,317

Chapter 1: The Whole Numbers

1099 R 2 21. 7 7695 −7 06 −0 69 −63 65 −63 2 111 R 1 22. 9 1000 −9 10 −9 10 −9 1 663 R 24 23. 32 21, 240 −19 2 2 04 −1 92 120 −96 24 1 076 R 60 24. 65 70, 000 −65 50 −0 5 00 −4 55 450 −390 60

25.

4000 − 2963 1037

26. 10, 000 − 101 9 899

Chapter 1: The Whole Numbers 27.

ISM: Prealgebra 38. 432,198 rounded to the nearest ten is 432,200. 432,198 rounded to the nearest hundred is 432,200. 432,198 rounded to the nearest thousand is 432,000.

303 × 101 303 30 300 30, 603

39. 6 + 6 + 6 + 6 = 24 6 × 6 = 36 The perimeter is 24 feet and the area is 36 square feet.

28. (475)(100) = 47,500 1

29.

62 + 9 71 The total of 62 and 9 is 71.

30.

62 × 9 558 The product of 62 and 9 is 558.

40. 14 + 7 + 14 + 7 = 42 14 × 7 98 The perimeter is 42 inches and the area is 98 square inches. 41.

6 R8 31. 9 62 −54 8 The quotient of 62 and 9 is 6 R 8.

32.

62 − 9 53 The difference of 62 and 9 is 53.

33.

200 − 17 183 17 subtracted from 200 is 183.

34.

432 − 201 231 The difference of 432 and 201 is 231.

35. 9735 rounded to the nearest ten is 9740. 9735 rounded to the nearest hundred is 9700. 9735 rounded to the nearest thousand is 10,000.

13 9 + 6 28 The perimeter is 28 miles.

42. The unknown vertical side has length 4 + 3 = 7 meters. The unknown horizontal side has length 3 + 3 = 6 meters. 3 4 3 7 6 +3 26 The perimeter is 26 meters. 3

43.

Average =

36. 1429 rounded to the nearest ten is 1430. 1429 rounded to the nearest hundred is 1400. 1429 rounded to the nearest thousand is 1000. 37. 20,801 rounded to the nearest ten is 20,800. 20,801 rounded to the nearest hundred is 20,800. 20,801 rounded to the nearest thousand is 21,000. 22

24 5 120 −10 20 −20 0

19 15 25 37 + 24 120

120 = 24 5

ISM: Prealgebra

Chapter 1: The Whole Numbers

124 4 496 −4 09 −8 16 −16 0

44.

108 131 98 + 159 496

Average =

45.

46.

496 = 124 4

28,547 − 26,372 2 175 The Twin Span Bridge is longer by 2175 feet. 329 × 18 2632 3290 5922 The amount spent on toys is $5922.

Section 1.7 Practice Exercises

12. 36 ÷ [20 − (4 ⋅ 2)] + 43 − 6 = 36 ÷ [20 − 8] + 43 − 6 = 36 ÷ 12 + 43 − 6 = 36 ÷ 12 + 64 − 6 = 3 + 64 − 6 = 61

13.

25 + 8 ⋅ 2 − 33 25 + 8 ⋅ 2 − 27 = 2(3 − 2) 2(1) 25 + 16 − 27 = 2 14 = 2 =7

14. 36 ÷ 6 ⋅ 3 + 5 = 6 ⋅ 3 + 5 = 18 + 5 = 23 15. Area = (side) 2 = (12 centimeters)2 = 144 square centimeters The area of the square is 144 square centimeters.

Calculator Explorations 1. 46 = 4096

1. 8 ⋅ 8 ⋅ 8 ⋅ 8 = 84

2. 56 = 15, 625

2. 3 ⋅ 3 ⋅ 3 = 33

3. 55 = 3125

3. 10 ⋅10 ⋅ 10 ⋅10 ⋅ 10 = 105

4. 76 = 117, 649

4. 5 ⋅ 5 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 = 52 ⋅ 46

5. 211 = 2048

5. 4 2 = 4 ⋅ 4 = 16

6. 68 = 1,679,616

6. 73 = 7 ⋅ 7 ⋅ 7 = 343

7. 7 4 + 53 = 2526

7. 111 = 11

8. 124 − 84 = 16, 640

8. 2 ⋅ 32 = 2 ⋅ 3 ⋅ 3 = 18

9. 63 ⋅ 75 − 43 ⋅ 10 = 4295

9. 9 ⋅ 3 − 8 ÷ 4 = 27 − 8 ÷ 4 = 27 − 2 = 25 2

10. 8 ⋅ 22 + 7 ⋅ 16 = 288

10. 48 ÷ 3 ⋅ 2 = 48 ÷ 3 ⋅ 4 = 16 ⋅ 4 = 64

11. 4(15 ÷ 3 + 2) − 10 ⋅ 2 = 8

11. (10 − 7)4 + 2 ⋅ 32 = 34 + 2 ⋅ 32 = 81 + 2 ⋅ 9 = 81 + 18 = 99

12. 155 − 2(17 + 3) + 185 = 300

Chapter 1: The Whole Numbers

ISM: Prealgebra

Vocabulary, Readiness & Video Check 1.7 1. In 25 = 32, the 2 is called the base and the 5 is called the exponent. 2. To simplify 8 + 2 ⋅ 6, which operation should be performed first? multiplication 3. To simplify (8 + 2) ⋅ 6, which operation should be performed first? addition

24. 81 = 8 26. 54 = 5 ⋅ 5 ⋅ 5 ⋅ 5 = 625 28. 33 = 3 ⋅ 3 ⋅ 3 = 27 30. 43 = 4 ⋅ 4 ⋅ 4 = 64 32. 83 = 8 ⋅ 8 ⋅ 8 = 512

4. To simplify 9(3 − 2) ÷ 3 + 6, which operation should be performed first? subtraction

34. 112 = 11 ⋅11 = 121

5. To simplify 8 ÷ 2 ⋅ 6, which operation should be performed first? division

36. 103 = 10 ⋅10 ⋅ 10 = 1000

6. exponent; base

38. 141 = 14

7. 1

40. 45 = 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 = 1024

8. division, multiplication, addition

42. 5 ⋅ 32 = 5 ⋅ 3 ⋅ 3 = 45

9. The area of a rectangle is length ⋅ width. A square is a special rectangle where length = width. Thus, the area of a square is

44. 2 ⋅ 7 2 = 2 ⋅ 7 ⋅ 7 = 98

side ⋅ side, or (side)2 .

48. 100 ÷ 10 ⋅ 5 + 4 = 10 ⋅ 5 + 4 = 50 + 4 = 54

Exercise Set 1.7 2. 5 ⋅ 5 ⋅ 5 ⋅ 5 = 5

46. 24 + 6 ⋅ 3 = 24 + 18 = 42

50. 42 ÷ 7 − 6 = 6 − 6 = 0 4

4. 6 ⋅ 6 ⋅ 6 ⋅ 6 ⋅ 6 ⋅ 6 ⋅ 6 = 6

52. 32 +

8 = 32 + 4 = 36 2

54. 3 ⋅ 4 + 9 ⋅ 1 = 12 + 9 = 21

6. 10 ⋅10 ⋅10 = 103

6+9÷3

6+3 9 = =1 9 9

8. 4 ⋅ 4 ⋅ 3 ⋅ 3 ⋅ 3 = 4 2 ⋅ 33

56.

10. 7 ⋅ 4 ⋅ 4 ⋅ 4 = 7 ⋅ 43

58. 62 ⋅ (10 − 8) = 62 ⋅ 2 = 36 ⋅ 2 = 72

12. 4 ⋅ 6 ⋅ 6 ⋅ 6 ⋅ 6 = 4 ⋅ 64

60. 53 ÷ (10 + 15) + 92 + 33 = 53 ÷ 25 + 92 + 33 = 125 ÷ 25 + 81 + 27 = 5 + 81 + 27 = 113

14. 6 ⋅ 6 ⋅ 2 ⋅ 9 ⋅ 9 ⋅ 9 ⋅ 9 = 62 ⋅ 2 ⋅ 94 16. 6 2 = 6 ⋅ 6 = 36

40 + 8

48 48 = =3 25 − 9 16

18. 63 = 6 ⋅ 6 ⋅ 6 = 216

62.

20. 35 = 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = 243

64. (9 − 7) ⋅ (12 + 18) = 2 ⋅ 30 = 60

22. 112 = 1 ⋅1 ⋅1 ⋅1 ⋅1 ⋅1 ⋅1 ⋅1 ⋅1 ⋅1 ⋅1 ⋅1 = 1 24

66.

5 −3

5(12 − 7) − 4 2

5 − 18

5(5) − 4 25 − 4 21 = = =3 25 − 18 25 − 18 7

ISM: Prealgebra

Chapter 1: The Whole Numbers

68. 18 − 7 ÷ 0 = undefined

84. [15 ÷ (11 − 6) + 22 ] + (5 − 1) 2 = [15 ÷ 5 + 22 ] + 42 = [15 ÷ 5 + 4] + 42

70. 23 ⋅ 3 − (100 ÷ 10) = 23 ⋅ 3 − 10 = 8 ⋅ 3 − 10 = 24 − 10 = 14

= [3 + 4] + 42

72. [40 − (8 − 2)] − 25 = [40 − 6] − 25

= 34 − 25 = 34 − 32 =2 74. (18 ÷ 6) + [(3 + 5) ⋅ 2] = (18 ÷ 6) + (8 ⋅ 2) = 3 + (8 ⋅ 2) = 3 + 16 = 19 76. 35 ÷ [32 + (9 − 7) − 22 ] + 10 ⋅ 3 = 35 ÷ [32 + 2 − 22 ] + 10 ⋅ 3 = 35 ÷ [9 + 2 − 4] + 10 ⋅ 3 = 35 ÷ 7 + 10 ⋅ 3 = 5 + 10 ⋅ 3 = 5 + 30 = 35

78.

80.

3+9

3(10 − 6) − 2 − 1

86. 29 − {5 + 3[8 ⋅ (10 − 8)] − 50} = 29 − {5 + 3[8 ⋅ 2] − 50} = 29 − {5 + 3(16) − 50} = 29 − {5 + 48 − 50} = 29 − 3 = 26 88. Area of a square = (side) 2 = (9 centimeters)2 = 81 square centimeters Perimeter = 4(side) = 4(9 centimeters) = 36 centimeters

90. Area of a square = (side) 2

52 − 23 + 14 25 − 8 + 1 18 18 = = = =9 10 ÷ 5 ⋅ 4 ⋅1 ÷ 4 2 ⋅ 4 ⋅1 ÷ 4 8 ÷ 4 2 2

= 7 + 42 = 7 + 16 = 23

3 + 81

3(4) − 22 − 1 84 = 3(4) − 4 − 1 84 = 12 − 4 − 1 84 = 8 −1 84 = 7 = 12

82. 10 ÷ 2 + 33 ⋅ 2 − 20 = 10 ÷ 2 + 27 ⋅ 2 − 20 = 5 + 27 ⋅ 2 − 20 = 5 + 54 − 20 = 39

= (41 feet) 2 = 1681 square feet Perimeter = 4(side) = 4(41 feet) = 164 feet

92. The statement is true. 94. 49 = 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 The statement is false. 96. (2 + 3) ⋅ (6 − 2) = (5) ⋅ (4) = 20 98. 24 ÷ (3 ⋅ 2 + 2) ⋅ 5 = 24 ÷ (6 + 2) ⋅ 5 = 24 ÷ 8 ⋅ 5 = 3⋅5 = 15 100. The total perimeter is 1260 feet. 4 × 1260 = 5040 The total charge is $5040. 102. 253 ⋅ (45 − 7 ⋅ 5) ⋅ 5 = 253 ⋅ (45 − 35) ⋅ 5 = 253 ⋅ (10) ⋅ 5 = 15, 625 ⋅10 ⋅ 5 = 156, 250 ⋅ 5 = 781, 250

104. answers may vary

Chapter 1: The Whole Numbers

ISM: Prealgebra

Section 1.8 Practice Exercises 1. x − 2 = 7 − 2 = 5 2. y(x − 3) = 4(8 − 3) = 4(5) = 20 3.

y + 6 18 + 6 24 = = =4 x 6 6

4. 25 − z 3 + x = 25 − 23 + 1 = 25 − 8 + 1 = 18 5.

5( F − 32) 5(41 − 32) 5(9) 45 = = = =5 9 9 9 9

6. 3( y − 6) = 6 3(8 − 6) 0 6 3(2) 0 6 6 = 6 True Yes, 8 is a solution. 7. 5n + 4 = 34 Let n be 10. 5(10) + 4 0 34 50 + 4 0 34 54 = 34 False No, 10 is not a solution. Let n be 6. 5(6) + 4 0 34 30 + 4 0 34 34 = 34 True Yes, 6 is a solution. Let n be 8. 5(8) + 4 0 34 40 + 4 0 34 44 = 34 False No, 8 is not a solution. 8. a.

Twice a number is 2x.

b. 8 increased by a number is 8 + x or x + 8. c.

10 minus a number is 10 − x.

d. 10 subtracted from a number is x − 10. e.

The quotient of 6 and a number is 6 ÷ x or

6 . x

Vocabulary, Readiness & Video Check 1.8 1. A combination of operations on letters (variables) and numbers is an expression. 2. A letter that represents a number is a variable. 26

ISM: Prealgebra

Chapter 1: The Whole Numbers

3. 3x − 2y is called an expression and the letters x and y are variables. 4. Replacing a variable in an expression by a number and then finding the value of the expression is called evaluating the expression. 5. A statement of the form “expression = expression” is called an equation. 6. A value for the variable that makes an equation a true statement is called a solution. 7. When a letter and a variable are next to each other, the operation is an understood multiplication. 8. When first replacing f with 8, we don’t know if the statement is true or false. 9. decreased by Exercise Set 1.8 2.

a+b

a−b

a⋅b

a÷b

24 + 6 = 30

24 − 6 = 18

24 ⋅ 6 = 144

24 ÷ 6 = 4

a+b

a−b

a⋅b

a÷b

298

298 + 0 = 298

298 − 0 = 298

298 ⋅ 0 = 0

298 ÷ 0 is undefined.

a+b

a−b

a⋅b

a÷b

82 + 1 = 83

82 − 1 = 81

82 ⋅ 1 = 82

82 ÷ 1 = 82

8. 7 + 3z = 7 + 3(3) = 7 + 9 = 16 10. 4yz + 2x = 4(5)(3) + 2(2) = 60 + 4 = 64 12. x + 5y − z = 2 + 5(5) − 3 = 2 + 25 − 3 = 24 14. 2y + 5z = 2(5) + 5(3) = 10 + 15 = 25 16. y 3 − z = 53 − 3 = 125 − 3 = 122 18. 3 yz 2 + 1 = 3(5)(3)2 + 1 = 3⋅5⋅9 +1 = 135 + 1 = 136 20. 3 + (2 y − 4) = 3 + (2 ⋅ 5 − 4) = 3 + (10 − 4) = 3+ 6 =9 22. x 4 − ( y − z ) = 24 − (5 − 3) = 24 − 2 = 16 − 2 = 14

Chapter 1: The Whole Numbers

ISM: Prealgebra

24.

8 yz 8 ⋅ 5 ⋅ 3 120 = = =8 15 15 15

26.

6 + 3 x 6 + 3(2) 6 + 6 12 = = = =4 z 3 3 3

28.

2 z + 6 2 ⋅ 3 + 6 6 + 6 12 = = = =4 3 3 3 3

30.

70 15 70 15 70 15 − = − = − = 7−5 = 2 2 y z 2 ⋅ 5 3 10 3

32. 3x 2 + 2 x − 5 = 3 ⋅ 22 + 2 ⋅ 2 − 5 = 3⋅ 4 + 2 ⋅ 2 − 5 = 12 + 4 − 5 = 11 34. (4 y + 3 z )2 = (4 ⋅ 5 + 3 ⋅ 3)2

= (20 + 9)2 = 292 = 841 36. ( xz − 5) 4 = (2 ⋅ 3 − 5) 4 = (6 − 5)4 = 14 = 1 38. 3x( y + z ) = 3 ⋅ 2(5 + 3) = 3 ⋅ 2(8) = 6(8) = 48 40. xz (2 y + x − z ) = 2 ⋅ 3(2 ⋅ 5 + 2 − 3) = 2 ⋅ 3(10 + 2 − 3) = 2 ⋅ 3(9) = 6(9) = 54 42.

44.

6z + 2 y 6 ⋅ 3 + 2 ⋅ 5 = 4 4 18 + 10 = 4 28 = 4 =7 F

5( F − 32) 9

5(50 − 32) 5(18) = = 10 9 9

5(59 − 32) 5(27) = = 15 9 9

5(68 − 32) 5(36) = = 20 9 9

5(77 − 32) 5(45) = = 25 9 9

ISM: Prealgebra 46. Let n be 9. n−2 = 7 9−20 7 7 = 7 True Yes, 9 is a solution. 48. Let n be 50. 250 = 5n 250 0 5(50) 250 = 250 True Yes, 50 is a solution. 50. Let n be 8. 11n + 3 = 91 11(8) + 3 0 91 88 + 3 0 91 91 = 91 True Yes, 8 is a solution. 52. Let n be 0. 5(n + 9) = 40 5(0 + 9) 0 40 5(9) 0 40 45 = 40 False No, 0 is not a solution. 54. Let x be 2. 3 x − 6 = 5 x − 10 3(2) − 6 0 5(2) − 10 6 − 6 0 10 − 10 0 = 0 True Yes, 2 is a solution. 56. Let x be 5. 8 x − 30 = 2 x 8(5) − 30 0 2(5) 40 − 300 10 10 = 10 True Yes, 5 is a solution. 58. n + 3 = 16 Let n be 9. 9 + 3 0 16 12 = 16 False Let n be 11. 11 + 3 0 16 14 = 16 False Let n be 13. 13 + 3 0 16 16 = 16 True 13 is a solution.

Chapter 1: The Whole Numbers 60. 3n = 45 Let n be 15. 3 ⋅15 0 45 45 = 45 True Let n be 30. 3 ⋅ 30 0 45 90 = 45 False Let n be 45. 3 ⋅ 45 0 45 135 = 45 False 15 is a solution. 62. 4n + 4 = 24 Let n be 0. 4 ⋅ 0 + 4 0 24 0 + 4 0 24 4 = 24 False Let n be 5. 4 ⋅ 5 + 40 24 20 + 4 0 24 24 = 24 True Let n be 10. 4 ⋅10 + 4 0 24 40 + 4 0 24 44 = 24 False 5 is a solution. 64. 6(n + 2) = 23 Let n be 1. 6(1 + 2) 0 23 6(3) 0 23 18 = 23 False Let n be 3. 6(3 + 2) 0 23 6(5) 0 23 30 = 23 False Let n be 5. 6(5 + 2) 0 23 6(7) 0 23 42 = 23 False None are solutions. 66. 9x − 15 = 5x + 1 Let x be 2. 9 ⋅ 2 − 15 0 5 ⋅ 2 + 1 18 − 15 0 10 + 1 3 = 11 False Let x be 4. 9 ⋅ 4 − 15 0 5 ⋅ 4 + 1 36 − 15 0 20 + 1 21 = 21 True

Chapter 1: The Whole Numbers

ISM: Prealgebra

Let x be 11. 9 ⋅11 − 15 0 5 ⋅11 + 1 99 − 15 0 55 + 1 84 = 56 False 4 is a solution.

100. As F gets larger,

Chapter 1 Vocabulary Check 1. The whole numbers are 0, 1, 2, 3, ...

68. The sum of three and a number is 3 + x. 70. The difference of a number and five hundred is x − 500. 72. A number less thirty is x − 30.

76. A number divided by 11 is x ÷ 11 or

x . 11

78. The quotient of twenty and a number, decreased 20 by three is − 3. x 80. The difference of twice a number, and four is 2x − 4. 82. Twelve subtracted from a number is x − 12. 84. The sum of a number and 7 is x + 7. 86. The product of a number and 7 is 7x. 88. Twenty decreased by twice a number is 20 − 2x. 90. Replace x with 3 and y with 5. 6y + 3x = 6(5) + 3(3) = 30 + 9 = 39

3. The position of each digit in a number determines its place value.

5. To find the area of a rectangle, multiply length times width. 6. The digits used to write numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. 7. A letter used to represent a number is called a variable. 8. An equation can be written in the form “expression = expression.” 9. A combination of operations on variables and numbers is called an expression. 10. A solution of an equation is a value of the variable that makes the equation a true statement. 11. A collection of numbers (or objects) enclosed by braces is called a set. 12. The 21 above is called the sum.

92. Replace x with 3 and y with 5. x3 + 4 y = 33 + 4(5) = 27 + 20 = 47

13. The 5 above is called the divisor. 14. The 35 above is called the dividend.

94. 2( x + y )2 = 2(23 + 72)2 = 2(95)2 = 2(9025) = 18, 050

15. The 7 above is called the quotient. 16. The 3 above is called a factor.

96. 16 y − 20 x + x3 = 16 ⋅ 72 − 20 ⋅ 23 + 233 = 1152 − 460 + 12,167 = 12,859

2. The perimeter of a polygon is its distance around or the sum of the lengths of its sides.

4. An exponent is a shorthand notation for repeated multiplication of the same factor.

74. A number times twenty is 20x.

98.

5( F − 32) gets larger. 9

x is the smallest; answers may vary. 3

17. The 6 above is called the product. 18. The 20 above is called the minuend. 19. The 9 above is called the subtrahend. 20. The 11 above is called the difference. 21. The 4 above is called an addend.

ISM: Prealgebra

Chapter 1: The Whole Numbers

Chapter 1 Review

16.

583 − 279 304

17.

428 + 21 449

18.

819 + 21 840

1. The place value of 4 in 7640 is tens. 2. The place value of 4 in 46,200,120 is tenmillions. 3. 7640 is written as seven thousand, six hundred forty. 4. 46,200,120 is written as forty-six million, two hundred thousand, one hundred twenty.

5. 3158 = 3000 + 100 + 50 + 8 6. 403,225,000 = 400,000,000 + 3,000,000 + 200,000 + 20,000 + 5000 7. Eighty-one thousand, nine hundred in standard form is 81,900.

19.

10. Locate Middle East in the first column and read across to the number in the 2022 column. There were 212,000,000 Internet users in Middle East in 2022.

21.

18 + 49 67 28 + 39 67

15.

462 − 397 65

91 3623 + 497 4211 82 1647 + 238 1967 11

23.

74 342 + 918 1334 The sum of 74, 342, and 918 is 1334. 2

24.

49 529 + 308 886 The sum of 49, 529, and 308 is 886.

25.

25,862 − 7 965 17,897 7965 subtracted from 25,862 is 17,897.

14.

8000 92 7908

22.

13.

−

121

11. Locate the smallest number in the 2016 column. Oceania/Australia had the fewest Internet users in 2016. 12. Locate the largest number in the 2013 column. Asia had the greatest number of Internet users in 2013.

4000 86 3914

20.

8. Six billion, three hundred four million in standard form is 6,304,000,000. 9. Locate Europe in the first column and read across to the number in the 2022 column. There were 750,000,000 Internet users in Europe in 2022.

−

Chapter 1: The Whole Numbers

ISM: Prealgebra

26.

39, 007 − 4 349 34, 658 4349 subtracted from 39,007 is 34,658.

27.

205 + 7318 7523 The total distance is 7523 miles.

11 1

28.

62,589 65,340 + 69, 770 197, 699 Their total earnings were $197,699.

29. 40 + 52 + 52 + 72 = 216 The perimeter is 216 feet. 30. 11 + 20 + 35 = 66 The perimeter is 66 kilometers. 31.

32.

653 − 341 312 The number of Internet users in Africa increased by 312 million or 312,000,000. 543 − 350 193 There were 193 million or 193,000,000 more Internet users in Latin America/Caribbean than in North America in 2022.

33. Find the shortest bar. The balance was the least in May. 34. Find the tallest bar. The balance was the greatest in August. 35.

280 − 170 110 The balance decreased by $110 from February to April.

36.

490 − 250 240 The balance increased by $240 from June to August.

37. To round 43 to the nearest ten, observe that the digit in the ones place is 3. Since this digit is less than 5, we do not add 1 to the digit in the tens place. The number 43 rounded to the nearest ten is 40. 38. To round 45 to the nearest ten, observe that the digit in the ones place is 5. Since this digit is at least 5, we add 1 to the digit in the tens place. The number 45 rounded to the nearest ten is 50. 39. To round 876 to the nearest ten, observe that the digit in the ones place is 6. Since this digit is at least 5, we add 1 to the digit in the tens place. The number 876 rounded to the nearest ten is 880. 40. To round 493 to the nearest hundred, observe that the digit in the tens place is 9. Since this digit is at least 5, we add 1 to the digit in the hundreds place. The number 493 rounded to the nearest hundred is 500. 41. To round 3829 to the nearest hundred, observe that the digit in the tens place is 2. Since this digit is less than 5, we do not add 1 to the digit in the hundreds place. The number 3829 rounded to the nearest hundred is 3800. 42. To round 57,534 to the nearest thousand, observe that the digit in the hundreds place is 5. Since this digit is at least 5, we add 1 to the digit in the thousands place. The number 57,534 rounded to the nearest thousand is 58,000. 43. To round 39,583,819 to the nearest million, observe that the digit in the hundred-thousands place is 5. Since this digit is at least 5, we add 1 to the digit in the millions place. The number 39,583,819 rounded to the nearest million is 40,000,000. 44. To round 768,542 to the nearest hundredthousand, observe that the digit in the tenthousands place is 6. Since this digit is at least 5, we add 1 to the digit in the hundred-thousands place. The number 768,542 rounded to the nearest hundred-thousand is 800,000.

ISM: Prealgebra

45.

46.

47.

Chapter 1: The Whole Numbers

3785 648 + 2866

rounds to rounds to rounds to

3800 600 + 2900 7300

5925 − 1787

rounds to rounds to

5900 − 1800 4100

630 192 271 56 703 454 + 329

rounds to rounds to rounds to rounds to rounds to rounds to rounds to

600 200 300 100 700 500 + 300 2700

54. 25(9)(4) = 225(4) = 900 or 25(4)(9) = 100(9) = 900 55. 26 ⋅ 34 ⋅ 0 = 0 56. 62 ⋅ 88 ⋅ 0 = 0 57.

586 × 29 5 274 11 720 16,994

58.

242 × 37 1694 7260 8954

59.

642 × 177 4 494 44 940 64 200 113, 634

60.

347 × 129 3 123 6 940 34 700 44, 763

61.

1026 401 1 026 410 400 411, 426

They traveled approximately 2700 miles. 48.

837, 427, 045 − 664, 099,841

837,000,000 − 664,000,000 173,000,000 The population of Europe was approximately 837,000,000 and the population of Latin America/Caribbean was approximately 664,000,000. The difference in population was about 173,000,000.

49.

276 × 8 2208

50.

349 × 4 1396

51.

57 × 40 2280

52.

69 × 42 138 2760 2898

rounds to rounds to

62.

2107 302 4 214 632 100 636,314 ×

53. 20(7)(4) = 140(4) = 560

Chapter 1: The Whole Numbers

ISM: Prealgebra

63. “Product” indicates multiplication. 250 × 6 1500 The product of 6 and 250 is 1500.

4 R2 72. 4 18 −16 2

64. “Product” indicates multiplication. 820 × 6 4920 The product of 6 and 820 is 4920.

73. 918 ÷ 0 is undefined.

65.

32 × 15 160 320 480

38 × 11 38 380 418

74. 0 ÷ 668 = 0

17, 265 × 20 345,300 The total cost for 20 students is $345,300.

33 R 2 75. 5 167 −15 17 −15 2

19 R 7 76. 8 159 −8 79 −72 7

Check: 19 × 8 + 7 = 159 77. 26

67. Area = (length)(width) = (13 miles)(7 miles) = 91 square miles 68. Area = (length)(width) = (25 centimeters)(20 centimeters) = 500 square centimeters 49 =7 7

Check:

7 ×7 49

70.

36 =4 9

Check:

9 ×4 36

24 R 2 626 −52 106 −104 2

Check: 24 × 26 + 2 = 626 78. 19

69.

35 R 15 680 −57 110 −95 15

Check: 35 × 19 + 15 = 680

5 R2 71. 5 27 −25 2

Check: 5 × 5 + 2 = 27 34

Check: 0 ⋅ 668 = 0

Check: 33 × 5 + 2 = 167

480 + 418 898 The total cost is $898.

66.

Check: 4 × 4 + 2 = 18

ISM: Prealgebra

Chapter 1: The Whole Numbers

18 R 2 83. 5 92 −5 42 −40 2 The quotient of 92 and 5 is 18 R 2.

506 R 10 79. 47 23, 792 −23 5 29 −0 292 −282 10

21 R 2 86 −8 06 −4 2 The quotient of 86 and 4 is 21 R 2.

Check: 506 × 47 + 10 = 23,792 80. 53

84. 4

907 R 40 48,111 −47 7 41 −0 411 −371 40

Check: 907 × 53 + 40 = 48,111 2793 R 140 81. 207 578, 291 −414 164 2 −144 9 19 39 −18 63 761 −621 140 Check: 2793 × 207 + 140 = 578,291 2012 R 60 82. 306 615, 732 −612 37 −0 3 73 −3 06 672 −612 60

27 85. 24 648 −48 168 −168 0 27 boxes can be filled with cans of corn. 13 22,880 −17 60 5 280 −5 280 0 There are 13 miles in 22,880 yards.

86. 1760

87. Divide the sum by 4. 76 49 32 + 47 204 The average is 51.

51 204 −20 04 −4 0

88. Divide the sum by 4.

Check: 2012 × 306 + 60 = 615,732

23 85 62 + 66 236 The average is 59.

59 236 −20 36 −36 0

Chapter 1: The Whole Numbers

ISM: Prealgebra

89. 82 = 8 ⋅ 8 = 64

105. (6 − 4)3 ⋅ [102 ÷ (3 + 17)] = (6 − 4)3 ⋅ [102 ÷ 20] = (6 − 4)3 ⋅ [100 ÷ 20]

90. 5 = 5 ⋅ 5 ⋅ 5 = 125

= 23 ⋅ 5 = 8⋅5 = 40

91. 5 ⋅ 9 2 = 5 ⋅ 9 ⋅ 9 = 405 92. 4 ⋅10 2 = 4 ⋅ 10 ⋅10 = 400

106. (7 − 5)3 ⋅ [92 ÷ (2 + 7)] = (7 − 5)3 ⋅ [92 ÷ 9]

93. 18 ÷ 2 + 7 = 9 + 7 = 16

= (7 − 5)3 ⋅ [81 ÷ 9] = 23 ⋅ 9 = 8⋅9 = 72

94. 12 − 8 ÷ 4 = 12 − 2 = 10 95.

96.

5(62 − 3) 2

3 +2

7(16 − 8) 23

5(36 − 3) 5(33) 165 = = = 15 9+2 11 11

7(8) 56 = =7 8 8

107.

108.

97. 48 ÷ 8 ⋅ 2 = 6 ⋅ 2 = 12

5⋅7 − 3⋅5 2

2(11 − 3 ) 4 ⋅ 8 − 1 ⋅11 3

3(9 − 2 )

35 − 15 20 20 = = =5 2(11 − 9) 2(2) 4

32 − 11 21 21 = = =7 3(9 − 8) 3(1) 3

98. 27 ÷ 9 ⋅ 3 = 3 ⋅ 3 = 9

109. Area = (side) 2 = (7 meters) 2 = 49 square meters

99. 2 + 3[15 + (20 − 17) ⋅ 3] + 5 ⋅ 2

110. Area = (side) 2 = (3 inches) 2 = 9 square inches

= 2 + 3[1 + 3 ⋅ 3] + 5 ⋅ 2 = 2 + 3[1 + 3 ⋅ 3] + 5 ⋅ 2 = 2 + 3[1 + 9] + 5 ⋅ 2 = 2 + 3 ⋅10 + 5 ⋅ 2 = 2 + 30 + 10 = 42 4

100. 21 − [2 − (7 − 5) − 10] + 8 ⋅ 2 = 21 − [24 − 2 − 10] + 8 ⋅ 2 = 21 − [16 − 2 − 10] + 8 ⋅ 2 = 21 − 4 + 8 ⋅ 2 = 21 − 4 + 16 = 33

101. 19 − 2(32 − 22 ) = 19 − 2(9 − 4) = 19 − 2(5) = 19 − 10 =9 102. 16 − 2(42 − 32 ) = 16 − 2(16 − 9) = 16 − 2(7) = 16 − 14 =2 103. 4 ⋅ 5 − 2 ⋅ 7 = 20 − 14 = 6

111.

2 x 2 ⋅ 5 10 = = =5 z 2 2

112. 4x − 3 = 4 ⋅ 5 − 3 = 20 − 3 = 17 113.

x+7 5+7 is undefined. = 0 y

114.

y 0 0 = = =0 5 x 5 ⋅ 5 25

115. x3 − 2 z = 53 − 2 ⋅ 2 = 125 − 2 ⋅ 2 = 125 − 4 = 121 116.

7 + x 7 + 5 12 = = =2 3z 3⋅ 2 6

117. ( y + z ) 2 = (0 + 2) 2 = 22 = 4 118.

100 y 100 0 + = + = 20 + 0 = 20 x 3 5 3

119. Five subtracted from a number is x − 5. 120. Seven more than a number is x + 7.

104. 8 ⋅ 7 − 3 ⋅ 9 = 56 − 27 = 29 36

ISM: Prealgebra

Chapter 1: The Whole Numbers

121. Ten divided by a number is 10 ÷ x or 122. The product of 5 and a number is 5x. 123. Let n be 5. n + 12 = 20 − 3 5 + 12 0 20 − 3 17 = 17 True Yes, 5 is a solution. 124. Let n be 23. n − 8 = 10 + 6 23 − 8 0 10 + 6 15 = 16 False No, 23 is not a solution. 125. Let n = 14. 30 = 3(n − 3) 30 0 3(14 − 3) 30 0 3(11) 30 = 33 False No, 14 is not a solution. 126. Let n be 20. 5(n − 7) = 65 5(20 − 7) 0 65 5(13) 0 65 65 = 65 True Yes, 20 is a solution. 127. 7n = 77 Let n be 6. 7 ⋅ 6 0 77 42 = 77 False Let n be 11. 7 ⋅11 0 77 77 = 77 True Let n be 20. 7 ⋅ 20 0 77 140 = 77 False 11 is a solution. 128. n − 25 = 150 Let n be 125. 125 − 25 0 150 100 = 150 False Let n be 145. 145 − 25 0 150 120 = 150 False Let n be 175.

175 − 25 0 150 150 = 150 True 175 is a solution.

10 . x

129. 5(n + 4) = 90 Let n be 14. 5(14 + 4) 0 90 5(18) 0 90 90 = 90 True Let n be 16. 5(16 + 4) 0 90 5(20) 0 90 100 = 90 False Let n be 26. 5(26 + 4) 0 90 5(30) 0 90 150 = 90 False 14 is a solution. 130. 3n − 8 = 28 Let n be 3. 3(3) − 8 0 28 9 − 8 0 28 1 = 28 False Let n be 7. 3(7) − 8 0 28 21 − 8 0 28 13 = 28 False Let n be 15. 3(15) − 8 0 28 45 − 80 28 37 = 28 False None are solutions. 131.

485 − 68 417

132.

729 − 47 682

133.

732 × 3 2196

134.

629 × 4 2516

Chapter 1: The Whole Numbers

135.

374 29 + 698 1101 21

136.

593 52 + 766 1411

458 R 8 137. 13 5962 −52 76 −65 112 −104 8 237 R 1 138. 18 4267 −36 66 −54 127 −126 1

139.

1968 × 36 11 808 59 040 70,848

140.

5324 × 18 42 592 53 240 95,832

141.

2000 − 356 1644

142.

9000 − 519 8481

ISM: Prealgebra 143. To round 842 to the nearest ten, observe that the digit in the ones place is 2. Since this digit is less than 5, we do not add 1 to the digit in the tens place. The number 842 rounded to the nearest ten is 840. 144. To round 258,371 to the nearest hundredthousand, observe that the digit in the tenthousands place is 5. Since this digit is at least 5, we add 1 to the digit in the hundred-thousands place. The number 258,371 rounded to the nearest hundred-thousand is 300,000. 145. 24 ÷ 4 ⋅ 2 = 6 ⋅ 2 = 12 146.

(15 + 3) ⋅ (8 − 5) 3

2 +1

(18)(3) 54 = =6 8 +1 9

147. Let n be 9. 5n − 6 = 40 5 ⋅ 9 − 6 0 40 45 − 6 0 40 39 = 40 False No, 9 is not a solution. 148. Let n be 3. 2n − 6 = 5n − 15 2(3) − 6 0 5(3) − 15 6 − 6 0 15 − 15 0 = 0 True Yes, 3 is a solution. 53 1714 −160 114 −96 18 There are 53 full boxes with 18 left over.

149. 32

150.

27 × 2 54

8 ×4 32

54 + 32 86 The total bill before taxes is $86.

ISM: Prealgebra

Chapter 1: The Whole Numbers

Chapter 1 Getting Ready for the Test 1. In the number 28,690,357,004, the digit 5 is in the ten-thousands place; D. 2. In the number 28,690,357,004, the digit 8 is in the billions place; E.

496 × 30 14,880

6. 69

3. In the number 28,690,357,004, the digit 6 is in the hundred-millions place; F. 4. In the number 28,690,357,004, the digit 0 to the far left is in the millions place; B. 5. To simplify 6 − 3 ⋅ 2, the first operation to perform is multiplying 3 ⋅ 2; C. 6. To simplify (6 − 3) ⋅ 2, the first operation to perform is subtracting 3 from 6; B. 7. To simplify 6 ÷ 3 ⋅ 2, the first operation to perform is dividing 6 by 3; D. 8. To simplify 6 + 3 − 2, the first operation to perform is adding 6 and 3; A. 9. 5 ⋅ 23 = 5 ⋅ 8 = 40; C 10. Since 35 ÷ 5 = 7, the expression is a ÷ b; B. 11. since 35 ⋅ 5 = 175, the expression is ab; D. 12. Since 35 − 5 = 30, the expression is a − b; A. 13. Since 35 + 5 = 40, the expression is a + b; C. Chapter 1 Test 1. 82,426 in words is eighty-two thousand, four hundred twenty-six. 2. Four hundred two thousand, five hundred fifty in standard form is 402,550.

766 R 42 52,896 −48 3 4 59 −4 14 456 −414 42

7. 23 ⋅ 52 = 2 ⋅ 2 ⋅ 2 ⋅ 5 ⋅ 5 = 200 8. 98 ÷ 1 = 98 9. 0 ÷ 49 = 0 10. 62 ÷ 0 is undefined. 11. (24 − 5) ⋅ 3 = (16 − 5) ⋅ 3 = 11 ⋅ 3 = 33 12. 16 + 9 ÷ 3 ⋅ 4 − 7 = 16 + 3 ⋅ 4 − 7 = 16 + 12 − 7 = 28 − 7 = 21 13. 61 ⋅ 23 = 6 ⋅ 2 ⋅ 2 ⋅ 2 = 48 14. 2[(6 − 4)2 + (22 − 19) 2 ] + 10 = 2[22 + 32 ] + 10 = 2[4 + 9] + 10 = 2[13] + 10 = 26 + 10 = 36 15. 5698 ⋅ 1000 = 5,698,000 16. Divide the sum by 5.

59 + 82 141

600 − 487 113

62 79 84 90 + 95 410 The average is 82.

82 410 −40 10 −10 0

Chapter 1: The Whole Numbers

ISM: Prealgebra

17. To round 52,369 to the nearest thousand, observe that the digit in the hundreds place is 3. Since this digit is less than 5, we do not add 1 to the digit in the thousands place. The number 52,369 rounded to the nearest thousand is 52,000.

26.

45 × 8 360 There are 360 calories in 8 tablespoons of white granulated sugar.

18.

6289 5403 + 1957

rounds to rounds to rounds to

6 300 5 400 + 2 000 13,700

27.

19.

4267 − 2738

rounds to rounds to

4300 − 2700 1600

53 × 16 318 530 848

20.

21.

22.

107 − 15 92

15 + 107 122

Area = (side) 2 = (5 centimeters)2 = 25 square centimeters

107 × 15 535 1070 1605

29. Perimeter = (20 + 10 + 20 + 10) yards = 60 yards Area = (length)(width) = (20 yards)(10 yards) = 200 square yards 30. Replace x with 2. 5( x3 − 2) = 5(23 − 2) = 5(8 − 2) = 5(6) = 30

31. Replace x with 7 and y with 8. 3 x − 5 3(7) − 5 21 − 5 16 = = = =1 2y 2 ⋅8 16 16

17 493 −29 203 −203 0 Each can cost $17.

24. 29

848 + 1548 2396 The total cost is $2396.

28. Perimeter = (5 + 5 + 5 + 5) centimeters = 20 centimeters

7 R2 23. 15 107 −105 2

25.

129 × 12 258 1290 1548

32. a.

The quotient of a number and 17 is x ÷ 17 or x . 17

b. Twice a number, decreased by 20 is 2x − 20.

320 − 139 91 The higher-priced one is $91 more.

33. Replace n with 6. 5n − 11 = 19 5(6) − 11 0 19 30 − 11 0 19 19 = 19 True 6 is a solution.

ISM: Prealgebra

Chapter 1: The Whole Numbers

34. n + 32 = 4n + 2 Replace n with 0. 0 + 32 0 4(0) + 2 32 0 0 + 2 32 = 2 False Replace n with 10. 10 + 32 0 4(10) + 2 42 0 40 + 2 42 = 42 True Replace n with 20. 20 + 32 0 4(20) + 2 52 0 80 + 2 52 = 82 False 10 is a solution.

Chapter 2 Section 2.1 Practice Exercises 1. a.

If 0 represents the surface of the earth, then 3805 below the surface of the earth is −3805.

b. If zero degrees Fahrenheit is represented by 0°F, then 85 degrees below zero, Fahrenheit is represented by −85°F. 2. 3. a.

0 > −5 since 0 is to the right of −5 on a number line.

b. −3 < 3 since −3 is to the left of 3 on a number line. c. 4. a.

−7 > −12 since −7 is to the right of −12 on a number line. |−6| = 6 because −6 is 6 units from 0.

b. |4| = 4 because 4 is 4 units from 0. c. 5. a.

|−12| = 12 because −12 is 12 units from 0. The opposite of 14 is −14.

b. The opposite of −9 is −(−9) or 9. 6. a.

−|−7| = −7

b. −|4| = −4 c.

−(−12) = 12

7. −|x| = −|−6| = −6 8. The planet with the highest average daytime surface temperature is the one that corresponds to the bar that extends the furthest in the positive direction (upward). Venus has the highest average daytime surface temperature. Vocabulary, Readiness & Video Check 2.1 1. The numbers ...−3, −2, −1, 0, 1, 2, 3, ... are called integers. 2. Positive numbers, negative numbers, and zero together are called signed numbers. 3. The symbols “<” and “>” are called inequality symbols. 4. Numbers greater than 0 are called positive numbers while numbers less than 0 are called negative numbers. 5. The sign “<” means is less than and “>” means is greater than. 6. On a number line, the greater number is to the right of the lesser number. 7. A number’s distance from 0 on the number line is the number’s absolute value. 8. The numbers −5 and 5 are called opposites. 42

ISM: Prealgebra

Chapter 2: Integers and Introduction to Solving Equations

9. number of feet a miner works underground 10. The tick marks are labeled with the integers. 11. 0 will always be greater than any of the negative integers. 12. 8; |8| = 8 13. A negative sign can be translated into the phrase “opposite of.” 14. Lake Eyre Exercise Set 2.1 2. If 0 represents the surface of the water, then 25 feet below the surface of the water is −25. 4. If 0 represents sea level, then 282 feet below sea level is −282. 6. If 0 represents 0 degrees Fahrenheit, then 134 degrees above zero is +134. 8. If 0 represents the surface of the ocean, then 14,040 below the surface of the ocean is −14,040. 10. If 0 represents a loss of $0, then a loss of $400 million is −400 million. 12. If 0 represents 0° Celsius, then 10° below 0° Celsius is −10. Since 5° below 0° Celsius is −5 and −10 is less than −5, −10 (or 10° below 0° Celsius) is cooler. 14. From the blue map, the coldest temperature on record for the state of Arkansas is −29°F. 16. 18. 20. 22. 24. −8 < 0 since −8 is to the left of 0 on a number line. 26. −12 < −10 since −12 is to the left of −10 on a number line. 28. −27 > −29 since −27 is to the right of −29 on a number line. 30. 13 > −13 since 13 is to the right of −13 on a number line. 32. |7| = 7 since 7 is 7 units from 0 on a number line. 34. |−19| = 19 since −19 is 19 units from 0 on a number line. 36. |100| = 100 since 100 is 100 units from 0 on a number line. 38. |−10| = 10 since −10 is 10 units from 0 on a number line.

Chapter 2: Integers and Introduction to Solving Equations 40. The opposite of 8 is negative 8. −(8) = −8

ISM: Prealgebra

80. −|−8| = −8 −|−4| = −4 Since −8 < −4, −|−8| < −|−4|.

42. The opposite of negative 6 is 6. −(−6) = 6 44. The opposite of 123 is negative 123. −(123) = −123 46. The opposite of negative 13 is 13. −(−13) = 13 48. |−11| = 11

82. −(−38) = 38 Since −22 < 38, −22 < −(−38). 84. If the number is −13, then the absolute value of −13 is 13 and the opposite of −13 is 13. 86. If the opposite of a number is 90, then the number is −90 and its absolute value is 90. 88. The ‘bar’ that is equal to 0 corresponds to Lake Maracaibo, so Lake Maracaibo has an elevation at sea level.

50. −|43| = −43 52. −|−18| = −18

90. The bar that extends second to the farthest in the negative direction corresponds to Lake Eyre, so Lake Eyre has the second lowest elevation.

54. −(−27) = 27 56. −(−14) = 14

92. The smallest number on the graph is −269°C, which corresponds to helium.

58. −|−29| = −29 60. −|x| = −|−8| = −8

94. The number on the graph closest to +300°C is 280°C, which corresponds to phosphorus.

62. −|−x| = −|−10| = −10

96. 9 + 0 = 9

64. |x| = |32| = 32

98.

66. |−x| = |−1| = 1 68. −4 > −17 since −4 is to the right of −17 on a number line.

72. −|17| = −17 −(−17) = 17 Since −17 < 17, −|17| < −(−17).

−|−5|, −(−4), 23 , |10|.

76. −45 < 0 since −45 is to the left of 0 on a number line.

362 37 + 90 489

102. |10| = 10, 23 = 8, −|−5| = −5, and −(−4) = 4, so the numbers in order from least to greatest are

74. |−24| = 24 −(−24) = 24 Since 24 = 24, |−24| = −(−24).

78. |−45| = 45 |0| = 0 Since 45 > 0, |−45| > |0|.

100.

70. |−8| = 8 |−4| = 4 Since 8 > 4, |−8| > |−4|.

20 + 15 35

104. 14 = 1, −(−3) = 3, −|7| = −7, and |−20| = 20, so the numbers in order from least to greatest are −|7|, 14 , −(−3), |−20|. 106. 33 = 27, −|−11| = −11, −(−10) = 10, −4 = −4, −|2| = −2, so the numbers in order from least to greatest are −|−11|, −4, −|2|, −(−10), and 33.

ISM: Prealgebra 108. a.

Chapter 2: Integers and Introduction to Solving Equations

|0| = 0; since 0 < 4, then |0| > 4 is false.

9. −54 + 20 = −34

b. |−4| = 4; since 4 = 4, then |−4| > 4 is false.

10. 7 + (−2) = 5

|5| = 5; since 5 > 4, then |5| > 4 is true.

11. −3 + 0 = −3

d. |−100| = 100; since 100 > 4, then |−100| > 4 is true.

12. 18 + (−18) = 0 13. −64 + 64 = 0

110. (−|−(−7)|) = (−|7|) = −7 112. False; consider 0, where |0| = 0 and 0 is not positive. 114. True; zero is always less than a positive number since it is to the left of it on a number line. 116. No; b > a because b is to the right of a on the number line. 118. answers may vary

14. 6 + (−2) + (−15) = 4 + (−15) = −11 15. 5 + (−3) + 12 + (−14) = 2 + 12 + (−14) = 14 + (−14) =0 16. x + 3y = −6 + 3(2) = −6 + 6 = 0 17. x + y = −13 + (−9) = −22 18. Temperature at 8 a.m. = −7 + (+4) + (+7) = −3 + (+7) =4 The temperature was 4°F at 8 a.m.

120. no; answers may vary Section 2.2 Practice Exercises 1.

Calculator Explorations 1. −256 + 97 = −159 2. 811 + (−1058) = −247

3. 6(15) + (−46) = 44 4. −129 + 10(48) = 351

5. −108,650 + (−786,205) = −894,855 6. −196,662 + (−129,856) = −326,518

4. |−3| + |−19| = 3 + 19 = 22 The common sign is negative, so (−3) + (−19) = −22.

Vocabulary, Readiness & Video Check 2.2 1. If n is a number, then −n + n = 0.

5. −12 + (−30) = −42

2. Since x + n = n + x, we say that addition is commutative.

6. 9 + 4 = 13

3. If a is a number, then −(−a) = a.

7. |−1| = 1, |26| = 26, and 26 − 1 = 25 26 > 1, so the answer is positive. −1 + 26 = 25

4. Since n + (x + a) = (n + x) + a, we say that addition is associative.

8. |2| = 2, |−18| = 18, and 18 − 2 = 16 18 > 2, so the answer is negative. 2 + (−18) = −16

5. Negative; the numbers have different signs and the sign of the sum is the same as the sign of the number with the larger absolute value, −6.

Chapter 2: Integers and Introduction to Solving Equations

ISM: Prealgebra

6. Negative; the numbers have the same sign (both are negative) and we keep this common sign in the sum. 7. The diver’s current depth is 231 feet below the surface. Exercise Set 2.2 2.

−6 + (−5) = −11 4.

10 + (−3) = 7 6.

9 + (−4) = 5 8. 15 + 42 = 57 10. |−5| + |−4| = 5 + 4 = 9 The common sign is negative, so −5 + (−4) = −9. 12. −62 + 62 = 0 14. |8| − |−3| = 8 − 3 = 5 8 > 3, so the answer is positive. 8 + (−3) = 5 16. −8 + 0 = −8 18. |−9| − |5| = 9 − 5 = 4 9 > 5, so the answer is negative. 5 + (−9) = −4 20. |−6| + |−1| = 6 + 1 = 7 The common sign is negative, so −6 + (−1) = −7. 22. |−23| + |−23| = 23 + 23 = 46 The common sign is negative, so −23 + (−23) = −46. 24. |−400| + |−256| = 400 + 256 = 656 The common sign is negative, so −400 + (−256) = −656. 26. |24| − |−10| = 24 − 10 = 14 24 > 10, so the answer is positive. 24 + (−10) = 14

ISM: Prealgebra

Chapter 2: Integers and Introduction to Solving Equations

28. |−8| − |4| = 8 − 4 = 4 8 > 4, so the answer is negative. −8 + 4 = −4 30. |−89| − |37| = 89 − 37 = 52 89 > 37, so the answer is negative. −89 + 37 = −52 32. |62| − |−32| = 62 − 32 = 30 62 > 32, so the answer is positive. −32 + 62 = 30 34. |−375| − |325| = 375 − 325 = 50 375 > 325, so the answer is negative. 325 + (−375) = −50 36. |−56| + |−33| = 56 + 33 = 89 The common sign is negative, so −56 + (−33) = −89.

68. The sum of −49, −2, and 40 is −49 + (−2) + 40 = −51 + 40 = −11. 70. 0 + (−248) + 8 + (−16) + (−28) + 32 = −248 + 8 + (−16) + (−28) + 32 = −240 + (−16) + (−28) + 32 = −256 + (−28) + 32 = −284 + 32 = −252 The diver’s final depth is 252 meters below the surface. 72. Since −6 < −3, Smith won Round 4. 74. The bar for 2021 has a height of 94,680 so the net income in 2021 was $94,680,000,000. 76.

38. −1 + 5 + (−8) = 4 + (−8) = −4 40. −103 + (−32) + (−27) = −135 + (−27) = −162 42. 18 + (−9) + 5 + (−2) = 9 + 5 + (−2) = 14 + (−2) = 12 44. 34 + (−12) + (−11) + 213 = 22 + (−11) + 213 = 11 + 213 = 224 46. −12 + (−3) + (−5) = −15 + (−5) = −20 48. −35 + (−12) = −47

57, 410 94, 680 + 99,800 251,890 The total net income for 2020, 2021, and 2022 was $251,890,000,000.

78. 14 + (−5) + (−8) + 7 = 9 + (−8) + 7 = 1 + 7 = 8 Their total score was 8. 80. −10,412 + (−1786) + 15,395 + 31,418 = −12,198 + 15,395 + 31,418 = 3197 + 31,418 = 34,615 The net income for all the years shown is $34,615. 82. −60 + 43 = −17 Georgia’s all-time record low temperature is −17°F.

50. 3 + (−23) + 6 = −20 + 6 = −14 52. −100 + 70 = −30 54. (−45) + 22 + 20 = −23 + 20 = −3 56. −87 + 0 = −87

84. −10,924 + 3245 = −7679 The depth of the Aleutian Trench is −7679 meters. 86. 91 − 0 = 91

58. −16 + 6 + (−14) + (−20) = −10 + (−14) + (−20) = −24 + (−20) = −44 60. x + y = −1 + (−29) = −30

88.

400 − 18 382

90. answers may vary

62. 3x + y = 3(7) + (−11) = 21 + (−11) = 10 64. 3x + y = 3(13) + (−17) = 39 + (−17) = 22

92. −4 + 14 = 10 94. −15 + (−17) = −32

66. The sum of −30 and 15 is −30 + 15 = −15.

Chapter 2: Integers and Introduction to Solving Equations 96. True

6. −3 + 4 + (23) + (−10); all the subtraction operations are rewritten as additions in one step rather than changing one operation at a time as you work from left to right.

98. True 100. answers may vary

7. to follow the order of operations

Section 2.3 Practice Exercises

8. The warmest temperature is 263°F warmer than the coldest temperature.

1. 13 − 4 = 13 + (−4) = 9 2. −8 − 2 = −8 + (−2) = −10

Exercise Set 2.3

3. 11 − (−15) = 11 + 15 = 26

2. −6 − (−6) = −6 + 6 = 0

4. −9 − (−1) = −9 + 1 = −8

4. 15 − 12 = 15 + (−12) = 3

5. 6 − 9 = 6 + (−9) = −3

6. 2 − 5 = 2 + (−5) = −3

6. −14 − 5 = −14 + (−5) = −19

8. 12 − (−12) = 12 + 12 = 24

7. −3 − (−4) = −3 + 4 = 1

10. −25 − (−25) = −25 + 25 = 0

8. −15 − 6 = −15 + (−6) = −21

12. −2 − 42 = −2 + (−42) = −44

9. −6 − 5 − 2 − (−3) = −6 + (−5) + (−2) + 3 = −11 + (−2) + 3 = −13 + 3 = −10

14. 8 − 9 = 8 + (−9) = −1

10. 8 + (−2) − 9 − (−7) = 8 + (−2) + (−9) + 7 = 6 + (−9) + 7 = −3 + 7 =4 11. x − y = −5 − 13 = −5 + (−13) = −18 12. 3y − z = 3(9) − (−4) = 27 + 4 = 31 13. 29,028 − (−1312) = 29,028 + 1312 = 30,340 Mount Everest is 30,340 feet higher than the Dead Sea. Vocabulary, Readiness & Video Check 2.3 1. It is true that a − b = a + (−b). b

3. To evaluate x − y for x = −10 and y = −14, we replace x with −10 and y with −14 and evaluate −10 − (−14). d 4. The expression −5 − 10 equals −5 + (−10). c 5. additive inverse

16. 17 − 63 = 17 + (−63) = −46 18. 844 − (−20) = 844 + 20 = 864 20. −5 − 8 = −5 + (−8) = −13 22. −12 − (−5) = −12 + 5 = −7 24. 16 − 45 = 16 + (−45) = −29 26. −22 − 10 = −22 + (−10) = −32 28. −8 − (−13) = −8 + 13 = 5 30. −50 − (−50) = −50 + 50 = 0 32. −35 + (−11) = −46 34. 4 − 21 = 4 + (−21) = −17 36. −105 − 68 = −105 + (−68) = −173

2. The opposite of n is −n. a

ISM: Prealgebra

38. 86 − 98 = 86 + (−98) = −12 40. 8 − 4 − 1 = 8 + (−4) + (−1) = 4 + (−1) = 3 42. 30 − 18 − 12 = 30 + (−18) + (−12) = 12 + (−12) =0

ISM: Prealgebra

Chapter 2: Integers and Introduction to Solving Equations

44. −10 − 6 − (−9) = −10 + (−6) + 9 = −16 + 9 = −7 46. −15 + (−8) − 4 = −15 + (−8) + (−4) = −23 + (−4) = −27 48. 23 − (−17) + (−9) = 23 + 17 + (−9) = 40 + (−9) = 31 50. −(−9) − 14 + (−23) = 9 + (−14) + (−23) = −5 + (−23) = −28

74. −384 − (−505) = −384 + 505 = 121 The difference in elevation is 121 feet. 76. −236 − (−505) = −236 + 505 = 269 The difference in elevation is 269 feet. 78. 512 − (−92) = 512 + 92 = 604 The difference in elevation is 604 feet. 80. −52 − (−92) = −52 + 92 = 40 The difference in elevation is 40 feet. 82. 845 − (−162) = 845 + 162 = 1007 The difference in temperature is 1007°F.

52. −6 − (−8) + (−12) − 7 = −6 + 8 + (−12) + (−7) = 2 + (−12) + (−7) = −10 + (−7) = −17

84. 6037 − 4948 = 6037 + (−4948) = 1089 The trade balance was 1089 thousand barrels of crude oil per day that week.

54. 5 + (−18) − (−21) − 2 = 5 + (−18) + 21 + (−2) = −13 + 21 + (−2) = 8 + (−2) =6

88. Add a number and −36 is x + (−36).

86. The difference of −3 and a number is −3 − x.

90.

56. x − y = −7 − 1 = −7 + (−1) = −8 58. x − y = 9 − (−2) = 9 + 2 = 11 60. 2x − y = 2(8) − (−10) = 16 + 10 = 26 62. 2x − y = 2(14) − (−12) = 28 + 12 = 40 64. From the graph, the maximum and minimum temperatures for Texas are 120°F and −23°F. 120 − (−23) = 120 + 23 = 143 The temperature extreme for Texas is 143°F.

92.

96 = 32 3 32 3 96 −9 06 −6 0

51 × 89 459 4080 4539

66. The state with the hottest maximum is California. The maximum and minimum temperatures for California are 134°F and −45°F. 134 − (−45) = 134 + 45 = 179 The temperature extreme for California is 179°F.

94. answers may vary

68. 134 − (−80) = 134 + 80 = 214 Therefore, 134°F is 214°F warmer than −80°F.

100. |−12| − |−5| = 12 − 5 = 12 + (−5) = 7

70. 93 − 18 − 26 = 93 + (−18) + (−26) = 75 + (−26) = 49 $49 is now owed on the account. 72. 13,796 − (−21,857) = 13,796 + 21,857 = 35,653 The difference in elevation is 35,653 feet.

96. −4 − 8 = −4 + (−8) = −12 98. −3 − (−10) = −3 + 10 = 7

102. |−8| − |8| = 8 − 8 = 0 104. |−23| − |−42| = 23 − 42 = 23 + (−42) = −19 106. |−2 − (−6)| = |−2 + 6| = |4| = 4 |−2| − |−6| = 2 − 6 = 2 + (−6) = −4 Since 4 ≠ −4, the statement is false. 108. no; answers may vary

Chapter 2: Integers and Introduction to Solving Equations

ISM: Prealgebra 3. The quotient of two negative numbers is a positive number.

Section 2.4 Practice Exercises 1. −3 ⋅ 8 = −24 2. −5(−2) = 10

4. The quotient of a negative number and a positive number is a negative number.

3. 0 ⋅ (−20) = 0

5. The product of a negative number and zero is 0.

4. 10(−5) = −50

6. The quotient of 0 and a negative number is 0.

5. 8(−6)(−2) = −48(−2) = 96

7. The quotient of a negative number and 0 is undefined.

6. (−9)(−2)(−1) = 18(−1) = −18 7. (−3)(−4)(−5)(−1) = 12(−5)(−1) = −60(−1) = 60 8. (−2)4 = (−2)(−2)(−2)(−2) = 4(−2)(−2) = −8(−2) = 16

(−3)2 , the exponent applies to everything within the parentheses, so −3 is squared; in Video Notebook Number 4, −32 , the exponent does not apply to the sign and only 3 is squared.

9. −82 = −(8 ⋅ 8) = −64 10.

9. We can find out about sign rules for division because we know sign rules for multiplication.

42 = −6 −7

10. that ab means a ⋅ b 11. The phrase “lost four yards” in the example translates to the negative number −4.

11. −16 ÷ (−2) = 8 12.

−80 = −8 10

13.

−6 is undefined. 0

14.

0 =0 −7

8. When a negative sign is involved in an expression with an exponent, parentheses tell you whether or not the exponent applies to the negative sign. In Video Notebook Number 3,

Exercise Set 2.4 2. 5(−3) = −15 4. −7(−2) = 14 6. −9(7) = −63 8. −6(0) = 0

15. xy = 5 ⋅ (−8) = −40

10. −2(3)(−7) = −6(−7) = 42

x −12 16. = =4 y −3

12. −8(−3)(−3) = 24(−3) = −72

17. total score = 4 ⋅ (−13) = −52 The card player’s total score was −52. Vocabulary, Readiness & Video Check 2.4 1. The product of a negative number and a positive number is a negative number.

14. 2(−5)(−4) = −10(−4) = 40 16. 3(0)(−4)(−8) = 0 18. −2(−1)(3)(−2) = 2(3)(−2) = 6(−2) = −12 20. −24 = −(2)(2)(2)(2) = −4(2)(2) = −8(2) = −16

2. The product of two negative numbers is a positive number.

ISM: Prealgebra

Chapter 2: Integers and Introduction to Solving Equations

22. (−1)4 = (−1)(−1)(−1)(−1) = 1(−1)(−1) = −1(−1) =1

62.

0 =0 −14

64.

63 = −7 −9

24. −43 = −(4 ⋅ 4 ⋅ 4) = −64

66. 480 ÷ (−8) =

26. (−3) 2 = ( −3)(−3) = 9 28. 90 ÷ (−9) = −10 30.

56 = −7 −8

32.

−32 = −8 4

68.

0 =0 −15

38.

−24 =2 −12

72. (−11)2 = (−11)(−11) = 121 74. −1(2)(7)(−3) = −2(7)(−3) = −14(−3) = 42 76. (−1)33 = −1, since there are an odd number of factors. 78. −2(−2)(−3)(−2) = 4(−3)(−2) = −12(−2) = 24 80.

56 × 43 168 2240 2408 −56 ⋅ 43 = −2408

82.

23 70 1610 70 ⋅ (−23) = −1610

40. 0(−100) = 0 42. −6 ⋅ 2 = −12 44. −12(13) = −156

46. −9(−5) = 45 48. −7(−5)(−3) = 35(−3) = −105

84. ab = 5(−1) = −5

50. (−5)2 = (−5)(−5) = 25

86. ab = (−8)(8) = −64

30 52. − = −6 5

54. −

−36 = 12 −3

70. −23 = −(2 ⋅ 2 ⋅ 2) = −8

−13 34. is undefined. 0

36.

480 = −60 −8

88. ab = (−9)(−6) = 54

49 = −7 7

56. −15 ÷ 3 = −5

90.

x 9 = = −3 y −3

92.

x 0 = =0 y −5

94.

x −10 = =1 y −10

58. 6(−5)(−2) = −30(−2) = 60 60. −20 ⋅ 5 ⋅ (−5) ⋅ (−3) = −100 ⋅ (−5) ⋅ (−3) = 500 ⋅ (−3) = −1500

Chapter 2: Integers and Introduction to Solving Equations

120. 2015 to 2021 is 6 years. 1 969 6 11,814 −6 58 −5 4 41 −36 54 −54 0 1969 fewer birds would be banded per year, or −1969 birds per year.

96. xy = 20 ⋅ (−5) = −100 x 20 = = −4 y −5 98. xy = −3 ⋅ 0 = 0 x −3 is undefined. = y 0 100. −63 ÷ (−3) = 21 The quotient of −63 and −3 is 21. 102.

49 × 5 245 −49(5) = −245 The product of −49 and 5 is −245.

104. The quotient of −8 and a number is

ISM: Prealgebra

122. a. −8 or x

−8 ÷ x. 106. The sum of a number and −12 is x + (−12). 108. The difference of a number and −10 is x − (−10). 110. Multiply a number by −17 is x ⋅ (−17) or −17x. 112. A loss of $400 is represented by −400. 7 ⋅ (−400) = −2800 His total loss was $2800. 114. A drop of 5 degrees is represented by −5. 6 ⋅ (−5) = −30 The total drop in temperature was 30 degrees. 116. −1 ⋅ (−39) = 39 The melting point of rubidium is 39°C. 118. −11 ⋅ (−70) = 770 The melting point of strontium is 770°C.

27,200 − 12,700 = 14,500 There were about 14,500 more digital non3D movie screens in 2021 than in 2011. This is a change of 14,500 screens.

b. 2011 to 2021 is 10 years. 1 450 10 14,500 −10 45 −4 0 50 −50 00 The average change was 1450 screens per year. 124. 3 ⋅ (7 − 4) + 2 ⋅ 52 = 3 ⋅ 3 + 2 ⋅ 52 = 3 ⋅ 3 + 2 ⋅ 25 = 9 + 2 ⋅ 25 = 9 + 50 = 59 126. 12 ÷ (4 − 2) + 7 = 12 ÷ 2 + 7 = 6 + 7 = 13 128. −9(−11) = 99 130. −4 + (−3) + 21 = −7 + 21 = 14 132. −16 − (−2) = −16 + 2 = −14 134. The product of an even number of negative numbers is positive, so the product of ten negative numbers is positive.

ISM: Prealgebra

Chapter 2: Integers and Introduction to Solving Equations

136. (−1)50 and (−7) 20 are positive since there are an even number of factors. Note that (−7) 20 > (−1)50 since (−1)50 = 1. (−1)55 and ( −7) 23 are negative since there are an odd number of factors. Note that (−7) 23 < (−1)55 since (−1)55 = −1. 015 = 0 The numbers from least to greatest are

(−7)23 , (−1)55 , 015 , (−1)50 , (−7)20 .

138. answers may vary Mid-Chapter Review

17. −8(−6)(−1) = 48(−1) = −48 18. −18 ÷ 2 = −9 19. 65 + (−55) = 10 20. 1000 − 1002 = 1000 + (−1002) = −2 21. 53 − (−53) = 53 + 53 = 106 22. −2 − 1 = −2 + (−1) = −3 23.

0 =0 −47

24.

−36 =4 −9

1. Let 0 represent 0°F. Then 50 degrees below zero is represented by −50 and 122 degrees above zero is represented by +122 or 122.

25. −17 − (−59) = −17 + 59 = 42

26. −8 + (−6) + 20 = −14 + 20 = 6

3. 0 > −10 since 0 is to the right of −10 on a number line.

27.

4. −4 < 4 since −4 is to the left of 4 on a number line.

28. −9(100) = −900

5. −15 < −5 since −15 is to the left of −5 on a number line. 6. −2 > −7 since −2 is to the right of −7 on a number line. 7. |−3| = 3 because −3 is 3 units from 0. 8. |−9| = 9 because −9 is 9 units from 0.

11. The opposite of 11 is −11.

31.

−105 is undefined. 0

34. The sum of −17 and −27 is −17 + (−27) = −44.

12. The opposite of −3 is −(−3) = 3.

15. −3 + 15 = 12

30. −4 + (−8) − 16 − (−9) = −4 + (−8) + (−16) + 9 = −12 + (−16) + 9 = −28 + 9 = −19

33. Subtract −8 from −12 is −12 − (−8) = −12 + 8 = −4.

10. −(−5) = 5

14. The opposite of 0 is −0 = 0.

29. −12 − 6 − (−6) = −12 + (−6) + 6 = −18 + 6 = −12

32. 7(−16)(0)(−3) = 0 (since one factor is 0)

9. −|−4| = −4

13. The opposite of 64 is −64.

−95 = 19 −5

35. The product of −5 and −25 is −5(−25) = 125. 36. The quotient of −100 and −5 is 37. Divide a number by −17 is

16. −9 + (−11) = −20

−100 = 20. −5

x or x ÷ (−17). −17

38. The sum of −3 and a number is −3 + x.

Chapter 2: Integers and Introduction to Solving Equations

39. A number decreased by −18 is x − (−18). 40. The product of −7 and a number is −7 ⋅ x or −7x. 41. x + y = −3 + 12 = 9 42. x − y = −3 − 12 = −3 + (−12) = −15

ISM: Prealgebra

10. −4[−6 + 5(−3 + 5)] − 7 = −4[−6 + 5(2)] − 7 = −4[−6 + 10] − 7 = −4(4) − 7 = −16 − 7 = −23 11. x 2 = (−15) 2 = (−15)(−15) = 225

43. 2y − x = 2(12) − (−3) = 24 − (−3) = 24 + 3 = 27 44. 3y + x = 3(12) + (−3) = 36 + (−3) = 33 45. 5x = 5(−3) = −15 46.

− x 2 = −(−15)2 = −(−15)( −15) = −225

12. 5 y 2 = 5(4)2 = 5(16) = 80 5 y 2 = 5(−4) 2 = 5(16) = 80

y 12 = = −4 x −3

13. x 2 + y = (−6)2 + (−3) = 36 + (−3) = 33

Section 2.5 Practice Exercises

14. 4 − x 2 = 4 − (−8) 2 = 4 − 64 = −60

1. (−2)4 = (−2)(−2)(−2)(−2) = 16

15. average sum of numbers = number of numbers 17 + (−1) + (−11) + (−13) + (−16) + (−13) + 2 = 7 −35 = 7 = −5 The average of the temperatures is −5°F.

2. −24 = −(2)(2)(2)(2) = −16 3. 3 ⋅ 62 = 3 ⋅ (6 ⋅ 6) = 3 ⋅ 36 = 108 4.

−25 −25 = =5 5(−1) −5

−18 + 6 −12 = =3 −3 − 1 −4

Calculator Explorations

6. 30 + 50 + ( −4) = 30 + 50 + (−64) = 80 + (−64) = 16

−120 − 360 = 48 −10

4750 = −250 −2 + (−17)

−316 + (−458) = −258 28 + (−25)

−234 + 86 = 74 −18 + 16

7. −23 + (−4)2 + 15 = −8 + 16 + 1 = 8 + 1 = 9 8. 2(2 − 9) + (−12) − 3 = 2(−7) + (−12) − 3 = −14 + (−12) − 3 = −26 − 3 = −29 9. (−5) ⋅ −8 + (−3) + 23 = (−5) ⋅ 8 + (−3) + 23 = (−5) ⋅ 8 + (−3) + 8 = −40 + (−3) + 8 = −43 + 8 = −35

Vocabulary, Readiness & Video Check 2.5 1. To simplify −2 ÷ 2 ⋅ (3), which operation should be performed first? division 2. To simplify −9 − 3 ⋅ 4, which operation should be performed first? multiplication

ISM: Prealgebra

Chapter 2: Integers and Introduction to Solving Equations

3. The average of a list of numbers is sum of numbers . number of numbers 4. To simplify 5[−9 + (−3)] ÷ 4, which operation should be performed first? addition

26. 7 ⋅ 6 − 6 ⋅ 5 + (−10) = 42 − 6 ⋅ 5 + (−10) = 42 − 30 + (−10) = 12 + (−10) =2 28. 7 − (−5)2 = 7 − 25 = −18

5. To simplify −2 + 3(10 − 12) ⋅ (−8), which operation should be performed first? subtraction

30.

6. To evaluate x − 3y for x = −7 and y = −1, replace x with −7 and y with −1 and evaluate −7 − 3(−1).

32. 10 ⋅ 53 + 7 = 10 ⋅125 + 7 = 1250 + 7 = 1257

7. A fraction bar means divided by and it is a grouping symbol. 8. to make sure that the entire value of −2, including the sign, is squared 9. Finding an average is a good application of both order of operations and adding and dividing integers. Exercise Set 2.5 2. −24 = −(2)(2)(2)(2) = −16

−3 + 7 ⋅ 7 2 = 4 ⋅ 7 2 = 4 ⋅ 7 2 = 4 ⋅ 49 = 196

34. 82 − (5 − 2)4 = 82 − 34 = 64 − 81 = −17 36. |12 − 19| ÷ 7 = |−7| ÷ 7 = 7 ÷ 7 = 1 38. −(−2)3 = −(−8) = 8 40. (2 − 7)2 ÷ (4 − 3)4 = (−5) 2 ÷ 14 = 25 ÷ 1 = 25 42. 3 − 15 ⋅ (−4) ÷ (−16) = −12 ⋅ (−4) ÷ (−16) = 12 ⋅ (−4) ÷ (−16) = −48 ÷ (−16) =3

4. (−2)4 = (−2)(−2)(−2)(−2) = 16

44. (−20 − 5) ÷ 5 − 15 = (−25) ÷ 5 − 15 = −5 − 15 = −20

6. 5 ⋅ 23 = 5 ⋅ 8 = 40

46. 3 ⋅ (8 − 3) + (−4) − 10 = 3 ⋅ (5) + (−4) − 10 = 15 + (−4) − 10 = 11 − 10 =1

8. 10 − 23 − 12 = −13 − 12 = −25 10. −8 + 4(3) = −8 + 12 = 4 12. 7(−6) + 3 = −42 + 3 = −39 14. −12 + 6 ÷ 3 = −12 + 2 = −10 16. 5 + 9 ⋅ 4 − 20 = 5 + 36 − 20 = 41 − 20 = 21 18.

20 − 15 5 = = −5 −1 −1

20.

88 88 = = −8 −8 − 3 −11

22. 7(−4) − (−6) = −28 + 6 = −22 24. [9 + (−2)]3 = [7]3 = 343

48. (4 − 12) ⋅ (8 − 17) = (−8) ⋅ (−9) = 72 50. (−4 ÷ 4) − (8 ÷ 8) = (−1) − (1) = −2 52. (11 − 32 )3 = (11 − 9)3 = 23 = 8 54. −3(4 − 8)2 + 5(14 − 16)3 = −3(−4)2 + 5(−2)3 = −3(16) + 5(−8) = −48 + (−40) = −88 56. 12 − [7 − (3 − 6)] + (2 − 3)3 = 12 − [7 − (−3)] + (2 − 3)3 = 12 − (7 + 3) + (−1)3 = 12 − 10 + (−1) = 2 + (−1) =1

Chapter 2: Integers and Introduction to Solving Equations

58.

−10 − 6 10(−1) − (−2)(−3) = 2[−8 ÷ (−2 − 2)] 2[−8 ÷ (−4)] −16 = 2(2) −16 = 4 = −4

60. −2[6 + 4(2 − 8)] − 25 = −2[6 + 4(−6)] − 25 = −2[6 + (−24)] − 25 = −2(−18) − 25 = 36 − 25 = 11 62. x − y − z = −2 − 4 − (−1) = −2 − 4 + 1 = −6 + 1 = −5

94. 3 + 5 + 3 + 5 = 16 The perimeter is 16 centimeters.

104. answers may vary

72. − x = −(−3) = −9

106. (−17)6 = (−17)(−17)(−17)(−17)(−17)(−17) = 24,137,569

74. 3 x 2 = 3(−3)2 = 3(9) = 27 2

76. 3 − z = 3 − (−4) = 3 − 16 = −13 78. 3z 2 − x = 3(−4)2 − (−3) = 3(16) + 3 = 48 + 3 = 51

−18 + (−8) + (−1) + (−1) + 0 + 4 6 −24 = 6 = −4

98. (7 ⋅ 3 − 4) ⋅ 2 = (21 − 4) ⋅ 2 = 17 ⋅ 2 = 34

102. answers may vary

70. z = (−4) = 16

80. average =

96. 17 + 23 + 32 = 72 The perimeter is 72 meters.

100. 2 ⋅ (8 ÷ 4 − 20) = 2 ⋅ (2 − 20) = 2 ⋅ (−18) = −36

−20 + (−5) + (−5) −30 = = −10 3 3 The average of the scores is −10.

86. average =

92. 45 + 90 = 135

4 x 4(−2) −8 = = = −2 y 4 4

84. The two lowest scores are −20 and −5. −5 − (−20) = −5 + 20 = 15 The difference between the two lowest scores is 15.

90. 90 ÷ 45 = 2

66. x 2 + z = (−2) 2 + (−1) = 4 + (−1) = 3

−40 + (−20) + (−10) + (−15) + (−5) 5 −90 = 5 = −18

82. average =

88. no; answers may vary

64. 5 x − y + 4 z = 5(−2) − 4 + 4(−1) = −10 − 4 + (−4) = −14 + (−4) = −18

68.

ISM: Prealgebra

108. 3x 2 + 2 x − y = 3(−18)2 + 2(−18) − 2868 = 3(324) + (−36) − 2868 = 972 + (−36) − 2868 = 936 − 2868 = −1932 110. 5(ab + 3)b = 5(−2 ⋅ 3 + 3)3

= 5(−6 + 3)3 = 5(−3)3 = 5(−27) = −135

ISM: Prealgebra

Chapter 2: Integers and Introduction to Solving Equations

Section 2.6 Practice Exercises

−4 x − 3 = 5 −4(−2) − 3 0 5 8−30 5 5 = 5 True Since 5 = 5 is true, −2 is a solution of the equation.

y − 6 = −2 y − 6 + 6 = −2 + 6 y=4 Check: y − 6 = −2 4−60 −2 −2 = −2 True The solution is 4.

−2 = z + 8 −2 − 8 = z + 8 − 8 −10 = z Check: −2 = z + 8 −2 0 − 10 + 8 −2 = −2 True

The solution is −10. 4. x = −2 + 90 + (−100) x = 88 + (−100) x = −12 The solution is −12. 5.

3 y = −18 3 y −18 = 3 3 3 −18 ⋅y= 3 3 y = −6 Check: 3 y = −18 3(−6) 0 − 18 −18 = −18 True

The solution is −6. 6. −32 = 8 x −32 8 x = 8 8 −32 8 = ⋅x 8 8 −4 = x Check: −32 = 8 x −32 0 8(−4) −32 = −32 True

The solution is −4.

−3 y = −27 −3 y −27 = −3 −3 −3 −27 ⋅y= −3 −3 y=9 Check: −3 y = −27 −3 ⋅ 9 0 − 27 −27 = −27 True The solution is 9.

x =7 −4 x −4 ⋅ = −4 ⋅ 7 −4 −4 ⋅ x = −4 ⋅ 7 −4 x = −28 x =7 Check: −4 −28 07 −4 7 = 7 True The solution is −28.

Vocabulary, Readiness & Video Check 2.6 1. A combination of operations on variables and numbers is called an expression. 2. A statement of the form “expression = expression” is called an equation. 3. An equation contains an equal sign (=) while an expression does not. 4. An expression may be simplified and evaluated while an equation may be solved. 5. A solution of an equation is a number that when substituted for the variable makes the equation a true statement. 6. Equivalent equations have the same solution. 7. By the addition property of equality, the same number may be added to or subtracted from both sides of an equation without changing the solution of the equation. 8. By the multiplication property of equality, both sides of an equation may be multiplied or divided by the same nonzero number without changing the solution of the equation.

Chapter 2: Integers and Introduction to Solving Equations

ISM: Prealgebra 12.

9. an equal sign 10. We can add the same number to both sides of an equation and we’ll have an equivalent equation. Also, we can subtract the same number from both sides of an equation and have an equivalent equation. 11. To check a solution, we go back to the original equation, replace the variable with the proposed solution, and see if we get a true statement.

The solution is −8. 14.

Exercise Set 2.6 2. y − 16 = −7 9 − 16 0 − 7 −7 = −7 True Since −7 = −7 is true, 9 is a solution of the equation. 4.

a + 23 = −16 −7 + 23 0 − 16 16 = −16 False

Since 16 = −16 is false, −7 is not a solution of the equation. 6.

−3k = 12 − k −3(−6) 0 12 − (−6) 18 0 12 + 6 18 = 18 True Since 18 = 18 is true, −6 is a solution of the equation.

2(b − 3) = 10 2(2 − 3) 0 10 2(−1) 0 10 −2 = 10 False Since −2 = 10 is false, 2 is not a solution of the equation.

10.

f + 4 = −6 f + 4 − 4 = −6 − 4 f = −10

Check: f + 4 = −6 −10 + 0 − 6 −6 = −6 True The solution is −10.

s − 7 = −15 s − 7 + 7 = −15 + 7 s = −8 Check: s − 7 = −15 −8 − 7 0 − 15 −15 = −15 True

1= y+7 1− 7 = y + 7 − 7 −6 = y Check: 1 = y + 7 10 − 6 + 7 1 = 1 True The solution is −6.

16. −50 + 40 − 5 = z −10 − 5 = z −15 = z Check: −50 + 40 − 5 = z −50 + 40 − 5 0 − 15 −10 − 5 0 − 15 −15 = −15 True The solution is −15. 18.

6 y = 48 6 y 48 = 6 6 6 48 ⋅y= 6 6 y =8 Check: 6 y = 48 6(8) 0 48 48 = 48 True The solution is 8.

20.

−2 x = 26 −2 x 26 = −2 −2 26 −2 ⋅x = −2 −2 x = −13 Check: −2 x = 26 −2(−13) 0 26 26 = 26 True

The solution is −13.

ISM: Prealgebra

22.

24.

n = −5 11 n 11 ⋅ = 11 ⋅ (−5) 11 11 ⋅ n = 11 ⋅ (−5) 11 n = −55 n = −5 Check: 11 −55 0 −5 11 −5 = −5 True The solution is −55. 7 y = −21 7 y −21 = 7 7 7 −21 ⋅y= 7 7 y = −3 Check: 7 y = −21 7 ⋅ (−3) 0 − 21 −21 = −21 True

Chapter 2: Integers and Introduction to Solving Equations 30.

3 y = −27 3 y −27 = 3 3 3 −27 ⋅y= 3 3 y = −9 The solution is −9.

32.

n − 4 = −48 n − 4 + 4 = −48 + 4 n = −44 The solution is −44.

34.

−36 = y + 12 −36 − 12 = y + 12 − 12 −48 = y

The solution is −48. 36.

The solution is −3. 26.

28.

−9 x = 0 −9 x 0 = −9 −9 −9 0 ⋅x = −9 −9 x=0 Check: −9 x = 0 −9 ⋅ 0 0 0 0 = 0 True The solution is 0. −31x = −31 −31x −31 = −31 −31 −31 −31 ⋅x = −31 −31 x =1 Check: −31x = −31 −31 ⋅1 0 − 31 −31 = −31 True The solution is 1.

x = −9 −9 x −9 ⋅ = −9 ⋅ (−9) −9 −9 ⋅ x = −9 ⋅ (−9) −9 x = 81 The solution is 81.

38. z = −28 + 36 z=8 The solution is 8. 40.

42.

−11x = −121 −11x −121 = −11 −11 −11 −121 ⋅x = −11 −11 x = 11 The solution is 11. n = −20 5 n 5 ⋅ = 5 ⋅ (−20) 5 5 ⋅ n = 5 ⋅ (−20) 5 n = −100 The solution is −100.

Chapter 2: Integers and Introduction to Solving Equations

ISM: Prealgebra

44. −81 = 27 x −81 27 x = 27 27 −81 27 = ⋅x 27 27 −3 = x The solution is −3. 46. A number increased by −5 is x + (−5). 48. The quotient of a number and −20 is x ÷ (−20) or

x . −20

50. −32 multiplied by a number is −32 ⋅ x or −32x. 52. Subtract a number from −18 is −18 − x. 54.

56.

n + 961 = 120 n + 961 − 961 = 120 − 961 n = −841 The solution is −841.

y = 1098 −18 y −18 ⋅ = −18 ⋅1098 −18 −18 ⋅ y = −18 ⋅1098 −18 y = −19, 764 The solution is −19,764.

58. answers may vary 60. answers may vary Chapter 2 Vocabulary Check 1. Two numbers that are the same distance from 0 on a number line but are on opposite sides of 0 are called opposites. 2. The absolute value of a number is that number’s distance from 0 on a number line. 3. The integers are ..., −3, −2, −1, 0, 1, 2, 3, .... 4. The negative numbers are numbers less than zero. 5. The positive numbers are numbers greater than zero. 6. The symbols < and > are called inequality symbols. 7. A solution of an equation is a number that when substituted for the variable makes the equation a true statement. 8. The average of a list of numbers is

sum of numbers . number of numbers

ISM: Prealgebra

Chapter 2: Integers and Introduction to Solving Equations

9. A combination of operations on variables and numbers is called an expression. 10. A statement of the form “expression = expression” is called an equation. 11. The sign “<” means is less than and “>” means is greater than. 12. By the addition property of equality, the same number may be added to or subtracted from both sides of an equation without changing the solution of the equation. 13. By the multiplication property of equality, both sides of an equation may be multiplied or divided by the same nonzero number without changing the solution of the equation. Chapter 2 Review 1. If 0 represents sea level, then 1572 feet below sea level is −1572. 2. If 0 represents sea level, then an elevation of 11,239 feet is +11,239. 3. 4. 5. |−11| = 11 since −11 is 11 units from 0 on a number line. 6. |0| = 0 since 0 is 0 units from 0 on a number line. 7. −|8| = −8 8. −(−9) = 9 9. −|−16| = −16 10. −(−2) = 2 11. −18 > −20 since −18 is to the right of −20 on a number line. 12. −5 < 5 since −5 is to the left of 5 on a number line. 13. |−123| = 123 −|−198| = −198 Since 123 > −198, |−123| > −|−198|. 14. |−12| = 12 −|−16| = −16 Since 12 > −16, |−12| > −|−16|. 15. The opposite of −18 is 18. −(−18) = 18 16. The opposite of 42 is negative 42. −(42) = −42 17. False; consider a = 1 and b = 2, then 1 < 2. 18. True

Chapter 2: Integers and Introduction to Solving Equations 19. True

ISM: Prealgebra

37. |−43| + |−108| = 43 + 108 = 151 The common sign is negative, so −43 + (−108) = −151.

20. True 21. |y| = |−2| = 2

38. |−100| + |−506| = 100 + 506 = 606 The common sign is negative, so −100 + (−506) = −606.

22. |−x| = |−(−3)| = |3| = 3 23. −|−z| = −|−(−5)| = −|5| = −5

39. −15 + (−5) = −20 The temperature at 6 a.m. is −20°C.

24. −|−n| = −|−(−10)| = −|10| = −10 25. The bar that extends the farthest in the negative direction corresponds to elevator D, so elevator D extends the farthest below ground. 26. The bar that extends the farthest in the positive direction corresponds to elevator B, so elevator B extends the highest above ground. 27. |5| − |−3| = 5 − 3 = 2 5 > 3, so the answer is positive. 5 + (−3) = 2

40. −127 + (−23) = −150 The diver’s current depth is −150 feet. 41. −7 + ( −6) + (−2) + ( −2) = −13 + ( −2) + ( −2) = −15 + ( −2) = −17 Her total score was −17. 42. 19 − 10 = 19 + (−10) = 9 The team’s score was 9.

28. |18| − |−4| = 18 − 4 = 14 18 > 4, so the answer is positive. 18 + (−4) = 14 29. |16| − |−12| = 16 − 12 = 4 16 > 12, so the answer is positive. −12 + 16 = 4 30. |40| − |−23| = 40 − 23 = 17 40 > 23, so the answer is positive. −23 + 40 = 17

43. 12 − 4 = 12 + (−4) = 8 44. −12 − 4 = −12 + (−4) = −16 45. −7 − 17 = −7 + (−17) = −24 46. 7 − 17 = 7 + (−17) = −10 47. 7 − (−13) = 7 + 13 = 20 48. −6 − (−14) = −6 + 14 = 8 49. 16 − 16 = 16 + (−16) = 0

31. |−8| + |−15| = 8 + 15 = 23 The common sign is negative, so −8 + (−15) = −23.

50. −16 − 16 = −16 + (−16) = −32

32. |−5| + |−17| = 5 + 17 = 22 The common sign is negative, so −5 + (−17) = −22.

52. −5 − (−12) = −5 + 12 = 7

33. |−24| − |3| = 24 − 3 = 21 24 > 3, so the answer is negative. −24 + 3 = −21 34. |−89| − |19| = 89 − 19 = 70 89 > 19, so the answer is negative. −89 + 19 = −70 35. 15 + (−15) = 0 36. −24 + 24 = 0 62

51. −12 − (−12) = −12 + 12 = 0

53. −(−5) − 12 + (−3) = 5 + (−12) + (−3) = −7 + (−3) = −10 54. −8 + (−12) − 10 − (−3) = −8 + (−12) + (−10) + 3 = −20 + (−10) + 3 = −30 + 3 = −27 55. 600 − (−92) = 600 + 92 = 692 The difference in elevations is 692 feet.

ISM: Prealgebra

Chapter 2: Integers and Introduction to Solving Equations

56. 142 − 125 + 43 − 85 = 142 + (−125) + 43 + (−85) = 17 + 43 + (−85) = 60 + (−85) = −25 The balance in the account is −25.

74.

−72 = −9 8

75.

−38 = 38 −1

57. 223 − 245 = 223 + (−245) = −22 You are −22 feet or 22 feet below ground at the end of the drop.

76.

45 = −5 −9

58. 66 − (−16) = 66 + 16 = 82 The total length of the elevator shaft for elevator C is 82 feet.

77. A loss of 5 yards is represented by −5. (−5)(2) = −10 The total loss is 10 yards.

59. |−5| − |−6| = 5 − 6 = 5 + (−6) = −1 5 − 6 = 5 + (−6) = −1 |−5| − |−6| = 5 − 6 is true. 60. |−5 − (−6)| = |−5 + 6| = |1| = 1 5 + 6 = 11 Since 1 ≠ 11, the statement is false.

62. −6(3) = −18 63. −4(16) = −64

81. (−7) 2 = (−7)(−7) = 49

64. −5(−12) = 60

82. −7 2 = −(7 ⋅ 7) = −49

65. (−5)2 = (−5)(−5) = 25 66. (−1)5 = (−1)(−1)(−1)(−1)(−1) = −1

83. 5 − 8 + 3 = −3 + 3 = 0 84. −3 + 12 + (−7) − 10 = 9 + (−7) − 10 = 2 − 10 = −8

67. 12(−3)(0) = 0 68. −1(6)(2)(−2) = −6(2)(−2) = −12(−2) = 24

−24 70. =3 −8

79. A debt of $1024 is represented by −1024. −1024 ÷ 4 = −256 Each payment is $256. 80. A drop of 45 degrees is represented by −45. −45 = −5 or −45 ÷ 9 = −5 9 The average drop each hour is 5°F.

61. −3(−7) = 21

69. −15 ÷ 3 = −5

78. A loss of $50 is represented by −50. (−50)(4) = −200 The total loss is $200.

85. −10 + 3 ⋅ (−2) = −10 + (−6) = −16 86. 5 − 10 ⋅ (−3) = 5 − (−30) = 5 + 30 = 35 87. 16 ÷ (−2) ⋅ 4 = −8 ⋅ 4 = −32 88. −20 ÷ 5 ⋅ 2 = −4 ⋅ 2 = −8

71.

0 =0 −3

89. 16 + (−3) ⋅12 ÷ 4 = 16 + (−36) ÷ 4 = 16 + (−9) =7

72.

−46 is undefined. 0

90. −12 + 10 ÷ (−5) = −12 + (−2) = −14

73.

100 = −20 −5

91. 43 − (8 − 3) 2 = 43 − (5)2 = 64 − 25 = 39 92. (−3)3 − 90 = −27 − 90 = −117

Chapter 2: Integers and Introduction to Solving Equations

93.

(−4)(−3) − (−2)(−1) 12 − 2 10 = = = −2 −10 + 5 −5 −5

94.

4(12 − 18) 4(−6) −24 = = = −12 −10 ÷ (−2 − 3) −10 ÷ (−5) 2

−18 + 25 + (−30) + 7 + 0 + (−2) 6 −18 = 6 = −3

95. average =

−45 + (−40) + (−30) + (−25) 4 −140 = 4 = −35

ISM: Prealgebra 105.

10 x = −30 10 x −30 = 10 10 10 −30 ⋅x = 10 10 x = −3 The solution is −3.

106.

−8 x = 72 −8 x 72 = −8 −8 72 −8 ⋅x = −8 −8 x = −9 The solution is −9.

96. average =

97. 2x − y = 2(−2) − 1 = −4 − 1 = −5

108.

98. y 2 + x 2 = 12 + (−2)2 = 1 + 4 = 5 99.

3 x 3(−2) −6 = = = −1 6 6 6

5 y − x 5(1) − (−2) 5 + 2 7 = = = = −7 100. −y −1 −1 −1

101.

2n − 6 = 16 2( −5) − 6 0 16 −10 − 6 0 16 −16 = 16 False Since −16 = 16 is false, −5 is not a solution of the equation.

102.

2(c − 8) = −20 2(−2 − 8) 0 − 20 2(−10) 0 − 20 −20 = −20 True Since −20 = −20 is true, −2 is a solution of the equation.

103.

n − 7 = −20 n − 7 + 7 = −20 + 7 n = −13 The solution is −13.

104.

−5 = n + 15 −5 − 15 = n + 15 − 15 −20 = n The solution is −20.

107. −20 + 7 = y −13 = y The solution is −13.

109.

110.

x − 31 = −62 x − 31 + 31 = −62 + 31 x = −31 The solution is −31.

n = −11 −4 n −4 ⋅ = −4 ⋅ (−11) −4 −4 ⋅ n = −4 ⋅ (−11) −4 n = 44 The solution is 44. x = 13 −2 x −2 ⋅ = −2 ⋅13 −2 −2 ⋅ x = −2 ⋅13 −2 x = −26 The solution is −26.

111.

n + 12 = −7 n + 12 − 12 = −7 − 12 n = −19 The solution is −19.

112.

n − 40 = −2 n − 40 + 40 = −2 + 40 n = 38 The solution is 38.

ISM: Prealgebra

Chapter 2: Integers and Introduction to Solving Equations

113. −36 = −6 x −36 −6 x = −6 −6 −36 −6 = ⋅x −6 −6 6=x The solution is 6.

129.

7 + 2(−3)

−14 − 6 −20 = = −20 7 + (−6) 1

130. 5(7 − 6)3 − 4(2 − 3)2 + 24 = 5(1)3 − 4(−1)2 + 24 = 5(1) − 4(1) + 16 = 5 − 4 + 16 = 1 + 16 = 17

114. −40 = 8 y −40 8 y = 8 8 −40 8 = ⋅y 8 8 −5 = y The solution is −5.

131.

n − 9 = −30 n − 9 + 9 = −30 + 9 n = −21 The solution is −21.

132.

n + 18 = 1 n + 18 − 18 = 1 − 18 n = −17 The solution is −17.

133.

−4 x = −48 −4 x −48 = −4 −4 −4 −48 ⋅x = −4 −4 x = 12 The solution is 12.

134.

9 x = −81 9 x −81 = 9 9 9 −81 ⋅x = 9 9 x = −9 The solution is −9.

115. −6 + (−9) = −15 116. −16 − 3 = −16 + (−3) = −19 117. −4(−12) = 48 118.

− −14 − 6

84 = −21 −4

119. −76 − (−97) = −76 + 97 = 21 120. −9 + 4 = −5 121. −18 − 9 = −27 The temperature on Friday was −27°C. 122. −11 + 17 = 6 The temperature at noon on Tuesday was 6°C. 123. 12,923 − (−195) = 12,923 + 195 = 13,118 The difference in elevations is 13,118 feet. 124. −32 + 23 = −9 His financial situation can be represented by −9 or −$9. 125. (3 − 7) 2 ÷ (6 − 4)3 = (−4)2 ÷ (2)3 = 16 ÷ 8 = 2 126. 3(4 + 2) + (−6) − 32 = 3(6) + (−6) − 32 = 3(6) + (−6) − 9 = 18 + (−6) − 9 = 12 − 9 =3

135.

n = 100 −2 n −2 ⋅ = −2 ⋅100 −2 −2 ⋅ n = −2 ⋅100 −2 n = −200 The solution is −200.

127. 2 − 4 ⋅ 3 + 5 = 2 − 12 + 5 = −10 + 5 = −5 128. 4 − 6 ⋅ 5 + 1 = 4 − 30 + 1 = −26 + 1 = −25

Chapter 2: Integers and Introduction to Solving Equations

136.

y = −3 −1 y −1 ⋅ = −1(−3) −1 −1 ⋅ y = −1 ⋅ (−3) −1 y=3 The solution is 3.

ISM: Prealgebra

13.

14.

Chapter 2 Getting Ready For the Test 1. The opposite of −2 is −(−2) = 2; A. 2. The absolute value of −2 is |−2| = 2 because −2 is 2 units from 0 on a number line; A. 3. The absolute value of 2 is |2| = 2 because 2 is 2 units from 0 on a number line; A.

5. For xy, the operation is multiplication; C. 6. For −12(+3), the operation is multiplication; C. 7. For −12 + 3, the operation is addition; A. −12 8. For , the operation is division; D. +3

9. For 4 + 6 ⋅ 2, the operation of multiplication is performed first, since multiplication and division are performed before addition and subtraction; C. 10. For 4 + 6 ÷ 2, the operation of division is performed first, since multiplication and division are performed before addition and subtraction; D.

12.

x − 2 = −4 x − 2 + 2 = −4 + 2 x = −2 Choice B is correct.

x+2= 4 x+2−2 = 4−2 x=2 Choice A is correct.

Chapter 2 Test 1. −5 + 8 = 3 2. 18 − 24 = 18 + (−24) = −6 3. 5 ⋅ (−20) = −100

4. The opposite of 2 is −2; B.

11. −3 x = 6 −3 x 6 = −3 −3 x = −2 Choice B is correct.

x =6 −3  x  −3   = −3(6)  −3  x = −18 Choice D is correct.

4. −16 ÷ (−4) = 4 5. −18 + (−12) = −30 6. −7 − (−19) = −7 + 19 = 12 7. −5 ⋅ (−13) = 65 8.

−25 =5 −5

9. |−25| + (−13) = 25 + (−13) = 12 10. 14 − |−20| = 14 − 20 = 14 + (−20) = −6 11. |5| ⋅ |−10| = 5 ⋅ 10 = 50 12.

−10 − −5

10 = −2 −5

13. −8 + 9 ÷ (−3) = −8 + (−3) = −11 14. −7 + (−32) − 12 + 5 = −7 + (−32) + (−12) + 5 = −39 + (−12) + 5 = −51 + 5 = −46 15. (−5)3 − 24 ÷ (−3) = −125 − 24 ÷ (−3) = −125 − (−8) = −125 + 8 = −117

ISM: Prealgebra

Chapter 2: Integers and Introduction to Solving Equations

17. −(−7) 2 ÷ 7 ⋅ (−4) = −49 ÷ 7 ⋅ (−4) = −7 ⋅ (−4) = 28

28. Subtract the elevation of the Romanche Gap from the elevation of Mount Washington. 6288 − (−25,354) = 6288 + 25,354 = 31,642 The difference in elevations is 31,642 feet.

18. 3 − (8 − 2)3 = 3 − 63 = 3 − 216 = 3 + (−216) = −213

29. Subtract the depth of the lake from the elevation of the surface. 1495 − 5315 = 1495 + (−5315) = −3820 The deepest point of the lake is 3820 feet below sea level.

16. (5 − 9) 2 ⋅ (8 − 2)3 = (−4) 2 ⋅ (6)3 = 16 ⋅ 216 = 3456

19.

4 82 4 64 − = − = 2 − 4 = 2 + (−4) = −2 2 16 2 16

20.

−3(−2) + 12 6 + 12 18 = = =2 −1(−4 − 5) −1(−9) 9 2

31. a.

−5

32.

22. 5(−8) − [6 − (2 − 4)] + (12 − 16)2 = 5(−8) − [6 − (−2)] + (12 − 16) 2 = 5(−8) − (6 + 2) + (−4)2 = 5(−8) − 8 + (−4) 2 = 5(−8) − 8 + 16 = −40 − 8 + 16 = −48 + 16 = −32

33.

23. 7 x + 3 y − 4 z = 7(0) + 3(−3) − 4(2) = 0 + (−9) − 8 = −9 − 8 = −17 24. 10 − y 2 = 10 − (−3)2 = 10 − 9 = 1 25.

3z 3(2) 6 = = = −1 2 y 2(−3) −6

26. A descent of 22 feet is represented by −22. 4(−22) = −88 Tui is 88 feet below sea level. 27. 129 + (−79) + (−40) + 35 = 50 + (−40) + 35 = 10 + 35 = 45 Their new balance can be represented by 45 or $45.

−12 + (−13) + 0 + 9 −16 = = −4 4 4

The product of a number and 17 is 17 ⋅ x or 17x.

b. A number subtracted from 20 is 20 − x.

(5)2 25 21. = = = = −5 2(−6) + 7 −12 + 7 −5 −5 25 − 30

30. average =

34.

−9n = −45 −9n −45 = −9 −9 −9 −45 ⋅n = −9 −9 n=5 The solution is 5. n =4 −7 n −7 ⋅ = −7 ⋅ 4 −7 −7 ⋅ n = −7 ⋅ 4 −7 n = −28 The solution is −28. x − 16 = −36 x − 16 + 16 = −36 + 16 x = −20 The solution is −20.

35. −20 + 8 + 8 = x −12 + 8 = x −4 = x The solution is −4. Cumulative Review Chapters 1−2 1. The place value of 3 in 396,418 is hundredthousands. 2. The place value of 3 in 4308 is hundreds. 3. The place value of 3 in 93,192 is thousands.

Chapter 2: Integers and Introduction to Solving Equations 4. The place value of 3 is 693,298 is thousands. 5. The place value of 3 in 534,275,866 is tenmillions. 6. The place value of 3 in 267,301,818 is hundredthousands. 7. a.

−7 < 7 since −7 is to the left of 7 on a number line.

b. 0 > −4 since 0 is to the right of −4 on a number line. c.

−9 > −11 since −9 is to the right of −11 on a number line.

8. a.

12 > −4 since 12 is to the right of −4 on a number line.

b. −13 > −31 since −13 is to the right of −31 on a number line.

ISM: Prealgebra 14. Subtract the cost of the camera from the amount in her account. 762 − 237 525 She will have $525 left in her account after buying the camera. 15. To round 568 to the nearest ten, observe that the digit in the ones place is 8. Since this digit is at least 5, we add 1 to the digit in the tens place. The number 568 rounded to the nearest ten is 570. 16. To round 568 to the nearest hundred, observe that the digit in the tens place is 6. Since this digit is at least 5, we add 1 to the digit in the hundreds place. The number 568 rounded to the nearest hundred is 600. 17.

4725 − 2879

rounds to rounds to

4700 − 2900 1800

9. 13 + 2 + 7 + 8 + 9 = (13 + 7) + (2 + 8) + 9 = 20 + 10 + 9 = 39

18.

8394 − 2913

rounds to rounds to

8000 − 3000 5000

10. 11 + 3 + 9 + 16 = (11 + 9) + (3 + 16) = 20 + 19 = 39

19. a.

11.

12.

−82 < 79 since −82 is to the left of 79 on a number line.

7826 − 505 7321 Check:

3285 − 272 3013 Check:

b. 20(4 + 7) = 20 ⋅ 4 + 20 ⋅ 7 c. 7321 + 505 7826

20. a.

2(7 + 9) = 2 ⋅ 7 + 2 ⋅ 9 5(2 + 12) = 5 ⋅ 2 + 5 ⋅ 12

b. 9(3 + 6) = 9 ⋅ 3 + 9 ⋅ 6 c.

4(8 + 1) = 4 ⋅ 8 + 4 ⋅ 1

21.

631 × 125 3 155 12 620 63 100 78,875

22.

299 × 104 1 196 29 900 31, 096

3013 + 272 3285

13. Subtract 7257 from the radius of Jupiter. 43, 441 − 7 257 36,184 The radius of Saturn is 36,184 miles.

5(6 + 5) = 5 ⋅ 6 + 5 ⋅ 5

ISM: Prealgebra

23. a. b.

c. 24. a.

Chapter 2: Integers and Introduction to Solving Equations

42 ÷ 7 = 6 because 6 ⋅ 7 = 42. 64 = 8 because 8 ⋅ 8 = 64. 8

7 3 21 because 7 ⋅ 3 = 21. 35 = 7 because 7 ⋅ 5 = 35. 5

b. 64 ÷ 8 = 8 because 8 ⋅ 8 = 64. c.

12 4 48 because 12 ⋅ 4 = 48.

741 25. 5 3705 −35 20 −20 05 −5 0 Check: 741 × 5 3705 456 26. 8 3648 −32 44 −40 48 −48 0

Check: 456 × 8 3648

27. number of cards number of number of = ÷ for each person cards employees = 238 ÷ 19 12 R 10 19 238 −19 48 −38 10 Each employee will receive 12 gift cards. There will be 10 gift cards left over. 28.

cost of each total = ÷ number of tickets ticket cost = 324 ÷ 36 9 36 324 −324 0 Each ticket cost $9.

29. 9 2 = 9 ⋅ 9 = 81 30. 53 = 5 ⋅ 5 ⋅ 5 = 125 31. 61 = 6 32. 41 = 4 33. 5 ⋅ 62 = 5 ⋅ 6 ⋅ 6 = 180 34. 23 ⋅ 7 = 2 ⋅ 2 ⋅ 2 ⋅ 7 = 56 35.

36.

7 − 2 ⋅ 3 + 32 7 − 2 ⋅ 3 + 9 7 − 6 + 9 10 = = = =2 5(2 − 1) 5(1) 5 5 62 + 4 ⋅ 4 + 23 37 − 5

36 + 4 ⋅ 4 + 8 37 − 25 36 + 16 + 8 = 12 60 = 12 =5

37. x + 6 = 8 + 6 = 14 38. 5 + x = 5 + 9 = 14

Chapter 2: Integers and Introduction to Solving Equations

39. a.

|−9| = 9 because −9 is 9 units from 0.

ISM: Prealgebra

45. −7 ⋅ 3 = −21

b. |8| = 8 because 8 is 8 units from 0.

46. 5(−2) = −10

|0| = 0 because 0 is 0 units from 0.

47. 0 ⋅ (−4) = 0

40. a.

|4| = 4 because 4 is 4 units from 0.

48. −6 ⋅ 9 = −54

b. |−7| = 7 because −7 is 7 units from 0. 41. −2 + 25 = 23 42. 8 + (−3) = 5 43. 2a − b = 2(8) − (−6) = 16 − (−6) = 16 + 6 = 22 44. x − y = −2 − (−7) = −2 + 7 = 5

49. 3(4 − 7) + (−2) − 5 = 3(−3) + (−2) − 5 = −9 + (−2) − 5 = −11 − 5 = −16 50. 4 − 8(7 − 3) − (−1) = 4 − 8(4) − (−1) = 4 − 32 − (−1) = 4 − 32 + 1 = −28 + 1 = −27

Chapter 3 Section 3.1 Practice Exercises 1. a.

8m − 14m = (8 − 14)m = −6m

b. 6a + a = 6a + 1a = (6 + 1)a = 7a c.

− y 2 + 3 y 2 + 7 = −1 y 2 + 3 y 2 + 7 = (−1 + 3) y 2 + 7 = 2 y2 + 7

14. 4(2x) = (4 ⋅ 2)x = 8x The perimeter is 8x centimeters. 15. 3(12y + 9) = 3 ⋅ 12y + 3 ⋅ 9 = 36y + 27 The area of the garden is (36y + 27) square yards. Vocabulary, Readiness & Video Check 3.1 1. 14 y 2 + 2 x − 23 is called an expression while

2. 6 z + 5 + z − 4 = 6 z + 5 + 1z + (−4) = 6 z + 1z + 5 + (−4) = (6 + 1) z + 5 + (−4) = 7z +1

14 y 2 , 2x, and −23 are each called a term.

2. To multiply 3(−7x + 1), we use the distributive property.

3. 6y + 12y − 6 = 18y − 6

3. To simplify an expression like y + 7y, we combine like terms.

4. 7 y − 5 + y + 8 = 7 y + (−5) + 1 y + 8 = 7 y + 1 y + (−5) + 8 = 8y + 3

4. By the commutative properties, the order of adding or multiplying two numbers can be changed without changing their sum or product.

5. −7 y + 2 − 2 y − 9 x + 12 − x = −7 y + 2 + (−2 y ) + (−9 x) + 12 + (−1x) = −7 y + (−2 y ) + (−9 x) + (−1x) + 2 + 12 = −9 y − 10 x + 14

5. The term 5x is called a variable term while the term 7 is called a constant term.

6. 6(4a) = (6 ⋅ 4)a = 24a

7. By the associative properties, the grouping of adding or multiplying numbers can be changed without changing their sum or product.

7. −8(9x) = (−8 ⋅ 9)x = −72x 8. 8(y + 2) = 8 ⋅ y + 8 ⋅ 2 = 8y + 16 9. 3(7a − 5) = 3 ⋅ 7a − 3 ⋅ 5 = 21a − 15

6. The term z has an understood numerical coefficient of 1.

8. The terms −x and 5x are like terms and the terms 5x and 5y are unlike terms.

10. 6(5 − y) = 6 ⋅ 5 − 6 ⋅ y = 30 − 6y

9. For the term −3 x 2 y, −3 is called the numerical coefficient.

11. 5(2 y − 3) − 8 = 5(2 y ) − 5(3) − 8 = 10 y − 15 − 8 = 10 y − 23

10. because the original expression contains like terms and we can combine like terms 11. distributive property

12. −7( x − 1) + 5(2 x + 3) = −7( x) − (−7)(1) + 5(2 x) + 5(3) = −7 x + 7 + 10 x + 15 = 3x + 22

12. The 20 is outside the parentheses, so the distributive property does not apply to it. 13. addition; multiplication; P = perimeter; A = area

13. −( y + 1) + 3 y − 12 = −1( y + 1) + 3 y − 12 = −1 ⋅ y + (−1)(1) + 3 y − 12 = − y − 1 + 3 y − 12 = 2 y − 13

Exercise Set 3.1 2. 8y + 3y = (8 + 3)y = 11y 4. 7z − 10z = (7 − 10)z = −3z

Chapter 3: Solving Equations and Problem Solving

6. 5b − 8b − b = (5 − 8 − 1)b = −4b 8. 8y + y − 2y − 8y = (8 + 1 − 2 − 8)y = −1y = −y 10. 5b − 4b + b − 15 = (5 − 4 + 1)b − 15 = 2b − 15 12. 4(4x) = (4 ⋅ 4)x = 16x 14. −3(21z) = (−3 ⋅ 21)z = −63z

ISM: Prealgebra 42. a + 4 − 7 a − 5 = a − 7 a + 4 − 5 = (1 − 7)a + 4 − 5 = −6 a − 1 44. m − 8n + m + 8n = m + m − 8n + 8n = (1 + 1)m + (−8 + 8)n = 2m + 0n = 2m 46. 5( x − 1) + 18 = 5 ⋅ x − 5 ⋅1 + 18 = 5 x − 5 + 18 = 5 x + 13

16. 13(5b) = (13 ⋅ 5)b = 65b 18. 3(x + 1) = 3 ⋅ x + 3 ⋅ 1 = 3x + 3 20. 4(y − 6) = 4 ⋅ y − 4 ⋅ 6 = 4y − 24 22. −8(8y + 10) = −8 ⋅ 8y + (−8) ⋅ 10 = −64y − 80 24. 5(6 − y ) − 2 = 5(6) − 5( y ) − 2 = 30 − 5 y − 2 = −5 y + 30 − 2 = −5 y + 28

48. 3( x + 2) − 11x = 3 ⋅ x + 3 ⋅ 2 − 11x = 3x + 6 − 11x = 3x − 11x + 6 = −8 x + 6 50. −8(1 + v) + 6v = −8 ⋅1 + (−8) ⋅ v + 6v = −8 − 8v + 6v = −2v − 8

26. 10 + 4(6d − 2) = 10 + 4 ⋅ 6d − 4 ⋅ 2 = 10 + 24d − 8 = 24d + 10 − 8 = 24d + 2 28. −3(5 − 2b) − 4b = −3(5) − (−3)(2b) − 4b = −15 + 6b − 4b = 6b − 4b − 15 = 2b − 15 30. 8 z + 5(6 + z ) + 20 = 8 z + 5 ⋅ 6 + 5 ⋅ z + 20 = 8 z + 30 + 5 z + 20 = 8 z + 5 z + 30 + 20 = 13 z + 50 32. 3(5 x − 2) + 2(3x + 1) = 3 ⋅ 5 x − 3 ⋅ 2 + 2 ⋅ 3x + 2 ⋅1 = 15 x − 6 + 6 x + 2 = 15 x + 6 x − 6 + 2 = 21x − 4 34. −(5 x − 1) − 10 = −1(5 x − 1) − 10 = −1 ⋅ 5 x − (−1)(1) − 10 = −5 x + 1 − 10 = −5 x − 9 36. x + 12x = (1 + 12)x = 13x 38. −12x + 8x = (−12 + 8)x = −4x

52. 5 y − 4 + 9 y − y + 15 = 5 y + 9 y − y − 4 + 15 = (5 + 9 − 1) y − 4 + 15 = 13 y + 11 54. −2( x + 4) + 8(3x − 1) = −2 ⋅ x + (−2) ⋅ 4 + 8 ⋅ 3 x − 8 ⋅1 = −2 x − 8 + 24 x − 8 = −2 x + 24 x − 8 − 8 = 22 x − 16 56. 6 x − 4 + 2 x − x + 3 = 6 x + 2 x − x − 4 + 3 = (6 + 2 − 1) x − 4 + 3 = 7x −1 58. 5(c + 2) + 7c = 5 ⋅ c + 5 ⋅ 2 + 7c = 5c + 10 + 7c = 5c + 7c + 10 = 12c + 10 60. 8(m + 3) − 20 + m = 8 ⋅ m + 8 ⋅ 3 − 20 + m = 8m + 24 − 20 + m = 8m + m + 24 − 20 = 9m + 4 62. 5(2a + 3) − 10a = 5 ⋅ 2a + 5 ⋅ 3 − 10a = 10a + 15 − 10a = 10a − 10a + 15 = 15

40. 8r + s − 7s = 8r + (1 − 7)s = 8r − 6s 72

ISM: Prealgebra

Chapter 3: Solving Equations and Problem Solving

64. −(7 y − 2) + 6 = −1(7 y − 2) + 6 = −1 ⋅ 7 y − (−1)(2) + 6 = −7 y + 2 + 6 = −7 y + 8 66. −(12b − 20) + 5(3b − 2) = −1(12b − 20) + 5(3b − 2) = −1(12b) − (−1)(20) + 5(3b) − 5(2) = −12b + 20 + 15b − 10 = −12b + 15b + 20 − 10 = 3b + 10

84. Area = (length) ⋅ (width) = 50 ⋅ 40 = 2000 The area is 2000 square feet. 86. Perimeter = 2 ⋅ (length) + 2 ⋅ (width) = 2 ⋅120 + 2 ⋅ 80 = 240 + 160 = 400 The perimeter is 400 feet so the rancher needs 400 feet of fencing. 88. x + x + x − 14 = (1 + 1 + 1)x − 14 = 3x − 14 The perimeter is (3x − 14) inches.

68. b + 2(b − 5) = b + 2 ⋅ b − 2 ⋅ 5 = b + 2b − 10 = 3b − 10

90. −15 + 23 = 8

70. 3x − 4( x + 2) + 1 = 3 x − 4 ⋅ x − 4 ⋅ 2 + 1 = 3x − 4 x − 8 + 1 = −1x − 7 = −x − 7

94. 8 + (−8) = 0

92. −7 − (−4) = −7 + 4 = −3

72. 3x + x + 7 + 4x + 12 + 5x = 3x + x + 4x + 5x + 7 + 12 = (3 + 1 + 4 + 5)x + 7 + 12 = 13x + 19 The perimeter is (13x + 19) feet. 74. 3 z + 1 + 5 z + 1 = 3 z + 5 z + 1 + 1 = (3 + 5) z + 1 + 1 = 8z + 2 The perimeter is (8z + 2) meters. 76. 6(9y + 1) = 6 ⋅ 9y + 6 ⋅ 1 = 54y + 6 The perimeter is (54y + 6) kilometers. 78. Area = (length) ⋅ (width) = (5 x) ⋅ (8) = (5 ⋅ 8) x = 40 x The area is 40x square centimeters. 80. Area = (length) ⋅ (width) = 11( z − 6) = 11 ⋅ z − 11 ⋅ 6 = 11z − 66 The area is (11z − 66) square meters. 82. Area = (length) ⋅ (width) = 12(2 x + 3) = 12 ⋅ 2 x + 12 ⋅ 3 = 24 x + 36 The area is (24x + 36) square feet.

96. −2(4 x − 1) = −2(4 x) − (−2)(1) = −8 x − (−2) = −8 x + 2 The expressions are not equivalent. 98. −8(ab) = −8 ⋅ ab = −8ab The expressions are not equivalent. 100. 12 y − (3 y − 1) = 12 y + (−1)(3 y − 1) = 12 y + (−1)(3 y) + (−1)(−1) = 12 y − 3 y + 1 The expressions are equivalent. 102. 6( x + 5) + 2 = 6 ⋅ x + 6 ⋅ 5 + 2 = 6 x + 30 + 2 = 6 x + 32 The expressions are not equivalent. 104. The order of the terms is changed. This is the commutative property of addition. 106. The order of the terms is changed, but the grouping is not. This is the commutative property of addition. 108. Add the areas of each rectangle. 12(3x − 5) + 4(5 x − 1) = 12 ⋅ 3 x − 12 ⋅ 5 + 4 ⋅ 5 x − 4 ⋅1 = 36 x − 60 + 20 x − 4 = 36 x + 20 x − 60 − 4 = 56 x − 64 The total area is (56x − 64) square kilometers.

Chapter 3: Solving Equations and Problem Solving

110. 76(268x + 592) − 2960 = 76 ⋅ 268x + 76 ⋅ 592 − 2960 = 20,368x + 44,992 − 2960 = 20,368x + 42,032 112. answers may vary 114. answers may vary

ISM: Prealgebra 5. −4 − 10 = 4 y − 5 y −14 = − y −14 −1 y = −1 −1 14 = y 6.

13x = 4(3 x − 1) 13x = 4 ⋅ 3 x − 4 ⋅1 13x = 12 x − 4 13x − 12 x = 12 x − 4 − 12 x x = −4

5 y + 2 = 17 5 y + 2 − 2 = 17 − 2 5 y = 15 5 y 15 = 5 5 y=3 Check: 5 y + 2 = 17 5(3) + 2 0 17 15 + 2 0 17 17 = 17 True The solution is 3.

Section 3.2 Practice Exercises 1.

x + 6 = 1− 3 x + 6 = −2 x + 6 − 6 = −2 − 6 x = −8 Check: x + 6 = 1 − 3 −8 + 6 0 1 − 3 −2 = −2 True

The solution is −8. 2.

10 = 2m − 4m 10 = −2m 10 −2m = −2 −2 −5 = m Check: 10 = 2m − 4m 10 0 2(−5) − 4(−5) 10 0 − 10 − (−20) 10 0 − 10 + 20 10 = 10 True The solution is −5.

a 3 a −2 = 3

−8 + 6 =

3 ⋅ (−2) = 3 ⋅

8. −4( x + 2) − 60 = 2 − 10 −4 x − 8 − 60 = 2 − 10 −4 x − 68 = −8 −4 x − 68 + 68 = −8 + 68 −4 x = 60 −4 x 60 = −4 −4 x = −15 9. a.

a 3

The sum of −3 and a number is −3 + x.

b. −5 decreased by a number is −5 − x.

3 3 ⋅ (−2) = ⋅ a 3 −6 = a

4. −6 y − 1 + 7 y = 17 + 2 −6 y + 7 y − 1 = 17 + 2 y − 1 = 19 y − 1 + 1 = 19 + 1 y = 20

Three times a number is 3x.

d. A number subtracted from 83 is 83 − x.

The quotient of a number and −4 is x − . 4

10. a.

The product of 5 and a number, decreased by 25, is 5x − 25.

b. Twice the sum of a number and 3 is 2(x + 3).

x or −4

ISM: Prealgebra c.

Chapter 3: Solving Equations and Problem Solving

The quotient of 39 and twice a number is 39 39 ÷ (2x) or . 2x

Vocabulary, Readiness & Video Check 3.2 1. The equations −3x = 51 and

equivalent equations.

−3 x 51 are called = −3 −3

2. The difference between an equation and an expression is that an equation contains an equal sign, while an expression does not. 3. The process of writing −3x + 10x as 7x is called simplifying the expression.

8. 100 = 15 y + 5 y 100 = 20 y 100 20 y = 20 20 5= y 10. −3 x = 11 − 2 −3 x = 9 −3 x 9 = −3 −3 x = −3 12.

z −4 z −2 = −4

20 − 22 =

z −4

4. For the equation −5x − 1 = −21, the process of finding that 4 is the solution is called solving the equation.

−4 ⋅ (−2) = −4 ⋅

5. By the addition property of equality, x = −2 and x + 7 = −2 + 7 are equivalent equations. 6. By the multiplication property of equality, y = 8 and 3 ⋅ y = 3 ⋅ 8 are equivalent equations.

14. 5 y − 9 y = −14 + ( −14) −4 y = −28 −4 y −28 = −4 −4 y=7

7. Simplify the left side of the equation by combining like terms.

16.

8= z

8. Simplify the left side of the equation by using the distributive property. 9. addition property of equality 10. because order matters with subtraction Exercise Set 3.2 2.

x+7 = 2+3 x+7 =5 x+7−7 = 5−7 x = −2 1− 8 = n + 2 −7 = n + 2 −7 − 2 = n + 2 − 2 −9 = n

6. 10 y − y = 45 9 y = 45 9 y 45 = 9 9 y=5

y = 32 − 52 3 y = −20 3 y 3 ⋅ = 3 ⋅ (−20) 3 y = −60

18. −3 + 5 x − 4 x = 13 −3 + x = 13 −3 + x + 3 = 13 + 3 x = 16 20. −7 + 10 = 4 x − 6 − 3 x 3= x−6 3+ 6 = x −6+ 6 9=x 22.

6(3 x + 1) = 19 x 6 ⋅ 3x + 6 ⋅1 = 19 x 18 x + 6 = 19 x 18 x − 18 x + 6 = 19 x − 18 x 6=x

Chapter 3: Solving Equations and Problem Solving

24.

17 x = 4(4 x − 6) 17 x = 4 ⋅ 4 x − 4 ⋅ 6 17 x = 16 x − 24 17 x − 16 x = 16 x − 16 x − 24 x = −24

26.

28 z = 9(3 z − 2) 28 z = 9 ⋅ 3 z − 9 ⋅ 2 28 z = 27 z − 18 28 z − 27 z = 27 z − 27 z − 18 z = −18

28.

−2(−1 − 3 y ) = 7 y −2 ⋅ (−1) − (−2)(3 y ) = 7 y 2 + 6y = 7y 2 + 6y − 6y = 7y − 6y 2= y

30.

3 y − 12 = 0 3 y − 12 + 12 = 0 + 12 3 y = 12 3 y 12 = 3 3 y=4

32.

5m + 1 = 46 5m + 1 − 1 = 46 − 1 5m = 45 5m 45 = 5 5 m=9

34.

−11 = 3t − 2 −11 + 2 = 3t − 2 + 2 −9 = 3t −9 3t = 3 3 −3 = t

36.

4(3 y − 5) = 14 y 4 ⋅ 3 y − 4 ⋅ 5 = 14 y 12 y − 20 = 14 y 12 y − 12 y − 20 = 14 y − 12 y −20 = 2 y −20 2 y = 2 2 −10 = y

ISM: Prealgebra

38.

5( x − 6) = −2 − 8 5 ⋅ x − 5 ⋅ 6 = −10 5 x − 30 = −10 5 x − 30 + 30 = −10 + 30 5 x = 20 5 x 20 = 5 5 x=4

40.

−2( x + 5) − 2 = −8 − 4 −2 ⋅ x − 2 ⋅ 5 − 2 = −12 −2 x − 10 − 2 = −12 −2 x − 12 = −12 −2 x − 12 + 12 = −12 + 12 −2 x = 0 −2 x 0 = −2 −2 x=0

42.

x − 63 = 10 x x − x − 63 = 10 x − x −63 = 9 x −63 9 x = 9 9 −7 = x

44. 35 − (−3) = 3( x − 2) + 17 35 + 3 = 3 ⋅ x − 3 ⋅ 2 + 17 38 = 3 x − 6 + 17 38 = 3 x + 11 38 − 11 = 3 x + 11 − 11 27 = 3 x 27 3 x = 3 3 9=x 46. 7 − (−10) = x − 5 7 + 10 = x − 5 17 = x − 5 17 + 5 = x − 5 + 5 22 = x 48.

y − 8 = −5 − 1 y − 8 = −6 y − 8 + 8 = −6 + 8 y=2

ISM: Prealgebra

Chapter 3: Solving Equations and Problem Solving

50. y − 6 y = 20 −5 y = 20 −5 y 20 = −5 −5 y = −4

64.

52. 6 x = 5 − 35 6 x = −30 6 x −30 = 6 6 x = −5 54.

x −12 x −5 = −12 60 = x

56.

66. −10 x + 11x + 5 = 9 − 5 x+5 = 4 x+5−5 = 4−5 x = −1

9 − 14 =

−12 ⋅ (−5) = −12 ⋅

68.

x −12

8 x − 8 = 32 8 x − 8 + 8 = 32 + 8 8 x = 40 8 x 40 = 8 8 x=5

58. −30 = t + 9t −30 = 10t −30 10t = 10 10 −3 = t 60. −42 + 20 = −2 x + 13 x −22 = 11x −22 11x = 11 11 −2 = x 62.

10 x = 6(2 x − 3) 10 x = 6 ⋅ 2 x − 6 ⋅ 3 10 x = 12 x − 18 10 x − 12 x = 12 x − 12 x − 18 −2 x = −18 −2 x −18 = −2 −2 x=9

65 y = 8(8 y − 9) 65 y = 8 ⋅ 8 y − 8 ⋅ 9 65 y = 64 y − 72 65 y − 64 y = 64 y − 64 y − 72 y = −72

y = 6 − (−1) −6 y = 6 +1 −6 y =7 −6 y −6 ⋅ = −6 ⋅ 7 −6 y = −42

70. 8 x − 4 − 6 x = 12 − 22 8 x − 6 x − 4 = −10 2 x − 4 = −10 2 x − 4 + 4 = −10 + 4 2 x = −6 2 x −6 = 2 2 x = −3 72.

−4( x + 7) − 30 = 3 − 37 −4 ⋅ x − 4 ⋅ 7 − 30 = 3 − 37 −4 x − 28 − 30 = −34 −4 x − 58 = −34 −4 x − 58 + 58 = −34 + 58 −4 x = 24 −4 x 24 = −4 −4 x = −6

74. Negative eight plus a number is −8 + x. 76. A number subtracted from 12 is 12 − x. 78. Twice a number is 2x. 80. The quotient of negative six and a number is −6 6 or − . −6 ÷ x or x x

Chapter 3: Solving Equations and Problem Solving 82. Negative four times a number, increased by 18 is −4x + 18. 84. Twice a number, decreased by thirty is 2x − 30. 86. The product of 7 and a number, added to 100 is 100 + 7x. 88. The difference of −9 times a number, and 1 is −9x − 1. 90. Twice the sum of a number and −5 is 2[x + (−5)] or 2(x − 5). 92. The quotient of ten times a number and −4 is 10 x 10 x or − . 10x ÷ (−4) or −4 4 94. The quotient of −20 and a number, decreased by −20 20 three is (−20 ÷ x) − 3 or − 3 or − − 3. x x 96. The bars with length less than $500 correspond to cities with daily costs less than $500. The cities with daily costs less than $500 are Oslo and Riyadh. 98. For a business traveler, the daily cost is $524 for Stavanger, $497 for Oslo, and $528 for Stockholm. 524 + 497 + 528 = 1549 The combined daily cost is $1549. 100. answers may vary 102. no; answers may vary 104. answers may vary 106.

x = 46 − 57 −13 x = 4096 − 78,125 −13 x = −74, 029 −13 x −13 ⋅ = −13 ⋅ (−74, 029) −13 x = 962,377

ISM: Prealgebra

108.

y = (−8)2 − 20 + (−2) 2 10 y = 64 − 20 + 4 10 y = 64 − 20 + 4 10 y = 48 10 y 10 ⋅ = 10 ⋅ 48 10 y = 480

110. 4( x − 11) + 90 − −86 + 25 = 5 x 4( x − 11) + 90 − 86 + 32 = 5 x 4( x − 11) + 36 = 5 x 4 x − 44 + 36 = 5 x 4 x − 8 = 5x 4x − 4x − 8 = 5x − 4x −8 = x Mid-Chapter Review 1. 7x − 5y + 14 is an expression because it does not contain an equal sign. 2. 7x = 35 + 14 is an equation because it contains an equal sign. 3. 3(x − 2) = 5(x + 1) − 17 is an equation because it contains an equal sign. 4. −9(2x + 1) − 4(x − 2) + 14 is an expression because it does not contain an equal sign. 5. To simplify an expression, we combine any like terms. 6. To solve an equation, we use properties of equality to find any value of the variable that makes the equation a true statement. 7. 7x + x = (7 + 1)x = 8x 8. 6y − 10y = (6 − 10)y = −4y 9. 2a + 5a − 9a − 2 = (2 + 5 − 9)a − 2 = −2a − 2 10. 6a − 12 − a − 14 = 6a − a − 12 − 14 = (6 − 1)a − 12 − 14 = 5a − 26 11. −2(4x + 7) = −2 ⋅ 4x + (−2) ⋅ 7 = −8x − 14 12. −3(2x − 10) = −3(2x) − (−3)(10) = −6x + 30

ISM: Prealgebra

Chapter 3: Solving Equations and Problem Solving

13. 5( y + 2) − 20 = 5 ⋅ y + 5 ⋅ 2 − 20 = 5 y + 10 − 20 = 5 y − 10 14. 12 x + 3( x − 6) − 13 = 12 x + 3 ⋅ x − 3 ⋅ 6 − 13 = 12 x + 3 x − 18 − 13 = (12 + 3) x − 18 − 13 = 15 x − 31 15. Area = (length) ⋅ (width) = 3(4 x − 2) = 3 ⋅ 4x − 3⋅ 2 = 12 x − 6 The area is (12x − 6) square meters.

20. 6 − (−5) = x + 5 6+5 = x+5 11 = x + 5 11 − 5 = x + 5 − 5 6=x Check: 6 − (−5) = x + 5 6 − (−5) 0 6 + 5 6+50 6+5 11 = 11 True The solution is 6. 21.

16. Perimeter = x + x + 2 + 7 = 2x + 9 The perimeter is (2x + 9) feet. 17.

12 = 11x − 14 x 12 = −3 x 12 −3 x = −3 −3 −4 = x Check: 12 = 11x − 14 x 12 0 11(−4) − 14(−4) 12 0 − 44 + 56 12 = 12 True The solution is −4.

The solution is −15. 22.

18. 8 y + 7 y = −45 15 y = −45 15 y −45 = 15 15 y = −3 8 y + 7 y = −45 Check: 8(−3) + 7(−3) 0 − 45 −24 + (−21) 0 − 45 −45 = −45 True The solution is −3. 19.

x − 12 = −45 + 23 x − 12 = −22 x − 12 + 12 = −22 + 12 x = −10 x − 12 = −45 + 23 Check: −10 − 12 0 − 45 + 23 −22 = −22 True

The solution is −10.

x = −14 + 9 3 x = −5 3 x 3 ⋅ = 3(−5) 3 x = −15 x = −14 + 9 Check: 3 −15 0 − 14 + 9 3 −5 = −5 True

z = −23 − 7 4 z = −30 4 z 4 ⋅ = 4 ⋅ (−30) 4 z = −120 z = −23 − 7 Check: 4 −120 0 − 23 − 7 4 −30 = −30 True The solution is −120.

23. −6 + 2 = 4 x + 1 − 3 x −4 = x + 1 −4 − 1 = x + 1 − 1 −5 = x Check: −6 + 2 = 4 x + 1 − 3x −6 + 2 0 4(−5) + 1 − 3(−5) −4 0 − 20 + 1 + 15 −4 = −4 True The solution is −5.

Chapter 3: Solving Equations and Problem Solving

ISM: Prealgebra

24.

5 − 8 = 5 x + 10 − 4 x −3 = x + 10 −3 − 10 = x + 10 − 10 −13 = x Check: 5 − 8 = 5 x + 10 − 4 x 5 − 8 0 5(−13) + 10 − 4(−13) −3 0 − 65 + 10 + 52 −3 = −3 True The solution is −13.

28. −8 + (−14) = −80 y + 20 + 81 y −22 = y + 20 −22 − 20 = y + 20 − 20 −42 = y Check: −8 + (−14) = −80 y + 20 + 81y −8 + (−14) 0 − 80(−42) + 20 + 81(−42) −22 0 3360 + 20 − 3402 −22 = −22 True The solution is −42.

25.

6(3 x − 4) = 19 x 6 ⋅ 3 x − 6 ⋅ 4 = 19 x 18 x − 24 = 19 x 18 x − 18 x − 24 = 19 x − 18 x −24 = x 6(3x − 4) = 19 x Check: 6[3(−24) − 4] 0 19(−24) 6(−72 − 4) 0 − 456 6(−76) 0 − 456 −456 = −456 True

29.

3x − 16 = −10 3x − 16 + 16 = −10 + 16 3x = 6 3x 6 = 3 3 x=2 Check: 3x − 16 = −10 3(2) − 16 0 − 10 6 − 16 0 − 10 −10 = −10 True The solution is 2.

30.

4 x − 21 = −13 4 x − 21 + 21 = −13 + 21 4x = 8 4x 8 = 4 4 x=2 Check: 4 x − 21 = −13 4 ⋅ 2 − 21 0 − 13 8 − 21 0 − 13 −13 = −13 True The solution is 2.

The solution is −24. 26.

25 x = 6(4 x − 9) 25 x = 6 ⋅ 4 x − 6 ⋅ 9 25 x = 24 x − 54 25 x − 24 x = 24 x − 24 x − 54 x = −54 25 x = 6(4 x − 9) Check: 25(−54) 0 6[4( −54) − 9] −1350 0 6(−216 − 9) −1350 0 6(−225) −1350 = −1350 True

The solution is −54. 27. −36 x − 10 + 37 x = −12 − (−14) x − 10 = −12 + 14 x − 10 = 2 x − 10 + 10 = 2 + 10 x = 12 −36 x − 10 + 37 x = −12 − (−14) Check: −36(12) − 10 + 37(12) 0 − 12 − (−14) −432 − 10 + 444 0 − 12 + 14 2 = 2 True The solution is 12.

31. −8 z − 2 z = 26 − (−4) −10 z = 26 + 4 −10 z = 30 −10 z 30 = −10 −10 z = −3 −8 z − 2 z = 26 − (−4) Check: −8(−3) − 2(−3) 0 26 − (−4) 24 + 6 0 26 + 4 30 = 30 True The solution is −3.

ISM: Prealgebra

Chapter 3: Solving Equations and Problem Solving

32. −12 + (−13) = 5 x − 10 x −25 = −5 x −25 −5 x = −5 −5 5= x Check: −12 + (−13) = 5 x − 10 x −12 + (−13) 0 5(5) − 10(5) −25 0 25 − 50 −25 = −25 True The solution is 5. 33.

34.

38. The quotient of 10 and a number is 10 ÷ x or 10 . x 39. Five added to the product of −2 and a number is −2x + 5. 40. The product of −4 and the difference of a number and 1 is −4(x − 1). Section 3.3 Practice Exercises

−4( x + 8) − 11 = 3 − 26 −4 ⋅ x + (−4) ⋅ 8 − 11 = 3 − 26 −4 x − 32 − 11 = −23 −4 x − 43 = −23 −4 x − 43 + 43 = −23 + 43 −4 x = 20 −4 x 20 = −4 −4 x = −5 Check: −4( x + 8) − 11 = 3 − 26 −4(−5 + 8) − 11 0 3 − 26 −4(3) − 11 0 − 23 −12 − 11 0 − 23 −23 = −23 True

7 x + 12 = 3x − 4 7 x + 12 − 12 = 3x − 4 − 12 7 x = 3x − 16 7 x − 3x = 3x − 16 − 3x 4 x = −16 4 x −16 = 4 4 x = −4 7 x + 12 = 3 x − 4 Check: 7(−4) + 21 0 3(−4) − 4 −28 + 12 0 − 12 − 4 −16 = −16 True The solution is −4.

The solution is −5.

40 − 5 y + 5 = −2 y − 10 − 4 y 45 − 5 y = −6 y − 10 45 − 5 y − 45 = −6 y − 10 − 45 −5 y = −6 y − 55 −5 y + 6 y = −6 y − 55 + 6 y y = −55 40 − 5 y + 5 = −2 y − 10 − 4 y Check: 40 − 5(−55) + 5 0 − 2(−55) − 10 − 4(−55) 40 + 275 + 5 0 110 − 10 + 220 320 = 320 True The solution is −55.

6(a − 5) = 4a + 4 6a − 30 = 4a + 4 6a − 30 − 4a = 4a + 4 − 4a 2a − 30 = 4 2a − 30 + 30 = 4 + 30 2a = 34 2a 34 = 2 2 a = 17 The solution is 17.

−6( x − 2) + 10 = −4 − 10 −6 ⋅ x − ( −6)(2) + 10 = −4 − 10 −6 x + 12 + 10 = −14 −6 x + 22 = −14 −6 x + 22 − 22 = −14 − 22 −6 x = −36 −6 x −36 = −6 −6 x=6 Check: −6( x − 2) + 10 = −4 − 10 −6(6 − 2) + 10 0 − 4 − 10 −6(4) + 10 0 − 14 −24 + 10 0 − 14 −14 = −14 True The solution is 6.

35. The difference of a number and 10 is x − 10. 36. The sum of −20 and a number is −20 + x. 37. The product of 10 and a number is 10x.

Chapter 3: Solving Equations and Problem Solving

Vocabulary, Readiness & Video Check 3.3

4( x + 3) + 1 = 13 4 x + 12 + 1 = 13 4 x + 13 = 13 4 x + 13 − 13 = 13 − 13 4x = 0 4x 0 = 4 4 x=0 Check: 4( x + 3) + 1 = 13 4(0 + 3) + 1 0 13 12 + 1 0 13 13 = 13 True The solution is 0.

5. a.

1. An example of an expression is 3x − 9 + x − 16, while an example of an equation is 5(2x + 6) − 1 = 39. x = −10, we use the multiplication −7 property of equality.

2. To solve

3. To solve x − 7 = −10, we use the addition property of equality. 4. To solve 9x − 6x = 10 + 6, first combine like terms.

The difference of 110 and 80 is 30 translates to 110 − 80 = 30.

b. The product of 3 and the sum of −9 and 11 amounts to 6 translates to 3(−9 + 11) = 6. c.

The quotient of 24 and −6 yields −4 24 translates to = −4. −6

Calculator Explorations 1. Replace x with 12. 76(12 − 25) = −988 Yes

ISM: Prealgebra

5. To solve 5(x − 1) = 25, first use the distributive property. 6. To solve 4x + 3 = 19, first use the addition property of equality. 7. the addition property of equality; to make sure we get an equivalent equation 8. remove parentheses; the distributive property 9. gives; amounts to Exercise Set 3.3 2.

7 x − 1 = 8x + 4 7 x − 7 x − 1 = 8x − 7 x + 4 −1 = x + 4 −1 − 4 = x + 4 − 4 −5 = x

5 x − 3 = 2 x − 18 5 x − 3 + 3 = 2 x − 18 + 3 5 x = 2 x − 15 5 x − 2 x = 2 x − 2 x − 15 3 x = −15 3 x −15 = 3 3 x = −5

4 − 7m = −3m + 4 4 − 7m + 7m = −3m + 7m + 4 4 = 4m + 4 4 − 4 = 4m + 4 − 4 0 = 4m 0 4m = 4 4 0=m

2. Replace x with 35. −47 ⋅ 35 + 862 = −783 Yes 3. Replace x with −170. −170 + 562 = 392 3 ⋅ (−170) + 900 = 390 No 4. Replace x with −18. 55(−18 + 10) = −440 75 ⋅ (−18) + 910 = −440 Yes 5. Replace x with −21. 29 ⋅ (−21) − 1034 = −1643 61 ⋅ (−21) − 362 = −1643 Yes 6. Replace x with 25. −38 ⋅ 25 + 205 = −745 25 ⋅ 25 + 120 = 745 No 82

ISM: Prealgebra

Chapter 3: Solving Equations and Problem Solving

57 y + 140 = 54 y − 100 57 y + 140 − 54 y = 54 y − 54 y − 100 3 y + 140 = −100 3 y + 140 − 140 = −100 − 140 3 y = −240 3 y −240 = 3 3 y = −80

10. 2 x + 10 + 3 x = −12 − x − 20 2 x + 3x + 10 = −12 − 20 − x 5 x + 10 = −32 − x 5 x + x + 10 = −32 − x + x 6 x + 10 = −32 6 x + 10 − 10 = −32 − 10 6 x = −42 6 x −42 = 6 6 x = −7 12. 19 x − 2 − 7 x = 31 + 6 x − 15 19 x − 7 x − 2 = 31 − 15 + 6 x 12 x − 2 = 16 + 6 x 12 x − 6 x − 2 = 16 + 6 x − 6 x 6 x − 2 = 16 6 x − 2 + 2 = 16 + 2 6 x = 18 6 x 18 = 6 6 x=3 14. 22 − 42 = 4( x − 1) −20 = 4 x − 4 −20 + 4 = 4 x − 4 + 4 −16 = 4 x −16 4 x = 4 4 −4 = x 16. 2( x + 5) + 8 = 0 2 x + 10 + 8 = 0 2 x + 18 = 0 2 x + 18 − 18 = 0 − 18 2 x = −18 2 x −18 = 2 2 x = −9

18.

3( z + 2) = 5 z + 6 3z + 6 = 5 z + 6 3z − 3z + 6 = 5 z − 3z + 6 6 = 2z + 6 6 − 6 = 2z + 6 − 6 0 = 2z 0 2z = 2 2 0=z

20.

−1( y + 3) = 10 −1y − 3 = 10 − y − 3 = 10 − y − 3 + 3 = 10 + 3 − y = 13 − y 13 = −1 −1 y = −13

22.

−4 + 3c = 4(c + 2) −4 + 3c = 4c + 8 −4 + 3c − 3c = 4c − 3c + 8 −4 = c + 8 −4 − 8 = c + 8 − 8 −12 = c

24. 4(3t + 4) − 20 = 3 + 5t 12t + 16 − 20 = 3 + 5t 12t − 4 = 3 + 5t 12t − 5t − 4 = 3 + 5t − 5t 7t − 4 = 3 7t − 4 + 4 = 3 + 4 7t = 7 7t 7 = 7 7 t =1 26. −3 x = 51 −3 x 51 = −3 −3 x = −17 28.

y − 6 = −11 y − 6 + 6 = −11 + 6 y = −5

30.

7 − z = 15 7 − 7 − z = 15 − 7 −z = 8 −1 ⋅ (− z ) = −1 ⋅ 8 z = −8

Chapter 3: Solving Equations and Problem Solving

32.

z 10 z −12 = 10

−2 − 10 =

10 ⋅ (−12) = 10 ⋅ −120 = z

34.

36.

7 y + 3 = 38 7 y + 3 − 3 = 38 − 3 7 y = 35 7 y 35 = 7 7 y=5 −2b + 5 = −7 −2b + 5 − 5 = −7 − 5 −2b = −12 −2b −12 = −2 −2 b=6

5(3 y − 2) = 16 y 15 y − 10 = 16 y 15 y − 15 y − 10 = 16 y − 15 y −10 = y

42. −9 + 20 = 19 x − 4 − 18 x 11 = x − 4 11 + 4 = x − 4 + 4 15 = x 44. −9( x + 2) + 25 = −19 − 19 −9 x − 18 + 25 = −38 −9 x + 7 = −38 −9 x + 7 − 7 = −38 − 7 −9 x = −45 −9 x −45 = −9 −9 x=5

46.

6 y − 8 = 3y + 7 6 y − 3y − 8 = 3y − 3y + 7 3y − 8 = 7 3y − 8 + 8 = 7 + 8 3 y = 15 3 y 15 = 3 3 y=5

48.

7n + 5 = 12n − 10 7 n + 5 − 7 n = 12n − 10 − 7 n 5 = 5n − 10 5 + 10 = 5n − 10 + 10 15 = 5n 15 5n = 5 5 3=n

50.

10 w + 8 = w − 10 10 w − w + 8 = w − w − 10 9 w + 8 = −10 9 w + 8 − 8 = −10 − 8 9 w = −18 9 w −18 = 9 9 w = −2

52.

4 − 7m = −3m 4 − 7 m + 7m = −3m + 7 m 4 = 4m 4 4m = 4 4 1= m

54.

10 + 4v + 6 = 0 4v + 16 = 0 4v + 16 − 16 = 0 − 16 4v = −16 4v −16 = 4 4 v = −4

56.

2( z − 2) = 5 z + 17 2 z − 4 = 5 z + 17 2 z − 4 − 2 z = 5 z + 17 − 2 z −4 = 3 z + 17 −4 − 17 = 3 z + 17 − 17 −21 = 3 z −21 3 z = 3 3 −7 = z

z 10

38. 4 x − 11x = −14 − 14 −7 x = −28 −7 x −28 = −7 −7 x=4 40.

ISM: Prealgebra

58.

Chapter 3: Solving Equations and Problem Solving

4 + 3c = 2(c + 2) 4 + 3c = 2c + 4 4 + 3c − 2c = 2c − 2c + 4 4+c = 4 4+c−4 = 4−4 c=0

78. y3 + 3xyz = (−1)3 + 3(3)(−1)(0) = −1 + 0 = −1 80. (− y )3 + 3 xyz = [−(−1)]3 + 3(3)(−1)(0)

= 13 + 0 = 1+ 0 =1

60. 4(2t + 5) − 21 = 7t − 6 8t + 20 − 21 = 7t − 6 8t − 1 = 7t − 6 8t − 7t − 1 = 7t − 7t − 6 t − 1 = −6 t − 1 + 1 = −6 + 1 t = −5

82. The first step in solving 3x + 2x = −x − 4 is to add 3x and 2x, which is choice c. 84. The first step in solving 9 − 5x = 15 is to subtract 9 from both sides, which is choice c. 86. The error is in the second line. 37 x + 1 = 9(4 x − 7) 37 x + 1 = 36 x − 63 37 x + 1 − 1 = 36 x − 63 − 1 37 x = 36 x − 64 37 x − 36 x = 36 x − 64 − 36 x x = −64

62. 14 + 4( w − 5) = 6 − 2w 14 + 4w − 20 = 6 − 2w 4w − 6 = 6 − 2w 4w + 2w − 6 = 6 − 2w + 2w 6w − 6 = 6 6w − 6 + 6 = 6 + 6 6w = 12 6w 12 = 6 6 w=2 64.

88. 32 ⋅ x = (−9)3 9 x = −729 9 x −729 = 9 9 x = −81

6(5 + c) = 5(c − 4) 30 + 6c = 5c − 20 30 + 6c − 5c = 5c − 5c − 20 30 + c = −20 30 − 30 + c = −20 − 30 c = −50

90.

66. The difference of −30 and 10 equals −40 translates to −30 − 10 = −40. 68. The quotient of −16 and 2 yields −8 translates to −16 = −8. 2 70. Negative 2 times the sum of 3 and 12 is −30 translates to −2(3 + 12) = −30. 72. Seventeen subtracted from −12 equals −29 translates to −12 − 17 = −29. 74. From the height of the 2015 bar, 132 million or 132,000,000 returns were filed electronically in 2015.

x + 452 = 542 x + 2025 = 2916 x + 2025 − 2025 = 2916 − 2025 x = 891

92. no; answers may vary Section 3.4 Practice Exercises 1. a.

“Four times a number is 20” is 4x = 20.

b. “The sum of a number and −5 yields 32” is x + (−5) = 32. c.

“Fifteen subtracted from a number amounts to −23” is x − 15 = −23.

d. “Five times the difference of a number and 7 is equal to −8” is 5(x − 7) = −8. e.

“The quotient of triple a number and 5 gives 3x 1” is 3x ÷ 5 = 1 or = 1. 5

76. answers may vary

Chapter 3: Solving Equations and Problem Solving 2. “The sum of a number and 2 equals 6 added to three times the number” is x + 2 = 6 + 3x x − x + 2 = 6 + 3x − x 2 = 6 + 2x 2 − 6 = 6 − 6 + 2x −4 = 2 x −4 2 x = 2 2 −2 = x 3. Let x be the distance from Denver to San Francisco. Since the distance from Cincinnati to Denver is 71 miles less than the distance from Denver to San Francisco, the distance from Cincinnati to Denver is x − 71. Since the total of the two distances is 2399, the sum of x and x − 71 is 2399. x + x − 71 = 2399 2 x − 71 = 2399 2 x − 71 + 71 = 2399 + 71 2 x = 2470 2 x 2470 = 2 2 x = 1235 The distance from Denver to San Francisco is 1235 miles. 4. Let x be the amount her son receives. Since her husband receives twice as much as her son, her husband receives 2x. Since the total estate is $57,000, the sum of x and 2x is 57,000. x + 2 x = 57, 000 3 x = 57, 000 3x 57, 000 = 3 3 x = 19, 000 2x = 2(19,000) = 38,000 Her husband will receive $38,000 and her son will receive $19,000. Vocabulary, Readiness & Video Check 3.4 1. The phrase is “a number subtracted from −20” so −20 goes first and we subtract the number from that. 2. The phrase is “three times the difference of some number and 5.” The “difference of some number and 5” translates to the expression x − 5, and in order to multiply 3 times this expression, we need parentheses around the expression.

ISM: Prealgebra 3. The original application asks for the fastest speeds of a pheasant and a falcon. The value of x is the speed in mph for a pheasant, so the falcon’s speed still needs to be found. Exercise Set 3.4 2. “Five subtracted from a number equals 10” is x − 5 = 10. 4. “The quotient of 8 and a number is −2” is 8 = −2. x 6. “Two added to twice a number gives −14” is 2x + 2 = −14. 8. “Five times a number is equal to −75” is 5x = −75. 10. “Twice the sum of −17 and a number is −14” is 2(−17 + x) = −14. 12. “Twice a number, subtracted from 60 is 20” is 60 − 2 x = 20 60 − 60 − 2 x = 20 − 60 −2 x = −40 −2 x −40 = −2 −2 x = 20 14. “The sum of 7, 9, and a number is 40” is 7 + 9 + x = 40 16 + x = 40 16 − 16 + x = 40 − 16 x = 24 16. “Eight decreased by a number equals the quotient of 15 and 5” is 15 8− x = 5 8− x = 3 8−8− x = 3−8 − x = −5 − x −5 = −1 −1 x=5 18. “The product of a number and 3 is twice the sum of that number and 5” is 3 x = 2( x + 5) 3 x = 2 x + 10 3x − 2 x = 2 x − 2 x + 10 x = 10

ISM: Prealgebra

Chapter 3: Solving Equations and Problem Solving

20. “Five times the sum of a number and 2 is 11 less than the number times 8” is 5( x + 2) = 8 x − 11 5 x + 10 = 8 x − 11 5 x − 5 x + 10 = 8 x − 5 x − 11 10 = 3x − 11 10 + 11 = 3x − 11 + 11 21 = 3x 21 3x = 3 3 7=x 22. “Seven times the difference of some number and 1 gives the quotient of 70 and 10” is 70 7( x − 1) = 10 7( x − 1) = 7 7x − 7 = 7 7x − 7 + 7 = 7 + 7 7 x = 14 7 x 14 = 7 7 x=2 24. “Twice a number equals 25 less triple that same number” is 2 x = 25 − 3 x 2 x + 3 x = 25 − 3 x + 3x 5 x = 25 5 x 25 = 5 5 x=5 26. The equation is x + 4x = 50. x + 4 x = 50 5 x = 50 5 x 50 = 5 5 x = 10 Maryland has 10 electoral votes, and Texas has 4(10) = 40 electoral votes.

28. Let x be the width of the playing surface for wheelchair pickleball play. Since the length of the playing surface is 14 feet less than twice the width, the length of the playing surface is 2x − 14. Since the total of the width and length is 118 feet, the sum of x and 2x − 14 is 118. x + 2 x − 14 = 118 3x − 14 = 118 3x − 14 + 14 = 118 + 14 3 x = 132 3 x 132 = 3 3 x = 44 2x − 14 = 2(44) − 14 = 88 − 14 = 74 The width of the playing surface for wheelchair pickleball play is 44 feet and the length of the playing surface is 74 feet. 30. Let x be the width of the playing surface for new construction of pickleball courts. Since the length of the playing surface is the width increased by 30 feet, the length of the playing surface is x + 30. Since the total of the width and length is 98 feet, the sum of x and x + 30 is 98. x + x + 30 = 98 2 x + 30 = 98 2 x + 30 − 30 = 98 − 30 2 x = 68 2 x 68 = 2 2 x = 34 The width of the playing surface for new construction of a pickleball court is 34 feet and the length of the playing surface is 34 feet + 30 feet = 64 feet. 32. Let x be the number of rulers for Liechtenstein. Since Norway has had three times as many rulers as Liechtenstein, Norway has had 3x rulers. Since the total number of rulers for both countries is 56, the sum of x and 3x is 56. x + 3 x = 56 4 x = 56 4 x 56 = 4 4 x = 14 Thus, Liechtenstein has had 14 rulers and Norway has had 3(14) = 42 rulers.

Chapter 3: Solving Equations and Problem Solving

ISM: Prealgebra

34. Let x be the 2021 average daily circulation (in thousands) of The New York Times. Since the average daily circulation of The Wall Street Journal was 367 thousand more than that of The New York Times, the average daily circulation of The Wall Street Journal was x + 367. Since the total circulation of the two papers averaged 1027 thousand per day, the sum of x and x + 367 was 1027. x + x + 367 = 1027 2 x + 367 = 1027 2 x + 367 − 367 = 1027 − 367 2 x = 660 2 x 660 = 2 2 x = 330 The average daily circulation of The New York Times in 2021 was 330 thousand and the average daily circulation of The Wall Street Journal was 330 + 367 = 697 thousand.

40. Let x be the distance from Los Angeles to Tokyo. Since the distance from New York to London is 2001 miles less than the distance from Los Angeles to Tokyo, the distance from New York to London is x − 2001. Since the total of the two distances is 8939, the sum of x and x − 2001 is 8939. x + x − 2001 = 8939 2 x − 2001 = 8939 2 x − 2001 + 2001 = 8939 + 2001 2 x = 10,940 2 x 10,940 = 2 2 x = 5470 The distance from Los Angeles to Tokyo is 5470 miles.

36. Let x be the recognition value (in billions of dollars) of Google. Since the recognition value of Amazon is $23 billion more than that of Google, the recognition value of Amazon is x + 23. Since the total recognition value of the two brands is $527 billion, the sum of x and x + 23 is 527. x + x + 23 = 527 2 x + 23 = 527 2 x + 23 − 23 = 527 − 23 2 x = 504 x = 252 The recognition value of Google is $252 billion, and the recognition value of Amazon is $252 + $23 = $275 billion. 38. Let x be the price of Hasbro Game Night. Since the price of Mario Kart 8 is $44 more than the price of Hasbro Game Night, the price of Mario Kart 8 is x + 44. Since the total of the two prices is $74, the sum of x and x + 44 is 74. x + x + 44 = 74 2 x + 44 = 74 2 x + 44 − 44 = 74 − 44 2 x = 30 x = 15 The price of Hasbro Gam Night is $15, and the price of Mario Kart 8 is $15 + $44 = $59.

42. Let x be the top speed (in miles per hour) of an IndyCar Series car. Since the top speed of an NHRA Top Fuel dragster is 102 mph faster than the top speed of an IndyCar Series car, the top speed of an NHRA Top Fuel dragster is x + 102. Since the combined top speed for these two cars is 574 mph, the sum of x and x + 102 is 574. x + x + 102 = 574 2 x + 102 = 574 2 x + 102 − 102 = 574 − 102 2 x = 472 2 x 472 = 2 2 x = 236 The top speed of an IndyCar Series car is 236 mph, and the top speed of an NHRA Top Fuel dragster is 236 + 102 = 338 mph. 44. Let x be the 2022 Vermont population of Indigenous peoples. Since the 2022 California population of Indigenous peoples was 101 times that of Vermont, the 2022 California population of Indigenous peoples was 101x. Since the total of these two populations was 816,000, the sum of x and 101x is 816,000. x + 101x = 816, 000 102 x = 816, 000 102 x 816, 000 = 102 102 x = 8000 The 2022 Vermont population of Indigenous peoples was 8000, and the 2022 California population of Indigenous peoples was 101(8000) = 808,000.

ISM: Prealgebra

Chapter 3: Solving Equations and Problem Solving

46. Let x be the speed of the truck. Since the speed of the car is twice that of the truck, the speed of the car is 2x. Since the combined speed is 105 miles per hour, the sum of x and 2x is 105. x + 2 x = 105 3 x = 105 3 x 105 = 3 3 x = 35 The speed of the truck is 35 miles per hour and the speed of the car is 2(35) = 70 miles per hour. 48. Let x be the value of the plow. Since the tractor is worth seven times as much as the plow, the tractor is worth 7x. Since their combined worth is $1200, the sum of x and 7x is 1200. x + 7 x = 1200 8 x = 1200 8 x 1200 = 8 8 x = 150 The plow is worth $150 and the tractor is worth 7($150) = $1050. 50. Let x be the maximum length (in yards) of a soccer field for under-6-year-olds. Since the maximum length of a under-13-year-old soccer field is 80 yards longer than the length for under6-year-olds, the maximum length for a under-13-year-old soccer field is x + 80. Since the total length for these two categories of soccer fields is 140 yards, the sum of x and x + 80 is 140. x + x + 80 = 140 2 x + 80 = 140 2 x + 80 − 80 = 140 − 80 2 x = 60 2 x 60 = 2 2 x = 30 The maximum length of a soccer field for under6-year-olds is 30 yards, and the maximum length of a under-13-year-old soccer field is 30 + 80 = 110 yards. 52. Let x be the number of points scored by the University of Connecticut Huskies. Since the South Carolina Gamecocks scored 15 points more than the Huskies, the Gamecocks scored x + 15 points. Since the total number of points scored by the two teams was 113, the sum of x and x + 15 is 113.

x + x + 15 = 113 2 x + 15 = 113 2 x + 15 − 15 = 113 − 15 2 x = 98 2 x 98 = 2 2 x = 49 x + 15 = 49 + 15 = 64 The South Carolina Gamecocks scored 64 points.

54. Let x be the amount (in millions of dollars) that is estimated to be spent on TV advertising in 2026. Since it is estimated that the amount spent on digital advertising will be $320,830 million more than will be spent on TV advertising, the amount estimated to be spent on digital advertising is x + 320,830. Since the total amount estimated to be spent on digital and TV advertising in 2026 is $450,110 million, the sum and x + 320,830 is 450,110. x + x + 320,830 = 450,110 2 x + 320,830 = 450,110 2 x + 320,830 − 320,830 = 450,110 − 320,830 2 x = 129, 280 2 x 129, 280 = 2 2 x = 64, 640 x + 320,830 = 64,640 + 320,830 = 385,470 The amount estimated to be spent on TV advertising in 2026 is $64,640 million, and the amount estimated to be spent on digital advertising is $385,470 million. 56. To round 82 to the nearest ten, observe that the digit in the ones place is 2. Since this digit is less than 5, we do not add 1 to the digit in the tens place. 82 rounded to the nearest ten is 80. 58. To round 52,333 to the nearest thousand, observe that the digit in the hundreds place is 3. Since this digit is less than 5, we do not add 1 to the digit in the thousands place. 52,333 rounded to the nearest thousand is 52,000. 60. To round 101,552 to the nearest hundred, observe that the digit in the tens place is 5. Since this digit is at least 5, we add one to the digit in the hundreds place. 101,552 rounded to the nearest hundred is 101,600. 62. yes; answers may vary

Chapter 3: Solving Equations and Problem Solving 64. Use P = A + C, where P = 165,000 and A = 156,750. P = A+C 165, 000 = 156, 750 + C 165, 000 − 156, 750 = 156, 750 − 156, 750 + C 8250 = C The agent will receive $8250. 66. Use P = C + M where P = 12 and C = 7. P =C+M 12 = 7 + M 12 − 7 = 7 − 7 + M 5=M The markup is $5. Chapter 3 Vocabulary Check 1. An algebraic expression is simplified when all like terms have been combined. 2. Terms that are exactly the same, except that they may have different numerical coefficients, are called like terms. 3. A letter used to represent a number is called a variable. 4. A combination of operations on variables and numbers is called an algebraic expression. 5. The addends of an algebraic expression are called the terms of the expression. 6. The number factor of a variable term is called the numerical coefficient. 7. Replacing a variable in an expression by a number and then finding the value of the expression is called evaluating the expression for the variable. 8. A term that is a number only is called a constant. 9. An equation is of the form expression = expression. 10. A solution of an equation is a value for the variable that makes an equation a true statement. 11. To multiply −3(2x + 1), we use the distributive property. 12. By the multiplication property of equality, we may multiply or divide both sides of an equation by any nonzero number without changing the solution of the equation. 90

ISM: Prealgebra 13. By the addition property of equality, the same number may be added to or subtracted from both sides of an equation without changing the solution of the equation. Chapter 3 Review 1. 3y + 7y − 15 = (3 + 7)y − 15 = 10y − 15 2. 2 y − 10 − 8 y = 2 y − 8 y − 10 = (2 − 8) y − 10 = −6 y − 10 3. 8a + a − 7 − 15a = 8a + a − 15a − 7 = (8 + 1 − 15)a − 7 = −6 a − 7 4. y + 3 − 9 y − 1 = y − 9 y + 3 − 1 = (1 − 9) y + 3 − 1 = −8 y + 2 5. 2(x + 5) = 2 ⋅ x + 2 ⋅ 5 = 2x + 10 6. −3(y + 8) = −3 ⋅ y + (−3) ⋅ 8 = −3y − 24 7. 7 x + 3( x − 4) + x = 7 x + 3 ⋅ x − 3 ⋅ 4 + x = 7 x + 3 x − 12 + x = 7 x + 3 x + x − 12 = (7 + 3 + 1) x − 12 = 11x − 12 8. −(3m + 2) − m − 10 = −1(3m + 2) − m − 10 = −1 ⋅ 3m + (−1) ⋅ 2 − m − 10 = −3m − 2 − m − 10 = −3m − m − 2 − 10 = (−3 − 1)m − 2 − 10 = −4m − 12 9. 3(5a − 2) − 20a + 10 = 3 ⋅ 5a − 3 ⋅ 2 − 20a + 10 = 15a − 6 − 20a + 10 = 15a − 20a − 6 + 10 = (15 − 20)a − 6 + 10 = −5a + 4 10. 6 y + 3 + 2(3 y − 6) = 6 y + 3 + 2 ⋅ 3 y − 2 ⋅ 6 = 6 y + 3 + 6 y − 12 = 6 y + 6 y + 3 − 12 = (6 + 6) y + 3 − 12 = 12 y − 9

ISM: Prealgebra

Chapter 3: Solving Equations and Problem Solving

11. 6 y − 7 + 11 y − y + 2 = 6 y + 11 y − y − 7 + 2 = (6 + 11 − 1) y − 7 + 2 = 16 y − 5 12. 10 − x + 5 x − 12 − 3 x = − x + 5 x − 3 x + 10 − 12 = (−1 + 5 − 3) x + 10 − 12 = 1x − 2 = x−2

22.

23.

n + 18 = 10 − (−2) n + 18 = 10 + 2 n + 18 = 12 n + 18 − 18 = 12 − 18 n = −6

24.

c − 5 = −13 + 7 c − 5 = −6 c − 5 + 5 = −6 + 5 c = −1

13. Perimeter = 2(2x + 3) = 2 ⋅ 2x + 2 ⋅ 3 = 4x + 6 The perimeter is (4x + 6) yards. 14. Perimeter = 4 ⋅ 5y = 20y The perimeter is 20y meters. 15. Area = (length) ⋅ (width) = 3 ⋅ (2 x − 1) = 3 ⋅ 2 x − 3 ⋅1 = 6x − 3 The area is (6x − 3) square yards. 16. Add the areas of the two rectangles. 10( x − 2) + 7(5 x + 4) = 10 ⋅ x − 10 ⋅ 2 + 7 ⋅ 5 x + 7 ⋅ 4 = 10 x − 20 + 35 x + 28 = 10 x + 35 x − 20 + 28 = (10 + 35) x − 20 + 28 = 45 x + 8 The area is (45x + 8) square centimeters. 17.

z − 5 = −7 z − 5 + 5 = −7 + 5 z = −2

18.

3 x + 10 = 4 x 3 x − 3 x + 10 = 4 x − 3 x 10 = x

25. 7 x + 5 − 6 x = −20 x + 5 = −20 x + 5 − 5 = −20 − 5 x = −25 26.

17 x = 2(8 x − 4) 17 x = 2 ⋅ 8 x − 2 ⋅ 4 17 x = 16 x − 8 17 x − 16 x = 16 x − 16 x − 8 x = −8

27.

5 x + 7 = −3 5 x + 7 − 7 = −3 − 7 5 x = −10 5 x −10 = 5 5 x = −2

28.

−14 = 9 y + 4 −14 − 4 = 9 y + 4 − 4 −18 = 9 y −18 9 y = 9 9 −2 = y

19. 3 y = −21 3 y −21 = 3 3 y = −7 20. −3a = −15 −3a −15 = −3 −3 a=5 21.

x =2 −6 x −6 ⋅ = −6 ⋅ 2 −6 x = −12

y = −3 −15 y −15 ⋅ = −15 ⋅ (−3) −15 y = 45

29.

z = −8 − (−6) 4 z = −8 + 6 4 z = −2 4 z 4 ⋅ = 4 ⋅ (−2) 4 z = −8

Chapter 3: Solving Equations and Problem Solving

x 5 x −9 = 5

30. −1 + (−8) =

5 ⋅ (−9) = 5 ⋅ −45 = x

ISM: Prealgebra

37.

2( x + 4) − 10 = −2(7) 2 ⋅ x + 2 ⋅ 4 − 10 = −14 2 x + 8 − 10 = −14 2 x − 2 = −14 2 x − 2 + 2 = −14 + 2 2 x = −12 2 x −12 = 2 2 x = −6

38.

−3( x − 6) + 13 = 20 − 1 −3 ⋅ x − (−3) ⋅ 6 + 13 = 19 −3 x + 18 + 13 = 19 −3 x + 31 = 19 −3 x + 31 − 31 = 19 − 31 −3x = −12 −3x −12 = −3 −3 x=4

x 5

31. 6 y − 7 y = 100 − 105 − y = −5 − y −5 = −1 −1 y=5 32. 19 x − 16 x = 45 − 60 3 x = −15 3 x −15 = 3 3 x = −5 33.

34.

35.

36.

9(2 x − 7) = 19 x 9 ⋅ 2 x − 9 ⋅ 7 = 19 x 18 x − 63 = 19 x 18 x − 18 x − 63 = 19 x − 18 x −63 = x

39. The product of −5 and a number is −5x.

−5(3 x + 3) = −14 x −5 ⋅ 3 x − 5 ⋅ 3 = −14 x −15 x − 15 = −14 x −15 x + 15 x − 15 = −14 x + 15 x −15 = x

42. The quotient of −2 and a number is −2 ÷ x or

3x − 4 = 11 3x − 4 + 4 = 11 + 4 3 x = 15 3 x 15 = 3 3 x=5 6 y + 1 = 73 6 y + 1 − 1 = 73 − 1 6 y = 72 6 y 72 = 6 6 y = 12

40. Three subtracted from a number is x − 3. 41. The sum of −5 and a number is −5 + x. −2 x

2 or − . x

43. The product of −5 and a number, decreased by 50 is −5x − 50. 44. Eleven added to twice a number is 2x + 11. 45. The quotient of 70 and the sum of a number and 70 6 is 70 ÷ (x + 6) or . x+6 46. Twice the difference of a number and 13 is 2(x − 13). 47.

2 x + 5 = 7 x − 100 2 x − 2 x + 5 = 7 x − 2 x − 100 5 = 5 x − 100 5 + 100 = 5 x − 100 + 100 105 = 5 x 105 5 x = 5 5 21 = x

ISM: Prealgebra

48.

−6 x − 4 = x + 66 −6 x + 6 x − 4 = x + 6 x + 66 −4 = 7 x + 66 −4 − 66 = 7 x + 66 − 66 −70 = 7 x −70 7 x = 7 7 −10 = x

49.

2x + 7 = 6x −1 2x − 2x + 7 = 6x − 2x −1 7 = 4x −1 7 +1 = 4x −1 +1 8 = 4x 8 4x = 4 4 2=x

50.

5 x − 18 = −4 x 5 x − 5 x − 18 = −4 x − 5 x −18 = −9 x −18 −9 x = −9 −9 2=x

51.

5(n − 3) = 7 + 3n 5n − 15 = 7 + 3n 5n − 3n − 15 = 7 + 3n − 3n 2n − 15 = 7 2n − 15 + 15 = 7 + 15 2n = 22 2n 22 = 2 2 n = 11

52.

7(2 + x) = 4 x − 1 14 + 7 x = 4 x − 1 14 + 7 x − 4 x = 4 x − 4 x − 1 14 + 3x = −1 14 − 14 + 3x = −1 − 14 3x = −15 3x −15 = 3 3 x = −5

Chapter 3: Solving Equations and Problem Solving

53. 6 x + 3 − (− x) = −20 + 5 x − 7 6 x + 3 + x = −20 + 5 x − 7 7 x + 3 = −27 + 5 x 7 x − 5 x + 3 = −27 + 5 x − 5 x 2 x + 3 = −27 2 x + 3 − 3 = −27 − 3 2 x = −30 2 x −30 = 2 2 x = −15 54.

x − 25 + 2 x = −5 + 2 x − 10 −25 + 3 x = −15 + 2 x −25 + 3x − 2 x = −15 + 2 x − 2 x −25 + x = −15 −25 + 25 + x = −15 + 25 x = 10

55.

3( x − 4) = 5 x − 8 3x − 12 = 5 x − 8 3x − 3x − 12 = 5 x − 3 x − 8 −12 = 2 x − 8 −12 + 8 = 2 x − 8 + 8 −4 = 2 x −4 2 x = 2 2 −2 = x

56.

4( x − 3) = −2 x − 48 4 x − 12 = −2 x − 48 4 x + 2 x − 12 = −2 x + 2 x − 48 6 x − 12 = −48 6 x − 12 + 12 = −48 + 12 6 x = −36 6 x −36 = 6 6 x = −6

57. 6(2n − 1) + 18 = 0 12n − 6 + 18 = 0 12n + 12 = 0 12n + 12 − 12 = 0 − 12 12n = −12 12n −12 = 12 12 n = −1

Chapter 3: Solving Equations and Problem Solving

68. “The difference of a number and 3 is the quotient 8 of 8 and 4” is x − 3 = . 4

58. 7(3 y − 2) − 7 = 0 21 y − 14 − 7 = 0 21 y − 21 = 0 21 y − 21 + 21 = 0 + 21 21y = 21 21y 21 = 21 21 y =1 59.

95 x − 14 = 20 x − 10 + 10 x − 4 95 x − 14 = 30 x − 14 95 x − 14 + 14 = 30 x − 14 + 14 95 x = 30 x 95 x − 30 x = 30 x − 30 x 65 x = 0 65 x 0 = 65 65 x=0

60. 32 z + 11 − 28 z = 50 + 2 z − (−1) 4 z + 11 = 50 + 2 z + 1 4 z + 11 = 51 + 2 z 4 z − 2 z + 11 = 51 + 2 z − 2 z 2 z + 11 = 51 2 z + 11 − 11 = 51 − 11 2 z = 40 2 z 40 = 2 2 z = 20 61. The difference of 20 and −8 is 28 translates to 20 − (−8) = 28. 62. Nineteen subtracted from −2 amounts to −21 translates to −2 − 19 = −21. 63. The quotient of −75 and the sum of 5 and 20 is −75 equal to −3 translates to = −3. 5 + 20 64. Five times the sum of 2 and −6 yields −20 translates to 5[2 + (−6)] = −20. 65. “Twice a number minus 8 is 40” is 2x − 8 = 40. 66. “The product of a number and 6 is equal to the sum of the number and 2a” is 6x = x + 2a. 67. “Twelve subtracted from the quotient of a x number and 2 is 10” is − 12 = 10. 2 94

ISM: Prealgebra

69. “Five times a number subtracted from 40 is the same as three times the number” is 40 − 5 x = 3 x 40 − 5 x + 5 x = 3 x + 5 x 40 = 8 x 40 8 x = 8 8 5= x The number is 5. 70. “The product of a number and 3 is twice the difference of that number and 8” is 3 x = 2( x − 8) 3 x = 2 x − 16 3 x − 2 x = 2 x − 2 x − 16 x = −16 The number is −16. 71. Let x be the number of votes for the Independent candidate. Since the Democratic candidate received 272 more votes than the Independent candidate, the Democratic candidate received x + 272 votes. The total number of votes is 18,500. x + x + 272 + 14, 000 = 18,500 2 x + 14, 272 = 18,500 2 x + 14, 272 − 14, 272 = 18,500 − 14, 272 2 x = 4228 2 x 4228 = 2 2 x = 2114 The Democratic candidate received 2114 + 272 = 2386 votes. 72. Let x be the number of shows he watches on Hulu each month. Since he watches twice as many shows each month on Netflix, the number of shows he watches on Netflix each month is 2x. Since the total number of shows he watches each month is 126, the sum of x and 2x is 126. x + 2 x = 126 3 x = 126 3 x 126 = 3 3 x = 42 He watches 2(42) = 84 shows each month on Netflix. 73. 9x − 20x = (9 − 20)x = −11x

ISM: Prealgebra

Chapter 3: Solving Equations and Problem Solving

74. −5(7x) = (−5 ⋅ 7)x = −35x

84.

75. 12 x + 5(2 x − 3) − 4 = 12 x + 10 x − 15 − 4 = 22 x − 19 76. −7( x + 6) − 2( x − 5) = −7 x − 42 − 2 x + 10 = −7 x − 2 x − 42 + 10 = −9 x − 32 77.

c − 5 = −13 + 7 c − 5 = −6 c − 5 + 5 = −6 + 5 c = −1

78. 7 x + 5 − 6 x = −20 x + 5 = −20 x + 5 − 5 = −20 − 5 x = −25

85.

12 y − 10 = −70 12 y − 10 + 10 = −70 + 10 12 y = −60 12 y −60 = 12 12 y = −5

86.

4n − 8 = 2n + 14 4n − 2n − 8 = 2n − 2n + 14 2n − 8 = 14 2n − 8 + 8 = 14 + 8 2n = 22 2n 22 = 2 2 n = 11

87.

−6( x − 3) = x + 4 −6 x + 18 = x + 4 −6 x + 6 x + 18 = x + 6 x + 4 18 = 7 x + 4 18 − 4 = 7 x + 4 − 4 14 = 7 x 14 7 x = 7 7 2=x

79. −7 x + 3 x = −50 − 2 −4 x = −52 −4 x −52 = −4 −4 x = 13 80. − x + 8 x = −38 − 4 7 x = −42 7 x −42 = 7 7 x = −6 81. 9 x + 12 − 8 x = −6 + (−4) x + 12 = −10 x + 12 − 12 = −10 − 12 x = −22 82. −17 x + 14 + 20 x − 2 x = 5 − (−3) −17 x + 20 x − 2 x + 14 = 5 + 3 x + 14 = 8 x + 14 − 14 = 8 − 14 x = −6 83.

5(2 x − 3) = 11x 10 x − 15 = 11x 10 x − 10 x − 15 = 11x − 10 x −15 = x

y = −1 − 5 −3 y = −6 −3 y −3 ⋅ = −3 ⋅ (−6) −3 y = 18

88. 9(3 x − 4) + 63 = 0 27 x − 36 + 63 = 0 27 x + 27 = 0 27 x + 27 − 27 = 0 − 27 27 x = −27 27 x −27 = 27 27 x = −1

Chapter 3: Solving Equations and Problem Solving 89. −5 z + 3 z − 7 = 8 z − 1 − 6 −2 z − 7 = 8 z − 7 −2 z − 8 z − 7 = 8 z − 8 z − 7 −10 z − 7 = −7 −10 z − 7 + 7 = −7 + 7 −10 z = 0 0 −10 z = −10 −10 z=0 90.

The amount of energy consumption for Vermont was 126 trillion BTU, and the amount of energy consumption for the District of Columbia was 126 + 18 = 144 trillion BTU.

4 x − 3 + 6 x = 5 x − 3 − 30 10 x − 3 = 5 x − 33 10 x − 5 x − 3 = 5 x − 5 x − 33 5 x − 3 = −33 5 x − 3 + 3 = −33 + 3 5 x = −30 5 x −30 = 5 5 x = −6

91. “Three times a number added to twelve is 27” is 12 + 3x = 27 12 − 12 + 3x = 27 − 12 3x = 15 3x 15 = 3 3 x=5 The number is 5. 92. “Twice the sum of a number and four is ten” is 2( x + 4) = 10 2 x + 8 = 10 2 x + 8 − 8 = 10 − 8 2x = 2 2x 2 = 2 2 x =1 The number is 1. 93. Let x be the amount of energy consumption (in trillions of BTU) for Vermont. Since the District of Columbia used 18 trillion BTU more than Vermont, the amount of energy consumption for the District of Columbia was x + 18. Since the total number of BTU used by both entities was 270 trillion BTU, the sum of x and x + 18 is 270. x + x + 18 = 270 2 x + 18 = 270 2 x + 18 − 18 = 270 − 18 2 x = 252 2 x 252 = 2 2 x = 126 96

ISM: Prealgebra

94. Let x be the number of roadway miles in South Dakota. Since North Dakota has 4489 more roadway miles than South Dakota, North Dakota has x + 4489 roadway miles. Since the total number of roadway miles is 169,197, the sum of x and x + 4489 is 169,197. x + x + 4489 = 169,197 2 x + 4489 = 169,197 2 x + 4489 − 4489 = 169,197 − 4489 2 x = 164, 708 2 x 164, 708 = 2 2 x = 82,354 South Dakota has 82,354 roadway miles and North Dakota has 82,354 + 4489 = 86,843 roadway miles. Chapter 3 Getting Ready for the Test 1. x − 7x = 1x − 7x = (1 − 7)x = −6x Choice A 2. −2 x + 3 + 8 x − 3 = −2 x + 8 x + 3 − 3 = (−2 + 8) x + (3 − 3) = 6x + 0 = 6x Choice B 3. −3(2x) = (−3 ⋅ 2)x = −6x Choice A 4. 2(2x + 1) = 2 ⋅ 2x + 2 ⋅ 1 = 4x + 2 Choice C 5. −( x + 2) + 5 x + 4 = −1( x + 2) + 5 x + 4 = −1 ⋅ x + (−1)(2) + 5 x + 4 = − x − 2 + 5x + 4 = − x + 5x − 2 + 4 = −1x + 5 x − 2 + 4 = (−1 + 5) x + (−2 + 4) = 4x + 2 Choice C 6. 3(1 + 2 x) − 3 = 3 ⋅1 + 3 ⋅ 2 x − 3 = 3 + 6x − 3 = 6x + 3 − 3 = 6x + 0 = 6x Choice B

ISM: Prealgebra 7.

Chapter 3: Solving Equations and Problem Solving

5 x − 10 = 0 5 x − 10 + 10 = 0 + 10 5 x = 10 Choice B

8. 4(2 x + 1) − 8 = 14 x − 10 8 x + 4 − 8 = 14 x − 10 8 x − 4 = 14 x − 10 Choice A

9. The sum of −3 and a number is −3 + x; choice C. 10. The product of −3 and a number is −3 ⋅ x or −3x; choice A. 11. 5 subtracted from a number is x − 5; choice D. 12. 5 minus a number is 5 − x; choice E. 13. Twice a number gives 14 is 2x = 14; choice B. 14. A number divided by 6 yields 14 is

x = 14; 6

−4 x + 7 = 15 −4 x + 7 − 7 = 15 − 7 −4 x = 8 −4 x 8 = −4 −4 x = −2

10.

2( x − 6) = 0 2 x − 12 = 0 2 x − 12 + 12 = 0 + 12 2 x = 12 2 x 12 = 2 2 x=6

15. Two subtracted from a number equals 14 is x − 2 = 14; choice E.

Chapter 3 Test 1. 7 x − 5 − 12 x + 10 = 7 x − 12 x − 5 + 10 = (7 − 12) x − 5 + 10 = −5 x + 5 2. −2(3y + 7) = −2 ⋅ 3y + (−2) ⋅ 7 = −6y − 14 3. −(3 z + 2) − 5 z − 18 = −1(3 z + 2) − 5 z − 18 = −1 ⋅ 3 z + (−1) ⋅ 2 − 5 z − 18 = −3 z − 2 − 5 z − 18 = −3 z − 5 z − 2 − 18 = −8 z − 20 4. perimeter = 3(5x + 5) = 3 ⋅ 5x + 3 ⋅ 5 = 15x + 15 The perimeter is (15x + 15) inches.

x = −5 − (−2) 2 x = −5 + 2 2 x = −3 2 x 2 ⋅ = 2 ⋅ (−3) 2 x = −6

8. 5 x + 12 − 4 x − 14 = 22 x − 2 = 22 x − 2 + 2 = 22 + 2 x = 24

choice C.

16. Six times a number, added to 2 is 14 is 2 + 6x = 14; choice F.

12 = y − 3 y 12 = −2 y 12 −2 y = −2 −2 −6 = y

11. −4( x − 11) − 34 = 10 − 12 −4 x + 44 − 34 = 10 − 12 −4 x + 10 = −2 −4 x + 10 − 10 = −2 − 10 −4 x = −12 −4 x −12 = −4 −4 x=3

5. Area = (length) ⋅ (width) = 4 ⋅ (3 x − 1) = 4 ⋅ 3 x − 4 ⋅1 = 12 x − 4 The area is (12x − 4) square meters.

Chapter 3: Solving Equations and Problem Solving

12.

5 x − 2 = x − 10 5 x − x − 2 = x − x − 10 4 x − 2 = −10 4 x − 2 + 2 = −10 + 2 4 x = −8 4 x −8 = 4 4 x = −2

13.

4(5 x + 3) = 2(7 x + 6) 20 x + 12 = 14 x + 12 20 x + 12 − 14 x = 14 x + 12 − 14 x 6 x + 12 = 12 6 x + 12 − 12 = 12 − 12 6x = 0 6x 0 = 6 6 x=0

ISM: Prealgebra 20. Let x be the number of free throws Jo made. Since Thanh made twice as many free throws as Jo, Thanh made 2x free throws. Since the total number of free throws was 12, the sum of x and 2x is 12. x + 2 x = 12 3 x = 12 3 x 12 = 3 3 x=4 Thanh made 2(4) = 8 free throws. 21. Let x be the number children participating in the event. Since the number of adults participating in the event is 112 more than the number of children, the number of adults is x + 112. Since the total number of participants in the event is 600, the sum of x and x + 112 is 600. x + x + 112 = 600 2 x + 112 = 600 2 x + 112 − 112 = 600 − 112 2 x = 488 2 x 488 = 2 2 x = 244 244 children participated in the event.

14. 6 + 2(3n − 1) = 28 6 + 6n − 2 = 28 6n + 4 = 28 6n + 4 − 4 = 28 − 4 6n = 24 6n 24 = 6 6 n=4

Cumulative Review Chapters 1−3

15. The sum of −23 and a number translates to −23 + x.

1. 308,063,557 in words is three hundred eight million, sixty-three thousand, five hundred fiftyseven.

16. Three times a number, subtracted from −2 translates to −2 − 3x.

2. 276,004 in words is two hundred seventy-six thousand, four.

17. The sum of twice 5 and −15 is −5 translates to 2 ⋅ 5 + (−15) = −5.

3. 2 + 3 + 1 + 3 + 4 = 13 The perimeter is 13 inches.

18. Six added to three times a number equals −30 translates to 3x + 6 = −30.

4. 6 + 3 + 6 + 3 = 18 The perimeter is 18 inches.

19. The difference of three times a number and five times the same number is 4 translates to 3x − 5 x = 4 −2 x = 4 −2 x 4 = −2 − 2 x = −2 The number is −2.

900 − 174 726

Check:

726 + 174 900

17,801 − 8216 9585

Check:

9585 + 8216 17,801

7. To round 248,982 to the nearest hundred, observe that the digit in the tens place is 8. Since this digit is at least 5, we add 1 to the digit in the hundreds place. The number 248,982 rounded to the nearest hundred is 249,000.

ISM: Prealgebra

Chapter 3: Solving Equations and Problem Solving

8. To round 844,497 to the nearest thousand, observe that the digit in the hundreds place is 4. Since this digit is less than 5, we do not add 1 to the digit in the thousands place. The number 844,497 rounded to the nearest thousand is 844,000.

16. 2a 2 + 5 − c = 2 ⋅ 22 + 5 − 3 = 2⋅4 + 5 − 3 = 8+5−3 = 13 − 3 = 10

25 × 8 200

10.

395 74 1 580 27 650 29, 230

17. 2n − 30 = 10 Let n = 26. 2 ⋅ 26 − 30 0 10 52 − 30 0 10 22 = 10 False 26 is not a solution. Let n = 40. 2 ⋅ 40 − 30 0 10 80 − 30 0 10 50 = 10 False 40 is not a solution. Let n = 20. 2 ⋅ 20 − 30 0 10 40 − 30 0 10 10 = 10 True 20 is a solution.

208 11. 9 1872 −18 07 −0 72 −72 0 Check: 208 × 9 1872

18. a.

b. −(−7) = 7, so −(−7) > −8 because 7 is to the right of −8 on the number line. 19. 5 + (−2) = 3

86 12. 46 3956 −368 276 −276 0 Check: 86 × 46 516 3440 3956

20. −3 + (−4) = −7 21. −15 + (−10) = −25 22. 3 + (−7) = −4 23. −2 + 25 = 23 24. 21 + 15 + (−19) = 36 + (−19) = 17 25. −4 − 10 = −4 + (−10) = −14

13. 2 ⋅ 4 − 3 ÷ 3 = 8 − 3 ÷ 3 = 8 − 1 = 7 14. 8 ⋅ 4 + 9 ÷ 3 = 32 + 9 ÷ 3 = 32 + 3 = 35 15. x 2 + z − 3 = 52 + 4 − 3 = 25 + 4 − 3 = 29 − 3 = 26

−14 < 0 because −14 is to the left of 0 on the number line.

26. −2 − 3 = −2 + (−3) = −5 27. 6 − (−5) = 6 + 5 = 11 28. 19 − (−10) = 19 + 10 = 29 29. −11 − (−7) = −11 + 7 = −4 30. −16 − (−13) = −16 + 13 = −3

Chapter 3: Solving Equations and Problem Solving

31.

−12 = −2 6

32.

−30 =6 −5

ISM: Prealgebra 46. −3 y = 15 −3 y 15 = −3 −3 y = −5 47.

2 x − 6 = 18 2 x − 6 + 6 = 18 + 6 2 x = 24 2 x 24 = 2 2 x = 12

48.

3a + 5 = −1 3a + 5 − 5 = −1 − 5 3a = −6 3a −6 = 3 3 a = −2

33. −20 ÷ (−4) = 5 34. 26 ÷ (−2) = −13 35.

36.

48 = −16 −3

−120 = −10 12

37. (−3)2 = (−3)(−3) = 9 38. −25 = −(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2) = −32 39. −32 = −(3 ⋅ 3) = −9 40. (−5)2 = (−5)(−5) = 25 41. 2 y − 6 + 4 y + 8 = 2 y + 4 y − 6 + 8 = (2 + 4) y − 6 + 8 = 6y + 2 42. 6 x + 2 − 3 x + 7 = 6 x − 3 x + 2 + 7 = (6 − 3) x + 2 + 7 = 3x + 9 43.

3y +1 = 3 3(−1) + 1 0 3 −3 + 1 0 3 −2 = 3 False Since −2 = 3 is false, −1 is not a solution of the equation.

44.

5x − 3 = 7 5(2) − 3 0 7 10 − 3 0 7 7 = 7 True Since 7 = 7 is true, 2 is a solution of the equation.

49. Let x be the price of the software. Since the price of the computer system is four times the price of the software, the price of the computer system is 4x. Since the combined price is $2100, the sum of x and 4x is 2100. x + 4 x = 2100 5 x = 2100 5 x 2100 = 5 5 x = 420 The price of the software is $420 and the price of the computer system is 4($420) = $1680. 50. Let x be the number. “Two times the number plus four is the same amount as three times the number minus seven” translates to 2 x + 4 = 3x − 7 2 x − 2 x + 4 = 3x − 2 x − 7 4 = x−7 4+7 = x−7+7 11 = x The number is 11.

45. −12 x = −36 −12 x −36 = −12 −12 x=3 100

Chapter 4 Section 4.1 Practice Exercises

11 , the numerator is 11 and the 2 denominator is 2.

1. In the fraction

13. a.

10 y , the numerator is 10y and 17 the denominator is 17.

2. In the fraction

3 3. 3 out of 8 equal parts are shaded: 8 1 6

4. 1 out of 6 equal parts is shaded:

5. 7 out of 10 equal parts are shaded:

6. 9 out of 16 equal parts are shaded:

7 10 9 16

b. c. 14.

9 =1 9

15.

−6 =1 −6

16.

0 =0 −1

17.

4 =4 1

18.

−13 is undefined 0

19.

−13 = −13 1

7. answers may vary; for example, 8. answers may vary; for example,

→5 9. number of planets farther number of planets in our solar system → 8 5 of the planets in our solar system are farther 8 from the Sun than Earth is. 1 of a whole and there are 8 parts 3 shaded, or 2 wholes and 2 more parts. 8 2 ;2 3 3

20. a.

2 7 ⋅ 5 + 2 35 + 2 37 = = = 7 7 7 7

2 3 ⋅ 6 + 2 18 + 2 20 = = = 3 3 3 3

1 5 ⋅ 4 + 1 20 + 1 21 4 = = = 5 5 5 5

10. Each part is

1 of a whole and there are 5 parts 4 shaded, or 1 whole and 1 more part. 5 1 ;1 4 4

11. Each part is

12. a.

21. a.

9 10 ⋅10 + 9 100 + 9 109 = = = 10 10 10 10

1 59 5 4 9 4 =1 5 5

101

Chapter 4: Fractions and Mixed Numbers

2 9 23 18 5

8 is called an improper fraction, 3 3 the fraction is called a proper fraction, and 8 3 10 is called a mixed number. 8

3. The fraction

23 5 =2 9 9

12 4 48 4 8 8 0

4. The value of an improper fraction is always ≥1 and the value of a proper fraction is always <1. 5. When the numerator is greater than or equal to the denominator, you have an improper fraction. 6. Each shape is divided into 3 equal parts.

48 = 12 4

7. how many equal parts to divide each whole number into

4 13 62 52 10

8. The fraction is equal to 1. 9. The operation of addition is understood in mixed 1 1 number notation; for example, 1 means 1 + _ 3. 3

62 10 =4 13 13

10. division

7 7 51 49 2

Exercise Set 4.1 1 , the numerator is 1 and the 4 denominator is 4. Since 1 < 4, the fraction is

2. In the fraction

51 2 =7 7 7

ISM: Prealgebra

proper.

1 20 21 20 1

53 , the numerator is 53 and the 21 denominator is 21. Since 53 > 21, the fraction is improper.

4. In the fraction

21 1 =1 20 20

26 , the numerator is 26 and the 26 denominator is 26. Since 26 ≥ 26, the fraction is improper.

6. In the fraction

Vocabulary, Readiness & Video Check 4.1 17 is called a fraction. The number 31 31 is called its denominator and 17 is called its numerator.

1. The number

0 −9 2. If we simplify each fraction, = 0, = 1, −9 −4 −4 is undefined. and we say 0

102

8. 5 out of 6 equal parts are shaded:

5 6

1 of a whole and there are 10 parts 4 shaded, or 2 wholes and 2 more parts.

10. Each part is

10 4

ISM: Prealgebra

Chapter 4: Fractions and Mixed Numbers 28. answers may vary; for example,

2 4

1 of a whole and there are 11 parts 3 shaded, or 3 wholes and 2 more parts.

12. Each part is

11 3

30. answers may vary; for example,

32. answers may vary; for example,

2 3

14. 5 out of 8 equal parts are shaded:

5 8

5 16. 5 out of 12 equal parts are shaded: 12

18. 7 out of 8 equal parts are shaded:

7 8

34. medical degrees → 22 employees → 63 22 of the employees have medical degrees. 63

36. a.

1 of a whole and there are 6 parts 5 shaded, or 1 whole and 1 more part.

20. Each part is

b. a.

6 5 1

1 5

23 5 4

no medical degree → 41 employees → 63 41 of the employees do not have medical 63 degrees.

1 22. Each part is of a whole and there are 23 parts 5 shaded, or 4 wholes and 3 more parts.

number of employees who do not have medical degrees = 63 − 22 = 41

→4 38. planets with longer days number of planets in solar system → 8 4 of the planets in our solar system have longer 8 days than Earth has.

40. 5 of 12 inches is

3 5

5 of a foot. 12

42. 37 of 60 minutes is 24. 3 out of 8 equal parts are shaded:

3 8

26. 13 out of 16 equal parts are shaded:

13 16

37 of an hour. 60

44. number with white lettering → 9 number of teams in league → 20 9 of the teams in the league have white 20 lettering on their shirts.

103

Chapter 4: Fractions and Mixed Numbers 46. There are 50 states total. Consumer fireworks are legal in 46 states. a.

Consumer fireworks are legal in

46 of the states. 50

b. 50 − 46 = 4 Consumer fireworks are illegal in 4 states. c. 48. a.

Consumer fireworks are illegal in

4 of the states. 50

number of watercolors → 15 total number of pieces → 37 15 of the inventory are watercolor paintings. 37

number of oil paintings → 17 total number of pieces → 37 17 of the inventory are oil paintings. 37

37 − 15 − 17 = 5 There are 5 sculptures.

numbers of sculptures → 5 total number of pieces → 37 5 of the inventory are sculptures. 37

50.

52.

54.

56.

58.

−3 =1 −3

60.

−20 = −20 1

62.

0 =0 −8

104

ISM: Prealgebra

64.

−14 =1 −14

66.

−7 is undefined 0

68.

5 =1 5

70. 1

Chapter 4: Fractions and Mixed Numbers

14 88. 14 196 14 56 56 0 196 = 14 14

13 17 ⋅1 + 13 17 + 13 30 = = = 17 17 17 17

1 90. 143 149 143 6

5 9 ⋅ 2 + 5 18 + 5 23 72. 2 = = = 9 9 9 9

149 6 =1 143 143

3 8 ⋅ 7 + 3 56 + 3 59 74. 7 = = = 8 8 8 8

7 92. 123 901 861 40

76. 12

2 5 ⋅12 + 2 60 + 2 62 = = = 5 5 5 5

78. 10

14 27 ⋅10 + 14 270 + 14 284 = = = 27 27 27 27

2 7 ⋅114 + 2 798 + 2 800 80. 114 = = = 7 7 7 7 1 82. 7 13 7 6 13 6 =1 7 7 7 84. 9 64 63 1 64 1 =7 9 9 5 86. 12 65 60 5

901 40 =7 123 123

94. 43 = 4 ⋅ 4 ⋅ 4 = 64 96. 34 = 3 ⋅ 3 ⋅ 3 ⋅ 3 = 81 98. − 100.

21 −21 21 = = −4 4 4

45 −45 45 = =− −57 57 57

102. answers may vary 104. 11 11 is close to 1. 5 12 12 rounded to the nearest whole number is 6.

106. 11 is close to 12, so

108. 5 + 4 + 9 + 8 = 26 9 of the 26 national parks in these states are in 9 California: 26

65 5 =5 12 12

105

Chapter 4: Fractions and Mixed Numbers

ISM: Prealgebra

Section 4.2 Practice Exercises 1. a.

64 4 ⋅16 4 ⋅16 16 8. − =− =− =− 20 4⋅5 4 ⋅5 5

30 = 2 ⋅15 ↓ ↓ 2⋅3 ⋅ 5

10.

7 is in simplest form. 9

7 21 7 both simplify to , they are Since and 9 27 9 equivalent.

11. Not equivalent, since the cross products are not equal: 13 ⋅ 5 = 65 and 4 ⋅ 18 = 72 12.

2 ⋅ 2 ⋅ 3 ⋅ 5 = 22 ⋅ 3 ⋅ 5 11 3. 3 33

7 mountains over 8000 meters in Nepal 14 mountains over 8000 meters 7 = 14 7 ⋅1 = 7⋅2 1

3 99

3 297

30 15 ⋅ 2 15 2 2 2 = = ⋅ = 1⋅ = 45 15 ⋅ 3 15 3 3 3

39 x 3 ⋅13 ⋅ x 3 13 x 13 x 13 x = = ⋅ = 1⋅ = 51 3 ⋅17 3 17 17 17

already in simplest form. 49 7⋅7 7 7 7 7 = = ⋅ = 1⋅ = 112 7 ⋅16 7 16 16 16

1 2

1 of the mountains over 8000 meters tall are 2 located in Nepal.

Calculator Explorations

9 3⋅3 =− 50 2⋅5⋅5

Since 9 and 50 have no common factors, −

7 ⋅1 7 ⋅2

The prime factorization of 297 is 33 ⋅11.

106

21 3 ⋅ 7 3 ⋅ 7 7 = = = 27 3 ⋅ 9 3 ⋅ 9 9

2. 60 = 2 ⋅ 30 ↓ ↓ 2 ⋅ 2 ⋅15 ↓ ↓ ↓

72 = 2 ⋅ 36 ↓ ↓ 2 ⋅ 2 ⋅18 ↓ ↓ ↓ 2⋅ 2⋅ 2⋅9 ↓ ↓ ↓ ↓ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 = 23 ⋅ 32

6. −

7⋅a⋅a⋅a 7 ⋅a⋅ a ⋅ a a 9. = = = 2 7 ⋅8⋅ a ⋅ a 7 ⋅8⋅ a ⋅ a 8 56a

2 ⋅ 2 ⋅ 2 ⋅ 7 = 23 ⋅ 7

7a3

56 = 2 ⋅ 28 ↓ ↓ 2 ⋅ 2 ⋅14 ↓ ↓ ↓

9 is 50

128 4 = 224 7

231 7 = 396 12

340 20 = 459 27

ISM: Prealgebra

Chapter 4: Fractions and Mixed Numbers Exercise Set 4.2

999 37 = 1350 50

2. 12 = 2 ⋅ 6 ↓ ↓ 2 ⋅ 2 ⋅ 3 = 22 ⋅ 3

432 8 = 810 15

225 5 = 315 7

4. 75 = 3 ⋅ 25 ↓ ↓ 3 ⋅ 5 ⋅ 5 = 3 ⋅ 52

54 2 = 243 9

6. 64 =

455 35 = 689 53

8 ⋅ 8   2 ⋅ 4 ⋅ 2 ⋅ 4   ↓  2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 = 26

Vocabulary, Readiness & Video Check 4.2 1. The number 40 equals 2 ⋅ 2 ⋅ 2 ⋅ 5. Since each factor is prime, we call 2 ⋅ 2 ⋅ 2 ⋅ 5 the prime factorization of 40. 2. A natural number, other than 1, that is not prime is called a composite number. 3. A natural number that has exactly two different factors, 1 and itself, is called a prime number. 4. Since 11 and 48 have no common factors other 11 than 1, the fraction is in simplest form. 48 5. Fractions that represent the same portion of a whole are called equivalent fractions. 5 15 = , 5 ⋅ 36 and 12 ⋅ 15 are 12 36 called cross products.

6. In the statement

7. Check that every factor is a prime number and check that the product of the factors is the original number. 8. an equivalent form of 1 or a factor of 1 9. You can simplify the two fractions and then 3 6 1 compare them. and both simplify to , 9 18 3 so the original fractions are equivalent. 10.

10 5 is not in simplest form; 24 12

8. 128 = 2 ⋅ 64 ↓ ↓ 2 ⋅ 2 ⋅ 32 ↓ ↓ ↓ 2 ⋅ 2 ⋅ 2 ⋅16 ↓ ↓ ↓ ↓ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅8 ↓ ↓ ↓ ↓ ↓ 2⋅ 2⋅2⋅ 2⋅2⋅4 ↓ ↓ ↓ ↓ ↓ ↓ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 = 27

10. 130 = 2 ⋅ 65 ↓ ↓ 2 ⋅ 5 ⋅13 = 2 ⋅ 5 ⋅13 12. 93 = 3 ⋅ 31 14. 836 = 2 ⋅ 418 ↓ ↓ 2 ⋅ 2 ⋅ 209 ↓ ↓ ↓ 2 ⋅ 2 ⋅11 ⋅19 = 22 ⋅11 ⋅19 16. 504 = 2 ⋅ 252 ↓ ↓ 2 ⋅ 2 ⋅126 ↓ ↓ ↓ 2 ⋅ 2 ⋅ 2 ⋅ 63 ↓ ↓ ↓ ↓ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 21 ↓ ↓ ↓ ↓ ↓

107

Chapter 4: Fractions and Mixed Numbers

ISM: Prealgebra

18.

5 5 ⋅1 1 = = 30 5 ⋅ 6 6

48.

20.

9y 3⋅3⋅ y 3⋅ y 3y = = = 48 3 ⋅16 16 16

50.

144 y 18 ⋅ 8 ⋅ y 8 = = 162 y 18 ⋅ 9 ⋅ y 9

270 x 4 y 3 z 3 3 3 3

15 x y z

18 ⋅15 ⋅ x ⋅ x ⋅ x ⋅ x ⋅ y ⋅ y ⋅ y ⋅ z ⋅ z ⋅ z 1 ⋅15 ⋅ x ⋅ x ⋅ x ⋅ y ⋅ y ⋅ y ⋅ z ⋅ z ⋅ z 18 ⋅ x = 1 = 18 x =

22.

22 2 ⋅11 11 = = 34 2 ⋅17 17

24.

70 7 ⋅10 7 = = 80 8 ⋅10 8

52. Equivalent, since the cross products are equal: 3 ⋅ 10 = 30 and 6 ⋅ 5 = 30

26.

25 z 5 ⋅ 5 ⋅ z 5 = = 55 z 5 ⋅11 ⋅ z 11

54. Not equivalent, since the cross products are not equal: 2 ⋅ 11 = 22 and 5 ⋅ 4 = 20

28. −

56. Equivalent, since the cross products are equal: 4 ⋅ 15 = 60 and 10 ⋅ 6 = 60

21 7⋅3 3 =− =− 49 7⋅7 7

5⋅9⋅b 9 9 = = = 30. 2 5 ⋅16 ⋅ b ⋅ b 16 ⋅ b 16b 80b 45b

32.

32 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 = 63 3⋅3⋅ 7 32 is Since 32 and 63 have no common factors, 63 already in simplest form.

34.

28 4⋅7 7 =− =− 60 4 ⋅15 15

64.

60a 2 b 36ab

66.

5 ⋅12 ⋅ a ⋅ a ⋅ b 5⋅a 5a = = 3 ⋅12 ⋅ a ⋅ b ⋅ b ⋅ b 3 ⋅ b ⋅ b 3b 2

28abc 7 ⋅ 4 ⋅ a ⋅ b ⋅ c 7 ⋅ b 7b = = = 60ac 15 ⋅ 4 ⋅ a ⋅ c 15 15

46. −

200 caps 200 ⋅1 1 = = 2000 caps 200 ⋅ 10 10 1 of the total caps. 10

20 centimeters 1 ⋅ 20 1 = = 100 centimeters 5 ⋅ 20 5 1 20 centimeters is of a meter. 5

68. a.

15 3⋅5 5 = = 423 3 ⋅141 141 5 of the National Park Service areas can 141 be found in New Mexico.

b. 423 − 15 = 408 408 National Park Service areas are found outside New Mexico.

65 5 ⋅13 5 44. = = 234 18 ⋅13 18

108

62. Not equivalent, since the cross products are not equal: 6 ⋅ 35 = 210 and 14 ⋅ 21 = 294

200 caps represents

60 30 ⋅ 2 2 40. = = 150 30 ⋅ 5 5

42.

60. Not equivalent, since the cross products are not equal: 16 ⋅ 16 = 256 and 20 ⋅ 9 = 180

36 y 6⋅6⋅ y 6 6 = = = 42 yz 6 ⋅ 7 ⋅ y ⋅ z 7 ⋅ z 7 z

36. −

38.

58. Equivalent, since the cross products are equal: 2 ⋅ 28 = 56 and 8 ⋅ 7 = 56

78 6 ⋅13 13 13 =− =− =− 90 x 6 ⋅15 ⋅ x 15 ⋅ x 15 x

ISM: Prealgebra

70.

408 3 ⋅136 136 = = 423 3 ⋅141 141 136 of the National Park Service areas are 141 found outside New Mexico.

10 5 ⋅ 2 2 = = 35 5 ⋅ 7 7 2 of the students made an A on the first test. 7

72. a.

74.

Chapter 4: Fractions and Mixed Numbers

34,200 − 15,200 = 19,000 $19,000 was not covered by the trade-in.

$19, 000 3800 ⋅ 5 5 = = $34, 200 3800 ⋅ 9 9 5 of the purchase price was not covered by 9 the trade-in.

4300 employees 100 ⋅ 43 43 = = 20, 000 employees 100 ⋅ 200 200 43 of the employees work at the Hallmark 200 headquarters in Kansas City, Missouri.

76.

131, 625 = 34 ⋅ 53 ⋅13 90. answers may vary 92. yes; answers may vary 94. The piece representing Nursing is labeled

8 . 100

8 4⋅2 2 = = 100 4 ⋅ 25 25 2 of entering college freshmen plan to major 25 in Nursing.

96. answers may vary

y 3 (−5)3 −125 = = = −25 5 5 5

98. The piece representing Memorials is labeled 76 . 1000 76 4 ⋅19 19 = = 1000 4 ⋅ 250 250 19 of National Park Service areas are 250 Memorials.

78. −5a = −5(−4) = 20 80. answers may vary 82.

88. 131, 625 = 3 ⋅ 43,875 ↓ ↓ 3 ⋅ 3 ⋅14, 625 ↓ ↓ ↓ 3 ⋅ 3 ⋅ 3 ⋅ 4875 ↓ ↓ ↓ ↓ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅1625 ↓ ↓ ↓ ↓ ↓ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 5 ⋅ 325 ↓ ↓ ↓ ↓ ↓ ↓ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 5 ⋅ 5 ⋅ 65 ↓ ↓ ↓ ↓ ↓ ↓ ↓ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 13

9506 7 ⋅1358 7 = = 12, 222 9 ⋅1358 9

84. 37 + 7 = 44 blood donors have an O blood type. 44 donors 4 ⋅11 11 = = 100 donors 4 ⋅ 25 25 11 of blood donors have an O blood type. 25 86. 9 + 1 = 10 blood donors have a B blood type. 10 donors 1 ⋅10 1 = = 100 donors 10 ⋅10 10 1 of blood donors have a B blood type. 10

100. answers may vary 102. 1235, 2235, 85, 105, 900, and 1470 are divisible by 5 because each number ends with a 0 or 5. 8691, 786, 2235, 105, 222, 900, and 1470 are divisible by 3 because the sum of each number’s digits is divisible by 3. 2235, 105, 900, and 1470 are divisible by both 3 and 5. 104. 15; answers may vary

109

Chapter 4: Fractions and Mixed Numbers Section 4.3 Practice Exercises

13. −

3 5 3 ⋅ 5 15 ⋅ = = 7 11 7 ⋅11 77

1 1 1 ⋅1 1 ⋅ = = 3 9 3 ⋅ 9 27

6 7 6⋅7 2 ⋅3⋅ 7 3 3 ⋅ = = = = 77 8 77 ⋅ 8 7 ⋅11 ⋅ 2 ⋅ 4 11 ⋅ 4 44

4 3 4⋅3 1⋅ 4 ⋅ 3 1 1 ⋅ = = = = 27 8 27 ⋅ 8 3 ⋅ 9 ⋅ 4 ⋅ 2 9 ⋅ 2 18

1  11  1⋅11 11 ⋅ −  = − =− 2  28  2 ⋅ 28 56

 4  33  4 ⋅ 33 4 ⋅ 3 ⋅11 3 = = 6.  −  −  =  11  16  11⋅16 11 ⋅ 4 ⋅ 4 4 7.

1 3 25 1 ⋅ 3 ⋅ 25 ⋅ ⋅ = 6 10 16 6 ⋅10 ⋅16 1⋅ 3 ⋅ 5 ⋅ 5 = 2 ⋅ 3 ⋅ 2 ⋅ 5 ⋅16 1⋅ 5 = 2 ⋅ 2 ⋅16 5 = 64 2 3y 2 ⋅3⋅ y y ⋅ = = =y 3 2 3 ⋅ 2 ⋅1 1 a3 b

⋅

b a

a3 ⋅ b 2

b ⋅a

a ⋅ a ⋅ a ⋅b a = b ⋅b ⋅a ⋅a b

10. a.

3 3 3 3 ⋅ 3 ⋅ 3 27 3 =   = ⋅ ⋅ = 4 4 4 4 ⋅ 4 ⋅ 4 64 4 2

 4  4   4  4 ⋅ 4 16 =  −  =  − ⋅ −  =  5  5   5  5 ⋅ 5 25

11.

8 2 8 9 8 ⋅ 9 2 ⋅ 4 ⋅ 9 4 ⋅ 9 36 ÷ = ⋅ = = = = 7 9 7 2 7⋅2 7⋅2 7 7

12.

4 1 4 2 4⋅2 8 ÷ = ⋅ = = 9 2 9 1 9 ⋅1 9

110

ISM: Prealgebra

14.

10 2 10 9 ÷ =− ⋅ 4 9 4 2 10 ⋅ 9 =− 4⋅2 2⋅5⋅9 =− 4⋅2 5⋅9 =− 4 45 =− 4

3y 3 y 5 y3 ÷ 5 y3 = ÷ 4 4 1 3y 1 = ⋅ 4 5 y3 3 y ⋅1 = 4 ⋅ 5 y3 3 ⋅ y ⋅1 = 4⋅5⋅ y ⋅ y ⋅ y 3 ⋅1 = 4⋅5⋅ y ⋅ y 3 = 20 y 2

 2 9  7  2⋅9  7 15.  − ⋅  ÷ =  − ÷  3 14  15  3 ⋅14  15  2 ⋅3⋅3  7 = − ÷  3 ⋅ 7 ⋅ 2  15  3 7 = − ÷  7  15  3  15 = − ⋅  7 7 3 ⋅15 =− 7⋅7 45 =− 49

16. a.

3 9 3⋅9 27 xy = − ⋅ = − =− 4 2 4⋅2 8

ISM: Prealgebra

Chapter 4: Fractions and Mixed Numbers 6. The word “of” indicates multiplication.

3 9 x÷ y = − ÷ 4 2 3 2 =− ⋅ 4 9 3⋅ 2 =− 4⋅9 1⋅ 3 ⋅ 2 =− 2 ⋅ 2 ⋅3⋅3 1 =− 2⋅3 1 =− 6

7. We have a negative fraction times a positive fraction, and a negative number times a positive number is a negative number. 8. Yes; the negative sign is inside the parentheses and the exponent applies to everything in the parentheses. 9. When writing the reciprocal of a fraction, the denominator becomes the numerator, and the numerator becomes the denominator.

9 2x = − 17. 4 9  9 2 −  0 − 4  8 2  9 9 ⋅ −  0 − 1  8 4 2⋅9 9 − 0− 1⋅ 8 4 2⋅9 9 − 0− 1⋅ 2 ⋅ 4 4 9 9 − =− True 4 4 9 Yes, − is a solution of the equation. 8 18.

10. There are only prime numbers in the denominator, and neither 4 nor 9 has a factor that includes these prime numbers. 11. radius =

Exercise Set 4.3 2.

1 1 90 1 ⋅ 90 1 ⋅ 5 ⋅18 ⋅ 90 = ⋅ = = = 18 5 5 1 5 ⋅1 5 ⋅1 There are 18 roller coasters in Cedar Point

Vocabulary, Readiness & Video Check 4.3 1. To multiply two fractions, we write

a c a ⋅c . ⋅ = b d b⋅d

2. Two numbers are reciprocals of each other if their product is 1. 3. The expression

23 2 ⋅ 2 ⋅ 2 = while 7 7

2 2 2 2   = ⋅ ⋅ . 7 7 7 7

4. Every number has a reciprocal except 0. 5. To divide two fractions, we write

a c a⋅d . ÷ = b d b⋅c

1 ⋅ diameter 2

5 7 5 ⋅ 7 35 ⋅ = = 9 4 9 ⋅ 4 36

4 1 4 ⋅1 ⋅− =− 15 20 15 ⋅ 20 4 ⋅1 =− 15 ⋅ 4 ⋅ 5 1 =− 15 ⋅ 5 1 =− 75

6. −

3 11 3 ⋅11 3 ⋅11 ⋅1 1 ⋅− = = = 11 12 11 ⋅12 11 ⋅ 3 ⋅ 4 4

5 64 y 5 ⋅ 64 y 5 ⋅ 32 ⋅ 2 ⋅ y y ⋅ = = = 32 100 32 ⋅100 32 ⋅ 2 ⋅ 5 ⋅10 10

2 2 6 y3 10. − ⋅ 6 y 3 = − ⋅ 3 3 1 2 ⋅ 6 y3 =− 3 ⋅1 2 ⋅ 3 ⋅ 2 ⋅ y3 =− 3 ⋅1 2 ⋅ 2 ⋅ y3 =− 1 3 = −4 y

111

Chapter 4: Fractions and Mixed Numbers

12.

14.

a b

⋅

b a

a ⋅b 3

26.

b ⋅a 1⋅ a ⋅ b = b ⋅b ⋅b ⋅ a ⋅ a ⋅ a 1 = a ⋅ a ⋅b ⋅b 1 = 2 2 a b

13 x 5 13x ⋅ 5 ⋅ =− 20 6 x 20 ⋅ 6 x 13 ⋅ x ⋅ 5 =− 4⋅5⋅6⋅ x 13 =− 4⋅6 13 =− 24

30.

27 10 16 27 ⋅10 ⋅16 ⋅ ⋅ = 32 13 30 32 ⋅13 ⋅ 30 3 ⋅ 3 ⋅ 3 ⋅10 ⋅16 = 16 ⋅ 2 ⋅13 ⋅10 ⋅ 3 3⋅3 = 2 ⋅13 9 = 26

32.

8  8   8  8 ⋅ 8 64 = 20.   =     = 9  9   9  9 ⋅ 9 81 4

 1  1  1  1  1  22.  −  =  −   −   −   −   2  2  2  2  2  1 ⋅1 ⋅1 ⋅1 = 2⋅2⋅2⋅2 1 = 16 3

 3  1  3  3   3  1 24.  −  ⋅ =  −  −   −  ⋅  4  3  4  4   4  3 3 ⋅ 3 ⋅ 3 ⋅1 =− 4⋅ 4⋅ 4⋅3 3 ⋅ 3 ⋅1 =− 4⋅4⋅4 9 =− 64 112

5 3 5 4 5⋅ 4 5⋅4 5 5 ÷ = ⋅ = = = = 8 4 8 3 8⋅3 4⋅ 2⋅3 2⋅3 6

28. −

11 ⋅0 = 0 12

16. −

18.

ISM: Prealgebra

4 8 4 3 ÷− = − ⋅− 15 3 15 8 4⋅3 = 15 ⋅ 8 4 ⋅ 3 ⋅1 = 5⋅3⋅ 4 ⋅ 2 1 = 5⋅2 1 = 10

10 4 10 5 x ÷− = ⋅− 11 5 x 11 4 10 ⋅ 5 x =− 11 ⋅ 4 2⋅5⋅5⋅ x =− 11 ⋅ 2 ⋅ 2 5⋅5⋅ x =− 11 ⋅ 2 25 x =− 22 9 z 2 9 z 9 9 z ⋅ 9 81z ÷ = ⋅ = = 20 9 20 2 20 ⋅ 2 40

5 5 10 34. − ÷ 10 = − ÷ 6 6 1 5 1 =− ⋅ 6 10 5 ⋅1 =− 6 ⋅10 5 ⋅1 =− 6⋅5⋅2 1 =− 6⋅2 1 =− 12

36.

3 y2

9 y3

y3 3 ⋅ y3 3 ⋅ y ⋅ y ⋅ y y = = = y2 9 y2 ⋅ 9 y ⋅ y ⋅ 3 ⋅ 3 3

⋅

38.

8 5 8 ⋅ 5 40 ⋅ = = 11 9 11 ⋅ 9 99

40.

2 5y 2 11 2 ⋅11 22 ÷ = ⋅ = = 5 y 11 5 y 5 y 5 y ⋅ 5 y 25 y 2

ISM: Prealgebra

42.

44.

Chapter 4: Fractions and Mixed Numbers

12 y 4 y 12 y 7 ÷ = ⋅ 21 7 21 4 y 12 y ⋅ 7 = 21 ⋅ 4 y 4 ⋅ 3 ⋅ y ⋅ 7 ⋅1 = 7 ⋅ 3 ⋅ 4 ⋅ y ⋅1 =1

54.

24 5 24 ⋅ 5 8 ⋅ 3 ⋅ 5 ⋅1 1 ⋅− = − =− =− 45 8 45 ⋅ 8 5⋅3⋅3⋅8 3 5

 1  1  1  1  1  1  46.  −  =  −   −   −   −   −   2  2  2  2  2  2  1 ⋅1 ⋅1 ⋅1 ⋅1 =− 2⋅2⋅2⋅2⋅2 1 =− 32

48.

56. 16a 2 ⋅ −

b a3 b ⋅ a3 b⋅a⋅a⋅a a a ⋅ = = = = 2 3 2 3 a ⋅ a ⋅ b ⋅ b ⋅ b b ⋅ b b2 a b a ⋅b

50. −100 ÷

52. −7 x 2 ÷

1 100 1 =− ÷ 2 1 2 100 2 =− ⋅ 1 1 100 ⋅ 2 =− 1 ⋅1 = −200

1  5 1  1  5 12  ⋅ ÷  = ⋅ ⋅  2  6 12  2  6 1  1 ⋅ 5 ⋅12 = 2 ⋅ 6 ⋅1 1⋅ 5 ⋅ 6 ⋅ 2 = 2 ⋅ 6 ⋅1 1⋅ 5 = 1 =5

31 16a 2 31 = ⋅− 24a 1 24a 16a 2 ⋅ 31 =− 1 ⋅ 24a 2 ⋅ 8 ⋅ a ⋅ a ⋅ 31 =− 1⋅ 8 ⋅ 3 ⋅ a 2 ⋅ a ⋅ 31 =− 1⋅ 3 62a =− 3

1 6 1⋅ 6 6 58. − ⋅ − = = 5 7 5 ⋅ 7 35

60.

9 16 9 15 9 ⋅15 135 ÷ = ⋅ = = 2 15 2 16 2 ⋅16 32

62.

17 y 2 13 y 17 y 2 18 x ÷ = ⋅ 24 x 18 x 24 x 13 y

14 x 7 x 2 14 x =− ÷ 3 1 3 7 x2 3 =− ⋅ 1 14 x 7 x2 ⋅ 3 =− 1 ⋅14 x 7⋅ x ⋅ x ⋅3 =− 1⋅ 2 ⋅ 7 ⋅ x x ⋅3 =− 1⋅ 2 3x =− 2

17 y 2 ⋅18 x 24 x ⋅13 y 17 ⋅ y ⋅ y ⋅ 6 ⋅ 3 ⋅ x = 6 ⋅ 4 ⋅ x ⋅13 ⋅ y 17 ⋅ y ⋅ 3 = 4 ⋅13 51 y = 52 =

113

Chapter 4: Fractions and Mixed Numbers

2  5  33 2  5  64.  33 ÷  ⋅ =  ÷  ⋅ 11   9  1 11  9  33 11  5 =  ⋅ ⋅  1 2 9 33 ⋅11 5 = ⋅ 1⋅ 2 9 33 ⋅11 ⋅ 5 = 2⋅9 3 ⋅11 ⋅11 ⋅ 5 = 2 ⋅3⋅3 11 ⋅11 ⋅ 5 = 2⋅3 605 = 6

66. 15c3 ÷

72.

3c 2 15c3 3c 2 = ÷ 5 1 5 15c3 5 = ⋅ 1 3c 2 =

15c ⋅ 5

76. a.

xy =

78.

8 1 8 ⋅1 2 ⋅ 4 ⋅1 2 ⋅1 2 ⋅ = = = = 9 4 9⋅4 9⋅4 9 9

x÷ y =

xy =

8 1 8 4 8 ⋅ 4 32 ÷ = ⋅ = = 9 4 9 1 9 ⋅1 9

7 1 7 ⋅1 7 ⋅− = − =− 6 2 6⋅2 12

7 1 ÷− 6 2 7 2 = ⋅− 6 1 7⋅2 =− 6 ⋅1 7⋅2 =− 2 ⋅ 3 ⋅1 7 =− 3

x÷ y =

ac b3 ac ⋅ b3 a ⋅ c ⋅ b ⋅ b ⋅ b b ⋅ b b 2 ⋅ = = = = b a2c b ⋅ a2c b⋅a⋅a⋅c a a

 3 8  2  3⋅8  2 70.  ⋅  ÷ =  ÷  4 9  5  4⋅9  5  3⋅ 4 ⋅ 2  2 = ÷  4 ⋅3⋅3  5 2 2 = ÷ 3 5 2 5 = ⋅ 3 2 2⋅5 = 3⋅ 2 5 = 3

114

5 4 5  5  4⋅5  ÷ ⋅−  = ÷−  8  7 16  8  7 ⋅16  5  4⋅5  = ÷−  8  7⋅4⋅4  5  5  = ÷−  8  28  5  28  = ⋅ −  8  5  5 ⋅ 28 =− 8⋅5 5⋅7⋅4 =− 4⋅ 2⋅5 7 =− 2

74. a.

1 ⋅ 3c 2 5⋅3⋅ c ⋅ c ⋅ c ⋅5 = 1⋅ 3 ⋅ c ⋅ c 5⋅5⋅c = 1 = 25c

68.

ISM: Prealgebra

2 6 y= 3 11 2 9 6 ⋅ 0 3 11 11 2⋅9 6 0 3 ⋅11 11 2 ⋅ 3⋅ 3 6 0 3⋅11 11 2⋅3 6 0 11 11 6 6 = True 11 11 9 Yes, is a solution of the equation. 11

ISM: Prealgebra

80.

82.

84.

86.

Chapter 4: Fractions and Mixed Numbers

1 3 3 1 5⋅ 0 5 3 5 3 1 ⋅ 0 1 5 3 5⋅3 1 0 1⋅ 5 3 3 1 = False 1 3 3 No, is not a solution of the equation. 5 5x =

1 1 of 200 = ⋅ 200 5 5 1 200 = ⋅ 5 1 1 ⋅ 200 = 5⋅1 1 ⋅5⋅ 40 = 5 ⋅1 40 = 1 = 40

88.

90.

1 1 of 44 million = ⋅ 44, 000, 000 11 11 1 44, 000, 000 = ⋅ 11 1 1 ⋅11 ⋅ 4, 000, 000 = 11 ⋅1 = 4, 000, 000 The increase in the number of people who subscribed to Hulu was approximately 4 million. 3 3 of 8 = ⋅8 16 16 3 8 = ⋅ 16 1 3 ⋅8 = 16 ⋅1 3 ⋅8 = 8 ⋅ 2 ⋅1 3 = 2 ⋅1 3 = 2

The screw sinks

3 inches deep after eight turns. 2

92. d = 2 ⋅ r 7 = 2⋅ 20 2 7 = ⋅ 1 20 2⋅7 = 1 ⋅ 20 2⋅7 = 1 ⋅ 2 ⋅10 7 = 10

5 5 of 24 = ⋅ 24 8 8 5 24 = ⋅ 8 1 5 ⋅ 24 = 8⋅1 5 ⋅8⋅ 3 = 8 ⋅1 5⋅3 = 1 15 = 1 = 15

The diameter is

7 foot. 10

1 1 of 3000 = ⋅ 3000 5 5 1 3000 = ⋅ 5 1 1 ⋅ 3000 = 5 ⋅1 1 ⋅5 ⋅ 600 = 5 ⋅1 = 600 The diet can contain 600 calories from fat per day.

115

Chapter 4: Fractions and Mixed Numbers

94.

96.

3 3 of 355,000 = ⋅ 355, 000 50 50 3 355, 000 = ⋅ 50 1 3 ⋅ 355, 000 = 50 ⋅1 3 ⋅50 ⋅ 7100 = 50 ⋅1 3 ⋅ 7100 = 1 = 21,300 The family pays $21,300 to the real estate companies.

3 3 of 662 = ⋅ 662 331 331 3 662 = ⋅ 331 1 3 ⋅ 662 = 331 ⋅1 3 ⋅ 2 ⋅ 331 = 331 ⋅1 3⋅ 2 = 1 =6 In the first 44 seasons of Survivor, 6 contestants have appeared four times.

98. area = length ⋅ width =

The area is

100.

116

ISM: Prealgebra

102.

1 12, 000 1 ⋅12, 000 = ⋅ 100 1 100 1 ⋅120 ⋅100 = 100 ⋅1 = 120 The family drove 120 miles for medical.

104.

811 42 + 69 922

106.

882 − 773 109

108. answers may vary 2

4 1 =  ÷ 2 3  4  4  1 =    ÷  3  3  2  4⋅4  1 = ÷  3⋅3  2 16 1 = ÷ 9 2 16 2 = ⋅ 9 1 16 ⋅ 2 = 9 ⋅1 32 = 9

1 3 1⋅ 3 3 ⋅ = = 2 8 2 ⋅8 16

3 square mile. 16

3 3 of 12,000 = ⋅12, 000 25 25 3 12, 000 = ⋅ 25 1 3 ⋅12, 000 = 25⋅1 3 ⋅ 25⋅ 480 = 25 ⋅1 3 ⋅ 480 = 1 = 1440 The family drove 1440 miles for shopping.

 8 39 8  1  8 ⋅ 39 ⋅ 8  1 110.  ⋅ ⋅  ÷ =   ÷ 2  13 16 9  2  13 ⋅16 ⋅ 9  2  8 ⋅13 ⋅ 3 ⋅ 2 ⋅ 4  1 =  ÷ 2  13 ⋅ 8 ⋅ 2 ⋅ 3 ⋅ 3 

112.

13 13 of 3720 = ⋅ 3720 40 40 13 3720 = ⋅ 40 1 13 ⋅ 3720 = 40 ⋅1 13 ⋅ 40 ⋅ 93 = 40 ⋅1 13 ⋅ 93 = 1 = 1209 There were approximately 1209 two-year colleges in the U.S. in 2020.

ISM: Prealgebra

114.

Chapter 4: Fractions and Mixed Numbers

3 1 of 36 on the first bus and of 30 on the 3 4 second bus are wearing orange polo shirts. 3 1 3 36 1 30 ⋅ 36 + ⋅ 30 = ⋅ + ⋅ 4 3 4 1 3 1 3 ⋅ 36 1 ⋅ 30 = + 4 ⋅1 3⋅1 3 ⋅ 4 ⋅ 9 1 ⋅ 3⋅10 = + 4 ⋅1 3⋅1 3 ⋅ 9 10 = + 1 1 = 27 + 10 = 37 There are 37 students on the two buses wearing orange polo shirts.

Section 4.4 Practice Exercises 1.

6 2 6+2 8 + = = 13 13 13 13

20 6 7 20 + 6 + 7 33 + + = = or 3 11 11 11 11 11

11 6 11 − 6 5 − = = 12 12 12 12

7 2 7 − 2 5 1⋅ 5 1 − = = = = 15 15 15 15 3 ⋅ 5 3

13 11 13 − 11 2 1 ⋅ 2 1 − = = = = 4 4 4 4 2⋅2 2 1 The jogger ran mile farther on Monday. 2

2 7y 2 −7y − = 5 5 5

4 6 3 4 − 6 − 3 −5 5 − − = = or − 11 11 11 11 11 11 10 5 −10 + 5 −5 5 + = = or − 12 12 12 12 12

7 11 and is 8 16

16. 13. 30 ⋅ 1 = 30 30 ⋅ 2 = 60 30 ⋅ 3 = 90 30 ⋅ 4 = 120 30 ⋅ 5 = 150

The LCD of

8 4 −8 + 4 −4 4 + = = or − 17 17 17 17 17

9. x + y = −

11.

12. 16 is a multiple of 8, so the LCD of

5 1 5 +1 6 2⋅3 3 + = = = = 8x 8x 8x 8x 2 ⋅ 4x 4x

6. −

3 3 3 3 + + + 20 20 20 20 3+3+3+3 = 20 12 = 20 3⋅ 4 = 5⋅ 4 3 = 5 3 The perimeter is mile. 5

10. perimeter =

Not a multiple of 25. Not a multiple of 25. Not a multiple of 25. Not a multiple of 25. A multiple of 25. 23 1 and is 150. 25 30

14. 40 = 2 ⋅ 2 ⋅ 2 ⋅ 5 108 = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 LCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 5 = 1080 3 11 and is 1080. The LCD of − 40 108 15. 20 = 2 ⋅ 2 ⋅ 5 24 = 2 ⋅ 2 ⋅ 2 ⋅ 3 45 = 3 ⋅ 3 ⋅ 5 LCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 = 360 7 1 13 is 360. The LCD of , , and 20 24 45 16. y = y 11 = 11

The LCD of

7 6 and is 11y. y 11

117

Chapter 4: Fractions and Mixed Numbers

17.

11. Multiplying by 1 does not change the value of a fraction.

7 7 7 7 ⋅ 7 49 = ⋅ = = 8 8 7 8 ⋅ 7 56

Exercise Set 4.4

1 1 5 1⋅ 5 5 18. = ⋅ = = 4 4 5 4 ⋅ 5 20

19.

3 x 3 x 6 3 x ⋅ 6 18 x = ⋅ = = 7 7 6 7⋅6 42

20. 4 =

21.

ISM: Prealgebra

4 6 4 ⋅ 6 24 ⋅ = = 1 6 1⋅ 6 6

9 2 9 + 2 11 + = = 17 17 17 17

3 2 3 + 2 5 1⋅ 5 1 + = = = = 10 10 10 10 2 ⋅ 5 2

3 1 −3 + 1 −2 1⋅ 2 1 6. − + = = =− =− 8 8 8 8 4⋅2 4

9 9 9 9⋅9 81 = ⋅ = = 4 x 4 x 9 4 x ⋅ 9 36 x

8. −

Vocabulary, Readiness & Video Check 4.4 9 13 and are called like 11 11 3 1 fractions while and are called unlike 3 4 fractions.

1. The fractions

10.

3 2 3+ 2 5 1⋅ 5 1 + = = = = 10 y 10 y 10 y 10 y 2 ⋅ 5 ⋅ y 2 y

12. −

a c a+c a c a−c + = and − = . b b b b b b

3. As long as b is not 0,

5  7  −5 + (−7) −12 1 ⋅12 1 +−  = = =− =− 24  24  24 24 2 ⋅12 2

−a a a = =− . −b b b

7 z 3 z z −7 z + 3 z + 1z + + = 15 15 15 15 −3 z = 15 3⋅ z =− 3⋅5 z =− 5

4. The distance around a figure is called its perimeter.

14.

9 5 9−5 4 − = = 13 13 13 13

5. The smallest positive number divisible by all the denominators of a list of fractions is called the least common denominator (LCD).

16.

5 1 5 −1 4 2 ⋅ 2 2 − = = = = 6 6 6 6 3⋅ 2 3

18.

4 7 4 − 7 −3 3 − = = =− z z z z z

6. Fractions that represent the same portion of a whole are called equivalent fractions. 7. To add like fractions, we add the numerators and keep the same denominator. 8. The y-value is negative and follows the minus sign, so parentheses are used around the y-value to separate the two signs. 9. P =

5 7 5 7 + + + ; 2 meters 12 12 12 12

20. −

37  18  37 18 −−  = − + 45  45  45 45 −37 + 18 = 45 −19 = 45 19 =− 45

10. 45 is the smallest number that both denominators, 15 and 9, divide into evenly.

118

ISM: Prealgebra

22.

24.

Chapter 4: Fractions and Mixed Numbers

27 5 28 27 − 5 − 28 − − = 28 28 28 28 −6 = 28 3⋅ 2 =− 14 ⋅ 2 3 =− 14

42.

44.

18b 3 18b − 3 − = 5 5 5

26. − 28. −

15 85 −15 + 85 70 7 ⋅10 7 + = = = = 200 200 200 200 20 ⋅10 20

−15 − 15 15 15 − = 26 y 26 y 26 y −30 = 26 y 2 ⋅15 =− 2 ⋅13 ⋅ y 15 =− 13 y

30.

2x 7 2x + 7 + = 15 15 15

32.

3 15 3 − 15 −12 3⋅ 4 3 − = = =− =− 16 z 16 z 16 z 16 z 4⋅4⋅ z 4z

34.

1 15 2 1 − 15 + 2 −12 4⋅3 3 − + = = =− =− 8 8 8 8 8 4⋅2 2

36.

9y 2y 5y 4y 9y + 2y + 5y − 4y + + − = 8 8 8 8 8 12 y = 8 4 ⋅3⋅ y = 4⋅2 3y = 2

38. x − y =

7 9 7 − 9 −2 1⋅ 2 1 − = = =− =− 8 8 8 8 4⋅2 4

1 5 −1 + 5 4 2 ⋅ 2 2 40. x + y = − + = = = = 6 6 6 6 2⋅3 3

3 6 4 3 2 3 + 6 + 4 + 3 + 2 18 + + + + = = 13 13 13 13 13 13 13 18 feet. The perimeter is 13 1 1 1 1 1+1+1+1 4 2 ⋅ 2 2 + + + = = = = 6 6 6 6 6 6 2⋅3 3 2 The perimeter is centimeter. 3

46. To find the remaining distance, subtract the 10 16 miles already run from the total miles. 8 8 16 10 16 − 10 6 2 ⋅ 3 3 − = = = = 8 8 8 8 2⋅4 4 3 He must run mile more. 4 48. To find the fraction that had speed limits greater 21 than 70 mph, subtract the that have 70 mph 50 37 speed limits from the that have speed limits 50 70 mph or greater. 37 21 37 − 21 16 2 ⋅8 8 − = = = = 50 50 50 50 2 ⋅ 25 25 8 of the states had speed limits that were 25 greater than 70 mph. 30 of the world’s land area, 100 20 while Africa makes up of the world’s land 100 area. 30 20 30 + 20 50 1 ⋅ 50 1 + = = = = 100 100 100 100 2 ⋅ 50 2 1 of the world’s land area is within Asia and 2 Africa.

50. Asia makes up

30 of the world’s land area, 100 6 while Australia makes up of the world’s 100 land area.

52. Asia makes up

119

Chapter 4: Fractions and Mixed Numbers

30 6 30 − 6 24 6⋅4 6 − = = = = 100 100 100 100 25 ⋅ 4 25 6 greater than that of Asia’s land area is 25 Australia’s.

ISM: Prealgebra

78.

80.

54. Multiples of 20: 20 ⋅ 1 = 20, not a multiple of 12 20 ⋅ 2 = 40, not a multiple of 12 20 ⋅ 3 = 60, a multiple of 12 LCD: 60 56. Multiples of 90: 90 ⋅ 1 = 90, a multiple of 15 LCD: 90 58. 4 = 2 ⋅ 2 14 = 2 ⋅ 7 20 = 2 ⋅ 2 ⋅ 5 LCD = 2 ⋅ 2 ⋅ 5 ⋅ 7 = 140 60. y = y 70 = 2 ⋅ 5 ⋅ 7 LCD = 2 ⋅ 5 ⋅ 7 ⋅ y = 70y 62. 24 = 2 ⋅ 2 ⋅ 2 ⋅ 3 45 = 3 ⋅ 3 ⋅ 5 LCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 = 360 64. 11 = 11 33 = 3 ⋅ 11 121 = 11 ⋅ 11 LCD = 3 ⋅ 11 ⋅ 11 = 363 66.

5 5 4 5 ⋅ 4 20 = ⋅ = = 6 6 4 6 ⋅ 4 24

68.

3 3 20 3 ⋅ 20 60 = ⋅ = = 5 5 20 5 ⋅ 20 100

70.

1 1 10 1 ⋅10 10 = ⋅ = = 5 5 10 5 ⋅10 50

72.

19 z 19 z 6 19 z ⋅ 6 114 z = ⋅ = = 21 21 6 21 ⋅ 6 126

3 x 3 x 6 3 x ⋅ 6 18 x 74. = ⋅ = = 2 2 6 2⋅6 12

76. 120

60 is the largest fraction, so clothing has the 100 largest fraction bought online. 2 2 ⋅ 20 40 = = 5 5 ⋅ 20 100 40 . 100 Books and magazines, clothing, and shoes have 2 over of the goods bought online. 5

Find those goods whose fraction is over

82. 43 = 4 ⋅ 4 ⋅ 4 = 64 84. 34 = 3 ⋅ 3 ⋅ 3 ⋅ 3 = 81 86. 54 = 5 ⋅ 5 ⋅ 5 ⋅ 5 = 625 88. 42 ⋅ 5 = 4 ⋅ 4 ⋅ 5 = 80 90.

3 1 3 − 1 2 1⋅ 2 1 − = = = = 4 4 4 4 2⋅2 2

92. answers may vary 94.

96.

3 3 4 3 + 3 − 4 2 1⋅ 2 1 + − = = = = 8 8 8 8 8 4⋅2 4 1 He is mile from home. 4

108 108 19 108 ⋅19 2052 = ⋅ = = 215 y 215 y 19 215 y ⋅19 4085 y

98. answers may vary 100. False;

7 7 4 7 ⋅ 4 28 = ⋅ = = 12 12 4 12 ⋅ 4 48

Section 4.5 Practice Exercises 1. The LCD is 21. 2 8 2⋅3 8 6 8 14 2 ⋅ 7 2 + = + = + = = = 7 21 7 ⋅ 3 21 21 21 21 3 ⋅ 7 3 2. The LCD is 18. 5 y 2 y 5 y ⋅ 3 2 y ⋅ 2 15 y 4 y 19 y + = + = + = 6 9 6⋅3 9⋅2 18 18 18

7 7 6a 7 ⋅ 6a 42a = ⋅ = = 6 6 6a 6 ⋅ 6a 36a

ISM: Prealgebra

Chapter 4: Fractions and Mixed Numbers 10. The LCD is 30. 3 3 1 3 ⋅ 6 3 ⋅ 3 1⋅ 2 + + = + + 5 10 15 5 ⋅ 6 10 ⋅ 3 15 ⋅ 2 18 9 2 = + + 30 30 30 29 = 30 29 cubic yard of cement. The homeowners need 30

3. The LCD is 20. 1 9 1⋅ 4 9 − + =− + 5 20 5 ⋅ 4 20 4 9 =− + 20 20 5 = 20 1⋅ 5 = 4⋅5 1 = 4 4. The LCD is 70. 5 9 5 ⋅10 9 ⋅ 7 50 63 13 − = − = − =− 7 10 7 ⋅10 10 ⋅ 7 70 70 70 5. The LCD is 24. 5 1 1 5 ⋅ 3 1⋅ 8 1⋅ 2 − − = − − 8 3 12 8 ⋅ 3 3 ⋅ 8 12 ⋅ 2 15 8 2 = − − 24 24 24 5 = 24

11. The LCD is 12. 3 2 3⋅3 2 ⋅ 4 9 8 1 − = − = − = 4 3 4 ⋅ 3 3 ⋅ 4 12 12 12 1 foot. The difference in length is 12 Calculator Explorations

5 6. Recall that 5 = . The LCD is 4. 1 5 y 5 ⋅ 4 y 20 y 20 − y − = − = − = 1 4 1⋅ 4 4 4 4 4

7. The LCD is 40. Write each fraction as an equivalent fraction with a denominator of 40. 5 5 ⋅ 5 25 = = 8 8 ⋅ 5 40 11 11 ⋅ 2 22 = = 20 20 ⋅ 2 40 25 22 5 11 Since 25 > 22, > , so > . 40 40 8 20 8. The LCD is 20. Write each fraction as an equivalent fraction with a denominator of 20. 17 − 20 4 4⋅4 16 − =− =− 5 5⋅ 4 20 17 16 17 4 Since −17 < −16, − < − , so − <− . 20 20 20 5 9. The LCD is 99. 5 4 5 ⋅ 9 4 ⋅11 45 44 1 x− y = − = − = − = 11 9 11 ⋅ 9 9 ⋅11 99 99 99

1 2 37 + = 16 5 80

3 2 23 + = 20 25 100

4 7 95 + = 9 8 72

9 5 163 + = 11 12 132

10 12 394 + = 17 19 323

14 15 253 + = 31 21 217

Vocabulary, Readiness & Video Check 4.5 1. To add or subtract unlike fractions, we first write the fractions as equivalent fractions with a common denominator. The common denominator we use is called the least common denominator. 2. The LCD for

1 5 and is 24. 6 8

1 5 1 4 5 3 4 15 19 + = ⋅ + ⋅ = + = . 6 8 6 4 8 3 24 24 24

121

Chapter 4: Fractions and Mixed Numbers

1 5 1 4 5 3 4 15 11 − = ⋅ − ⋅ = − =− . 6 8 6 4 8 3 24 24 24

5. x − y is an expression while 3 x =

1 is an 5

equation. 6. Since −10 < −1, we know that −

10 1 < − . 13 13

7. They are unlike terms and so cannot be combined. 8. Once the fractions have the same denominator, we then just compare numerators. 9.

3 1 + ; 6 2 3

10.

34 4 cm and 2 cm 15 15

Exercise Set 4.5 2. The LCD is 12. 5 1 5 ⋅ 2 1 10 1 11 + = + = + = 6 12 6 ⋅ 2 12 12 12 12 4. The LCD is 12. 2 1 2 ⋅ 4 1⋅ 3 8 3 5 − = − = − = 3 4 3 ⋅ 4 4 ⋅ 3 12 12 12 6. The LCD is 9. 5 1 5 1⋅ 3 5 3 2 − + =− + =− + =− 9 3 9 3⋅3 9 9 9 8. The LCD is 15. 2 2 2 2⋅3 2 6 4 − = − = − =− 15 5 15 5 ⋅ 3 15 15 15 10. The LCD is 25. 2 y 3 y 2 y ⋅ 5 3 y 10 y 3 y 13 y + = + = + = 5 25 5 ⋅ 5 25 25 25 25

ISM: Prealgebra 12. The LCD is 20. y 5 y 5− = − 20 1 20 5 ⋅ 20 y = − 1 ⋅ 20 20 100 y = − 20 20 100 − y = 20 14. The LCD is 36. 7 5 7 ⋅ 3 5 ⋅ 2 21 10 11 − = − = − = 12 18 12 ⋅ 3 18 ⋅ 2 36 36 36 16. The LCD is 10. 7 10 7 −10 + = − + 10 1 10 10 ⋅10 7 =− + 1 ⋅10 10 100 7 =− + 10 10 93 =− 10 18. The LCD is 18. 7 x 2 x 7 x 2 x ⋅ 2 7 x 4 x 11x + = + = + = 18 9 18 9 ⋅ 2 18 18 18 20. The LCD is 12. 5 z 1 5 z ⋅ 2 1 10 z 1 10 z − 1 − = − = − = 6 12 6 ⋅ 2 12 12 12 12 22. The LCD is 5x. 2 3 2 ⋅ x 3 ⋅ 5 2 x 15 2 x + 15 + = + = + = 5 x 5 ⋅ x x ⋅ 5 5x 5x 5x 24. The LCD is 9. 5 1 5 1⋅ 3 5 3 8 − − =− − =− − =− 9 3 9 3⋅3 9 9 9 26. The LCD is 15. 4 2 4 ⋅ 3 2 12 2 10 2 ⋅ 5 2 − = − = − = = = 5 15 5 ⋅ 3 15 15 15 15 3 ⋅ 5 3 28. The LCD is 25. 2b 3 2b ⋅ 5 3 10b 3 10b − 3 − = − = − = 5 25 5 ⋅ 5 25 25 25 25 30. The LCD is 36. 5 7 5 ⋅ 2 7 ⋅ 3 10 21 11 − = − = − =− 18 12 18 ⋅ 2 12 ⋅ 3 36 36 36

122

ISM: Prealgebra

Chapter 4: Fractions and Mixed Numbers

32. The LCD is 40. 5 3 5⋅5 3⋅ 2 25 6 19 − = − = − = 8 20 8 ⋅ 5 20 ⋅ 2 40 40 40 34. The LCD is 130. 10 7 10 ⋅10 7 ⋅13 100 91 9 − = − = − = 13 10 13 ⋅10 10 ⋅13 130 130 130 36. The LCD is 18. 7 2 7 2 ⋅ 2 7 4 11 + = + = + = 18 9 18 9 ⋅ 2 18 18 18 38.

3 4 3− 4 1 − = =− 13 13 13 13

40. The LCD is 60. 1 1 2 1 ⋅ 20 1 ⋅15 2 ⋅12 − − + =− − + 3 4 5 3 ⋅ 20 4 ⋅15 5 ⋅12 20 15 24 =− − + 60 60 60 11 =− 60 42. The LCD is 26. 5 z 3 5 z ⋅ 2 3 10 z 3 10 z + 3 + = + = + = 13 26 13 ⋅ 2 26 26 26 26 44. The LCD is 58. 1 3 1 ⋅ 29 3 ⋅ 2 29 6 35 − − =− − =− − =− 2 29 2 ⋅ 29 29 ⋅ 2 58 58 58 46. The LCD is 16. z z 2z z ⋅ 4 z ⋅ 2 2z + + = + + 4 8 16 4 ⋅ 4 8 ⋅ 2 16 4z 2z 2z = + + 16 16 16 8z = 16 8⋅ z = 8⋅2 z = 2

52. The LCD is 60. 7 3 7⋅2 3⋅3 − = − 30 20 30 ⋅ 2 20 ⋅ 3 14 9 = − 60 60 5 = 60 1⋅ 5 = 12 ⋅ 5 1 = 12 54. The LCD is 12x. 1 5 1 ⋅ x 5 ⋅12 − = − 12 x 12 ⋅ x x ⋅12 60 x = − 12 x 12 x x − 60 = 12 x 56. The LCD is 20. 6 3 1 6 ⋅ 4 3 ⋅ 5 1 ⋅10 + − = + − 5 4 2 5 ⋅ 4 4 ⋅ 5 2 ⋅10 24 15 10 = + − 20 20 20 29 = 20 58. The LCD is 24. 11 7 11 ⋅ 2 7 22 7 15 3 ⋅ 5 5 − = − = − = = = 12 24 12 ⋅ 2 24 24 24 24 3 ⋅ 8 8 60. The LCD is 24x. 3 5 3 ⋅ 3x 5⋅ 2 + = + 8 12 x 8 ⋅ 3x 12 x ⋅ 2 9x 10 = + 24 x 24 x 9 x + 10 = 24 x

48. The LCD is 16. 9 3 9 3⋅ 2 9 6 3 − = − = − = 16 8 16 8 ⋅ 2 16 16 16 50. The LCD is 28. 3 y y 3 y ⋅ 7 y ⋅ 4 21 y 4 y 25 y + = + = + = 4 7 4⋅7 7⋅4 28 28 28

62. The LCD is 14. 5 3 1 5 3 ⋅ 2 1⋅ 7 − + − =− + − 14 7 2 14 7 ⋅ 2 2 ⋅ 7 5 6 7 =− + − 14 14 14 6 =− 14 3⋅ 2 =− 7⋅2 3 =− 7

123

Chapter 4: Fractions and Mixed Numbers 64. The LCD is 10. 9 x 1 x 9 x 1⋅ 5 x ⋅ 2 − + = − + 10 2 5 10 2 ⋅ 5 5 ⋅ 2 9x 5 2x = − + 10 10 10 11x − 5 = 10 66. The LCD is 99. Write each fraction as an equivalent fraction with a denominator of 99. 5 5 ⋅11 55 = = 9 9 ⋅11 99 6 6 ⋅ 9 54 = = 11 11 ⋅ 9 99 55 54 5 6 Since 55 > 54, > , so > . 99 99 9 11 68. The LCD is 48. Write each fraction as an equivalent fraction with a denominator of 48. 7 7⋅6 42 − =− =− 8 8⋅6 48 5 5⋅8 40 − =− =− 6 6 ⋅8 48 42 40 7 5 Since −42 < −40, − < − , so − < − . 48 48 8 6 70. The LCD is 117. Write each fraction as an equivalent fraction with a denominator of 117. 2 2 ⋅13 26 − =− =− 9 9 ⋅13 117 3 3⋅9 27 − =− =− 13 13 ⋅ 9 117 26 27 2 3 Since −26 > −27, − >− , so − > − . 117 117 9 13 72. The LCD is 12. 1 3 1⋅ 4 3 ⋅ 3 4 9 5 x− y = − = − = − =− 3 4 3 ⋅ 4 4 ⋅ 3 12 12 12 74. x ÷ y =

1 3 1 4 1⋅ 4 4 ÷ = ⋅ = = 3 4 3 3 3⋅3 9

ISM: Prealgebra

1 3 76. 2 x + y = 2   + 3 4 2 3 = + 3 4 2 ⋅ 4 3⋅3 = + 3⋅ 4 4⋅3 8 9 = + 12 12 17 = 12

78. The LCD is 8. 3 5 1 3 5 1⋅ 4 + + = + + 8 8 2 8 8 2⋅4 3 5 4 = + + 8 8 8 12 = 8 4⋅3 = 4⋅2 3 1 = or 1 2 2 3 1 The perimeter is miles or 1 miles. 2 2 80. The LCD is 21. 10 1 10 1 10 1 ⋅ 3 10 1 ⋅ 3 + + + = + + + 21 7 21 7 21 7 ⋅ 3 21 7 ⋅ 3 10 3 10 3 = + + + 21 21 21 21 26 5 = or 1 21 21 26 5 yards or 1 yards. The perimeter is 21 21 82. A number increased by −

 2 x +  − .  5 84. The difference of a number and x−

124

2 translates as 5

7 . 20

7 translates as 20

ISM: Prealgebra

Chapter 4: Fractions and Mixed Numbers

86. The LCD is 264. 1 5 1 ⋅ 66 5 66 5 61 − = − = − = 4 264 4 ⋅ 66 264 264 264 264 61 A killer bee will chase a person mile 264 farther than a domestic honeybee. 88.

96. The piece representing historical parks is labeled 8 and the piece representing memorials is 25 19 labeled . The LCD is 250. 250 8 19 8 ⋅10 19 + = + 25 250 25 ⋅10 250 80 19 = + 25 250 99 = 250 99 of the areas maintained by the National 250 Park Service are designated as historical parks or memorials.

3 1 1 3⋅ 2 1 1 − − = − − 4 8 8 4⋅2 8 8 6 1 1 = − − 8 8 8 4 = 8 1⋅ 4 = 2⋅4 1 = 2 1 The inner diameter is inch. 2

90. The LCD is 100. 67 1 67 1 ⋅ 20 67 20 47 − = − = − = 100 5 100 5 ⋅ 20 100 100 100 47 The eSIM (MFF2) card is mm thicker than 100 the eSIM (WLCSP) card.

100. 8 − 6 ⋅ 4 − 7 = 8 − 24 − 7 = −16 − 7 = −23 102. 50 ÷ (5 ⋅ 2) = 50 ÷ 10 = 5 104. a.

92. The LCD is 16. 11 5 11 11 5 2 11 + + = + ⋅ + 16 8 16 16 8 2 16 11 10 11 = + + 16 16 16 32 = 16 =2 The total width is 2 inches.

b. There seems to be an error. c.

94. The LCD is 25. 1 1 1 1 5 1 5 6 + = + ⋅ = + = 25 5 25 5 5 25 25 25

The Arctic and Indian Oceans account for the world’s water surface.

17 1000 17 983 = − = 1000 1000 1000 1000 983 of the areas maintained by the National 1000 Park Service are not national parkways or scenic trails.

98. 1 −

6 of 25

The LCD is 8. 2 5 3 5 3⋅ 2 5 6 1  − = − = − = −  not  8 4 8 4⋅2 8 8 8  4

106. The LCD is 600. 9 7 1 9 60 7 3 1 200 − − = ⋅ − ⋅ − ⋅ 10 200 3 10 60 200 3 3 200 540 21 200 = − − 600 600 600 519 200 = − 600 600 319 = 600

125

Chapter 4: Fractions and Mixed Numbers

ISM: Prealgebra 11. 315 = 3 ⋅105 ↓ ↓ 3 ⋅ 3 ⋅ 35 ↓ ↓ ↓ 3⋅3⋅5⋅ 7

108. The LCD is 1352. 19 968 19 ⋅ 52 968 − = − 26 1352 26 ⋅ 52 1352 988 968 = − 1352 1352 20 = 1352 5⋅4 = 338 ⋅ 4 5 = 338

315 = 32 ⋅ 5 ⋅ 7

12. 441 = 3 ⋅147 ↓ ↓ 3 ⋅ 3 ⋅ 49 ↓ ↓ ↓ 3⋅3⋅ 7 ⋅ 7

110. 1; answers may vary

441 = 32 ⋅ 7 2

Mid-Chapter Review 1. 3 out of 7 equal parts are shaded:

3 7

1 of a whole and there are 5 parts 4 5 1 shaded, or 1 whole and 1 more part: or 1 4 4

2. Each part is

3. number that get fewer than 8 → 73 total number of people → 85 73 of people get fewer than 8 hours of sleep 85 each night.

13.

2 2 ⋅1 1 = = 14 2 ⋅ 7 7

14.

24 6 ⋅ 4 6 = = 20 5 ⋅ 4 5

15. −

56 14 ⋅ 4 14 =− =− 60 15 ⋅ 4 15

16. −

72 8⋅9 9 =− =− 80 8 ⋅10 10

17.

54 x 27 ⋅ 2 ⋅ x 2 x = = 135 27 ⋅ 5 5

18.

90 30 ⋅ 3 3 = = 240 y 30 ⋅ 8 ⋅ y 8 y

19.

165 z 3 15 ⋅11 ⋅ z ⋅ z ⋅ z 11 ⋅ z ⋅ z 11z 2 = = = 210 z 15 ⋅14 ⋅ z 14 14

4. 5.

11 = −1 −11 17 = 17 1

0 7. =0 −3

7 is undefined 0

9. 65 = 5 ⋅ 13 10. 70 = 2 ⋅ 35 ↓ ↓ 2⋅ 5 ⋅ 7 70 = 2 ⋅ 5 ⋅ 7 126

20.

35 ⋅ 7 ⋅ a ⋅ b 35 ⋅ 11 ⋅ a ⋅ a ⋅b ⋅b ⋅b 385a b 7 = 11 ⋅ a ⋅ b ⋅ b 7 = 11ab 2 245ab

2 3

21. Not equivalent, since the cross products are not equal: 7 ⋅ 10 = 70, 8 ⋅ 9 = 72 22. Equivalent, since the cross products are equal: 10 ⋅ 18 = 180, 12 ⋅ 15 = 180

ISM: Prealgebra

Chapter 4: Fractions and Mixed Numbers

number not adjacent → 2 total number → 50

31.

1 3 1+ 3 4 + = = 5 5 5 5

2 1⋅ 2 1 of the states are not = = 50 25 ⋅ 2 25 adjacent to any other states.

32.

1 3 1 − 3 −2 2 − = = =− 5 5 5 5 5

b. 50 − 2 = 48; 48 states are adjacent to other states.

33.

1 3 1⋅ 3 3 ⋅ = = 5 5 5 ⋅ 5 25

48 2 ⋅ 24 24 of the states are adjacent = = 50 2 ⋅ 25 25 to other states.

34.

1 3 1 5 1⋅ 5 1 ÷ = ⋅ = = 5 5 5 3 5⋅3 3

35.

2 5 2 6 2⋅6 2⋅2⋅3 2⋅2 4 ÷ = ⋅ = = = = 3 6 3 5 3⋅5 3⋅5 5 5

36.

2a 5 2a ⋅ 5 2⋅a ⋅5 5 5 ⋅ = = = = 3 6a 3 ⋅ 6 a 3 ⋅ 2 ⋅ 3 ⋅ a 3 ⋅ 3 9

23. a.

24. a. digital 3-D screens in U.S./Canada → 14, 000 screens in U.S./Canada → 44, 000

14, 000 7 ⋅ 2000 7 == = of the screens in 44, 000 22 ⋅ 2000 22 the U.S./Canada are digital 3-D screens. b. 44,000 − 14,000 = 30,000 30,000 of the screens in the U.S./Canada are not digital 3-D. c.

30, 000 15 ⋅ 2000 15 = = of the screens in 44, 000 22 ⋅ 2000 22 the U.S./Canada are not digital 3-D screens.

25. 5 = 5 6=2⋅3 LCM = 2 ⋅ 3 ⋅ 5 = 30

37. The LCD is 6y. 2 5 2 2 5 − = ⋅ − 3y 6 y 3y 2 6 y 2⋅2 5 = − 3y ⋅ 2 6 y 4 5 = − 6y 6y 1 =− 6y

28.

7 7 4 7 ⋅ 4 28 = ⋅ = = 9 9 4 9 ⋅ 4 36

38. The LCD is 6. 2 x 5x 2 x 2 5x + = ⋅ + 3 6 3 2 6 2 x ⋅ 2 5x = + 3⋅ 2 6 4x 5x = + 6 6 9x = 6 3⋅3⋅ x = 3⋅ 2 3x = 2

29.

11 11 5 11 ⋅ 5 55 = ⋅ = = 15 15 5 15 ⋅ 5 75

1 7 1⋅ 7 1 39. − ⋅ − = = 7 18 7 ⋅18 18

30.

5 5 8 5 ⋅ 8 40 = ⋅ = = 6 6 8 6 ⋅ 8 48

4 3 4⋅3 4⋅3 4 4 40. − ⋅ − = = = = 9 7 9 ⋅ 7 3 ⋅ 3 ⋅ 7 3 ⋅ 7 21

26. 2 = 2 14 = 2 ⋅ 7 LCM = 2 ⋅ 7 = 14 27. 6 = 2 ⋅ 3 18 = 2 ⋅ 3 ⋅ 3 30 = 2 ⋅ 3 ⋅ 5 LCM = 2 ⋅ 3 ⋅ 3 ⋅ 5 = 90

127

Chapter 4: Fractions and Mixed Numbers

41. −

42. −

7z 7z 6z2 ÷ 6z2 = − ÷ 8 8 1 7z 1 =− ⋅ 8 6z2 7 z ⋅1 =− 8 ⋅ 6z2 7 ⋅ z ⋅1 =− 8⋅6⋅ z ⋅ z 7 =− 48 z

49. A number subtracted from −

50.

51.

9 9 5 9 1 9 ⋅1 9 ÷5 = − ÷ = − ⋅ = − =− 10 10 1 10 5 10 ⋅ 5 50

44. The LCD is 36. 5 1 5 ⋅ 3 1 ⋅ 4 15 4 11 − = − = − = 12 9 12 ⋅ 3 9 ⋅ 4 36 36 36 45. The LCD is 18. 2 1 1 2 ⋅ 2 1 1⋅ 6 + + = + + 9 18 3 9 ⋅ 2 18 3 ⋅ 6 4 1 6 = + + 18 18 18 11 = 18 46. The LCD is 50. 3 y y 6 3 y ⋅ 5 y ⋅10 6 ⋅ 2 + + = + + 10 5 25 10 ⋅ 5 5 ⋅10 25 ⋅ 2 15 y 10 y 12 = + + 50 50 50 25 y 12 = + 50 50 25 y + 12 = 50 2 2 2 of a number translates as ⋅ x or x. 3 3 3

48. The quotient of a number and −

 1 x ÷  − .  5

128

8 translates as 9

8 − − x. 9

43. The LCD is 40. 7 1 7 ⋅ 5 1⋅ 2 35 2 37 + = + = + = 8 20 8 ⋅ 5 20 ⋅ 2 40 40 40

47.

ISM: Prealgebra

1 translates as 5

6 6 increased by a number translates as + x. 11 11 2 2 1530 ⋅1530 = ⋅ 3 3 1 2 ⋅1530 = 3 ⋅1 2 ⋅ 3 ⋅ 510 = 3 ⋅1 2 ⋅ 510 = 1 = 1020 2 of 1530 is 1020. 3

3 18 3 = ÷ 4 1 4 18 4 = ⋅ 1 3 18 ⋅ 4 = 1⋅ 3 3⋅ 6 ⋅ 4 = 1⋅ 3 6⋅4 = 1 = 24 They can sell 24 lots.

52. 18 ÷

53. The LCD is 16. 7 1 1 7⋅2 1 1 − − = − − 8 16 16 8 ⋅ 2 16 16 14 1 1 = − − 16 16 16 12 = 16 4⋅3 = 4⋅4 3 = 4 3 The inner diameter is foot. 4

ISM: Prealgebra

Chapter 4: Fractions and Mixed Numbers

54. The LCD is 21. 17 5 17 5 ⋅ 3 17 15 2 − = − = − = 21 7 21 7 ⋅ 3 21 21 21

 1 1  5 1   1 ⋅ 5 1 ⋅ 2  5 1  + 6.  − +  +  =  −  +   2 5  8 8   2 ⋅ 5 5 ⋅ 2  8 8   5 2  5 1  =  − +  +   10 10   8 8   3  6  =  −    10   8  3⋅ 2⋅3 =− 2 ⋅5⋅8 3⋅3 =− 5 ⋅8 9 =− 40

1⋅3 + 1 1+1 2 6 = 2⋅3 6 3−2 3⋅3 − 2⋅4 4 3 4⋅3 3⋅4 3+1 = 6 6 9 − 8 12 12 4 = 6 1 12

3 3 3 2 7. − − xy = − − ⋅ 5 5 10 3 3 3⋅ 2 =− − 5 5⋅ 2⋅3 3 1 =− − 5 5 4 =− 5

2 of the 21 2022 FIA Formula One World Championship races.

Red Bull driver Sergio Perez won

Section 4.6 Practice Exercises 7y

7y 1 7y 5 7 y ⋅5 7 ⋅ y ⋅5 7y ÷ = ⋅ = = = 1. 10 = 1 10 5 10 1 10 ⋅1 2 ⋅ 5 ⋅1 2

4 1 ÷ 6 12 4 12 = ⋅ 6 1 4⋅2⋅6 = 6 ⋅1 8 = or 8 1 =

Vocabulary, Readiness & Video Check 4.6 1. A fraction whose numerator or denominator or both numerator and denominator contain fractions is called a complex fraction. 1 2 7 2. To simplify − + ⋅ , which operation do we 2 3 8 perform first? multiplication

3. The LCD is 12. 1 + 1 12 2 6 = 3−2 4 3 12

( 12 + 16 ) ( 34 − 23 )

1 2 7 3. To simplify − ÷ ⋅ , which operation do we 2 3 8 perform first? division

12 ⋅ 12 + 12 ⋅ 16

7 1 2 ⋅  −  , which operation do we 8 2 3 perform first? subtraction

12 ⋅ 34 − 12 ⋅ 23 6+2 = 9−8 8 = or 8 1

4. To simplify

4. The LCD is 20. 3 20 34 20 ⋅ 34 15 4 = = = x −1 x x x 4 − 20 20 5 − 1 20 ⋅ 5 − 20 ⋅1 5

(

( )

1 1  9 3 ÷ ⋅  +  , which operation 3 4  11 8  do we perform first? addition

5. To simplify

)

8 8 54 46 2 −2 = − =− 5.   − 2 = 27 27 27 27 3

 3 6. To simplify 9 −  −  , which operation do we  4 perform first? evaluate the exponential expression 7. distributive property

129

Chapter 4: Fractions and Mixed Numbers 8. They have the same denominator so they are like fractions.

ISM: Prealgebra

12.

9. Since x is squared and the replacement value is negative, we use parentheses to make sure the whole value of x is squared. Without parentheses, the exponent would not apply to the negative sign. Exercise Set 4.6 2.

5 5 15 5 12 5 ⋅12 1 ⋅ 5 ⋅ 6 ⋅ 2 2 6 = ÷ = ⋅ = = = 15 6 12 6 15 6 ⋅15 6⋅5⋅3 3 12

25 25 25 11 = ÷ 25 11 6 6

14.

25 6 = ⋅ 11 25 25 ⋅ 6 = 11 ⋅ 25 6 = 11

8. The LCD is 18. 7 + 2 18 ⋅ 7 + 2 6 3 6 3 = 3−8 18 ⋅ 32 − 89 2 9

( (

) )

18 ⋅ 32 − 18 ⋅ 89 21 + 12 = 27 − 16 33 = 11 =3

10. The LCD is 10y. 3 + 2 10 y ⋅ 3 + 2 10 10 = 2 10 y ⋅ 52y 5y

(

1 1 16. 32 −   = 9 − 4 2 9 1 = − 1 4 9⋅4 1 = − 1⋅ 4 4 36 1 = − 4 4 35 = 4  1 1  1 1   1 2 1  1 2 1  18.  −  +  =  ⋅ −  ⋅ +   5 10  5 10   5 2 10  5 2 10   2 1  2 1  =  −  +   10 10  10 10   1  3  =     10  10  1⋅ 3 = 10 ⋅10 3 = 100

18 ⋅ 76 + 18 ⋅ 23

7 1 1 7 4 1 7 ⋅ 4 ⋅1 1 ÷ ⋅ = ⋅ ⋅ = = 8 4 7 8 1 7 4 ⋅ 2 ⋅1 ⋅ 7 2 2

3y 11 = 3 y ÷ 1 = 3 y ⋅ 2 = 3 y ⋅ 2 = 6 y 1 11 2 11 1 11 ⋅1 11 2

1 1 1 1 1 ⋅1 + ⋅ = + 2 6 3 2 6⋅3 1 1 = + 2 18 1⋅ 9 1 = + 2 ⋅ 9 18 9 1 = + 18 18 10 = 18 2⋅5 = 2⋅9 5 = 9

)

3 + 10 y ⋅ 2 10 y ⋅ 10

4 3 y + 20 y = 4 23 y = 4 130

ISM: Prealgebra

Chapter 4: Fractions and Mixed Numbers

9 4  2 7 4 20.  − −  ÷ = − ÷ 3 9  3 3 9 3 4 =− ÷ 1 9 3 9 =− ⋅ 1 4 3⋅9 =− 1⋅ 4 27 =− 4

22.

2  1 2  5⋅ 2 1  ⋅  5 −  −1 = ⋅  −  −1 5  2 5  1⋅ 2 2  2  10 1  = ⋅  −  −1 5  2 2 2 9 = ⋅   −1 5 2 2⋅9 = −1 5⋅ 2 9 5 = − 5 5 4 = 5 2

3 4  3⋅3 4 ⋅ 2  26.  −  =  −  2 3  2 ⋅3 3⋅ 2  3 9 8 = −  6 6

1 =  6 1 1 1 =  ⋅ ⋅  6 6 6 1 = 216 3

3  6  =   +   6   20  3 2 1  3  =   +   2   10  1 9 = + 8 100 1 ⋅ 25 9⋅2 = + 8 ⋅ 25 100 ⋅ 2 25 18 = + 200 200 43 = 200

2   8   2⋅3 2  8  −  24.   ÷  2 −  =   ÷  3   9   1⋅ 3 3  9  2 8 6 2 =   ÷ −  9 3 3 2

8 4 =  ÷ 9 3 64 4 = ÷ 81 3 64 3 = ⋅ 81 4 16 ⋅ 4 ⋅ 3 = 27 ⋅ 3 ⋅ 4 16 = 27

1 1  2 3  1 1⋅ 2   2 ⋅ 3  28.  +  +  ⋅  =  +  +  6 3 5 4      6 3⋅ 2   5⋅ 4  3 2 1 2  6  =  +  +   6 6   20 

5  1 5 1 6 30. 2 z − x = 2   −  −  = + = = 2  6  3 3 3 3 2

32.

( 1)

y+x 5 + −3 = 5 z 6

( ) ()

30  52 + − 13   =  30 56 12 + (−10) 25 2 = 25 =

131

Chapter 4: Fractions and Mixed Numbers

 1 5 34. x 2 − z 2 =  −  −    3  6  1  1   5  5  =  −  −  −     3  3   6  6  1 25 = − 9 36 1 4 25 = ⋅ − 9 4 36 4 25 = − 36 36 21 =− 36 7 ⋅3 =− 12 ⋅ 3 7 =− 12   1   5  36. (1 − x)(1 − z ) = 1 −  −   1 −    3   6   1  5  = 1 +  1 −   3  6   3 1  6 5  =  +  −   3 3  6 6   4  1  =     3  6  2 ⋅ 2 ⋅1 = 3⋅ 2 ⋅3 2 = 9

38.

7 7 14 z 10 = ÷ 14 z 10 25 25

7 25 ⋅ 10 14 z 7 ⋅5⋅5 = 5⋅2⋅7⋅2⋅ z 5 = 2⋅2⋅ z 5 = 4z =

132

ISM: Prealgebra

 5 1 1 1  5 2 1 1 40.  ÷  +  ⋅  =  ⋅  +  ⋅   21 2   7 3   21 1   7 3  5 ⋅ 2 1 ⋅1 = + 21 ⋅1 7 ⋅ 3 10 1 = + 21 21 11 = 21 2

 3  3 9 3 9 3 ⋅ 2 9 6 15 = + = 42.  −  + = + = +  4  8 16 8 16 8 ⋅ 2 16 16 16 44.

( ) = 10 ⋅ 3 − 10 ⋅ 12 = 30 − 5 = 25 4 + 15 10 ( 4 + 1 ) 10 ⋅ 4 + 10 ⋅ 15 40 + 2 42 5

3 − 12

10 3 − 12

 1 3  1  1   3  3  46.  −  −   =  −   −  −      2 4  2  2   4  4  1 9 = − 4 16 1⋅ 4 9 = − 4 ⋅ 4 16 4 9 = − 16 16 5 =− 16 3  1   1 ⋅ 2 3  1 5  1 48.  +  − 1 =  +  −   10 20  5   10 ⋅ 2 20   5 5  3  1 5   2 =  +  −  20 20   5 5   5  4  =   −   20  5  5⋅ 4 =− 5⋅4⋅5 1 =− 5

ISM: Prealgebra

Chapter 4: Fractions and Mixed Numbers

5 2  5 2⋅3  50.  −  =  −  9 3  9 3⋅3  2 5 6 = −  9 9

60. 4 −

62. yes; answers may vary

 1 = −   9  1  1  =  −  −   9  9  1 = 81

52.

( (

64.

3+ 9 5 10

) )

14 + 10 16 + 15 24 = 31

( ) 3 2 + 8 8⋅(2 + 3 ) 8 =

+ 66. 6 9 = 2

1 2 1 2 1 2 1 7 54.  ⋅  −  ÷  =  ⋅  −  ⋅  2 7 2 7 2 7 2 2 1⋅ 2 1⋅ 7 = − 2⋅7 2⋅2 2 7 = − 14 4 2⋅2 7⋅7 = − 14 ⋅ 2 4 ⋅ 7 4 49 = − 28 28 45 =− 28 =

(

9 10 53 + 10

)

10(2) 9 10 ⋅ 53 + 10 ⋅ 10

10 ⋅ 2 6+9 = 20 15 = 20 5⋅3 = 5⋅ 4 3 = 4

7 +1 7 +1 20 10 2 10 2 = 4+3 4+3 20 5 4 5 4 7 + 20 ⋅ 1 20 ⋅ 10 2 = 20 ⋅ 54 + 20 ⋅ 34

1 − 4x

= =

56.

1 4 5 1 20 1 19 4 = ⋅ − = − = or 3 5 1 5 5 5 5 5 5

(

18 56 + 79

)

18(2) 18 ⋅ 56 + 18 ⋅ 79

18 ⋅ 2 15 + 14 = 36 29 = 36 68. no; answers may vary 70. False; the average cannot be less than the smallest number. 72. False, if the absolute value of the negative fraction is greater than the absolute value of the positive fraction, the sum is negative. 74. True

8 ⋅ 1 − 4x

76. No; answers may vary.

8 ⋅1 − 8 ⋅ 4x

78. Addition, multiplication, subtraction, division

8 ⋅ 2 + 8 ⋅ 83

80. Subtraction, multiplication, division, addition

8 − 2x 16 + 3 8 − 2x = 19 =

58. 2 +

2 2 3 2 6 2 8 2 = ⋅ + = + = or 2 3 1 3 3 3 3 3 3

133

Chapter 4: Fractions and Mixed Numbers

7 3 97 3 ⋅ 97 97 2 = ⋅ = = or 19 15 1 15 1 ⋅ 5 ⋅ 3 5 5 7 rounds to 6 and 3 ⋅ 6 = 18, so the Estimate: 6 15 answer is reasonable.

3  4 82. 4 x + y = 4 ⋅ +  −  4  7 4 3  4 = ⋅ +−  1 4  7 12 4 = − 4 7 12 ⋅ 7 4 ⋅ 4 = − 4⋅7 7⋅4 84 16 = − 28 28 68 = 28 4 ⋅17 = 4⋅7 17 = 7

84.

9 14

x+ y

5. 3 ⋅ 6

9 14

( − 74 ) 9 28 ( 14 ) = 28 ( 34 − 74 ) =

ISM: Prealgebra

3+ 4

28 ⋅ 34 − 28 ⋅ 74

18 21 − 16 18 = 5

Section 4.7 Practice Exercises 1. 2 11 5 11 5 ⋅11 11 2 2. 1 ⋅ = ⋅ = = or 1 3 15 3 15 3 ⋅ 5 ⋅ 3 9 9 5 5 18 5 ⋅18 5 ⋅ 6 ⋅ 3 15 ⋅18 = ⋅ = = = or 15 6 6 1 6 ⋅1 6 ⋅1 1

1 3 16 11 16 ⋅11 4 ⋅ 4 ⋅11 44 4 4. 3 ⋅ 2 = ⋅ = = = or 8 5 4 5 4 5⋅ 4 5⋅ 4 5 5 3 1 Estimate: 3 rounds to 3, 2 rounds to 3. 5 4 3 ⋅ 3 = 9, so the answer is reasonable.

134

8 4 8 19 ÷3 = ÷ 15 5 15 5 8 5 = ⋅ 15 19 8⋅5 = 15 ⋅19 8⋅5 = 5 ⋅ 3 ⋅19 8 = 57

2 3 23 31 ÷ 8. 3 ÷ 2 = 7 14 7 14 23 14 = ⋅ 7 31 23 ⋅ 7 ⋅ 2 = 7 ⋅ 31 46 15 = or 1 31 31

4 4 7 4 1 4 ⋅1 4 ÷7 = ÷ = ⋅ = = 9 9 1 9 7 9 ⋅ 7 63

10.

1 6 2 + 4 5

5 30 12 + 4 30 17 6 30 1 2 Estimate: 2 rounds to 2, 4 rounds to 4, and 6 5 2 + 4 = 6, so the answer is reasonable.

5 14 6 + 2 7 3

5 14 12 +2 14 3 3 17 = 5 +1 = 6 5 14 14 14 3

ISM: Prealgebra

11.

12.

13.

14.

6 3 7 1 + 2 5

30 3 35 7 + 2 35 37 2 2 17 = 17 + 1 = 18 35 35 35

7 9 5 − 16 18

14 18 5 − 16 18 9 1 16 = 16 18 2

7 15 3 − 4 5 9

25 − 10

15.

Chapter 4: Fractions and Mixed Numbers

2 9

1 4 5 − 19 12

7 15 9 −4 15 9

22 15 9 − 4 15 13 4 15

9 24 9 2 − 10 9 7 14 9 3 12 5 − 19 12 23

17. −9

3 7 ⋅9 + 3 66 =− =− 7 7 7

18. −5

10 11 ⋅ 5 + 10 65 =− =− 11 11 11

15 12 5 − 19 12 10 5 3 =3 12 6 The girth of the largest known American beech 5 tree is 3 feet larger than the girth of the largest 6 known sugar maple tree. 23

1 22 16. 44 ÷ 3 = 44 ÷ 7 7 44 7 = ⋅ 1 22 2 ⋅ 22 ⋅ 7 = 1 ⋅ 22 14 = or 14 1 Therefore, 14 dresses can be made from 44 yards.

19. −

37 5 = −4 8 8

20. −

46 1 = −9 5 5

4 8 37 32 5 9 5 46 45 1

3  3  11  18  21. 2 ⋅  −3  = ⋅  −  4  5 4  5  11 ⋅18 =− 4⋅5 11 ⋅ 2 ⋅ 9 =− 2⋅ 2⋅5 99 9 =− or − 9 10 10 2 1 30 5 22. −4 ÷ 1 = − ÷ 7 4 7 4 30 4 =− ⋅ 7 5 30 ⋅ 4 =− 7⋅5 5⋅6⋅4 =− 7⋅5 24 3 or − 3 =− 7 7

135

Chapter 4: Fractions and Mixed Numbers

23.

3 4 2 −6 3 12

9 12 8 − 6 12 1 6 12

3 2. For 5 , the 5 is called the whole number part 4 3 and is called the fraction part. 4

4 9 14 11 + 30 14 1 15 = 40 39 14 14

4. The mixed number 2

2 3 1 6 − 12 = −6 3 4 12

2 24. 9 7 11 + 30 14

2 11 1 −9 − 30 = −40 7 14 14

5 280 = 11 11

2. 67

14 1019 = 15 15

3. 107

3 1 in. is than 3 in., 5 4 we subtract the mixed numbers, which includes subtracting the fractional parts.

9. We’re adding two mixed numbers with unlike signs, so the answer has the sign of the mixed number with the larger absolute value, which in this case is negative.

365 1 = 26 14 14

Exercise Set 4.7 2.

1 3

2769 3 7. = 92 30 10 3941 14 = 231 17 17

3 is called a mixed number. 4

9 10

1 rounds to 5. 6 5 3 rounds to 4. 7 5 ⋅ 4 = 20 The best estimate is d.

6. 5

Vocabulary, Readiness & Video Check 4.7

136

5. The denominator of the mixed number we’re 4 graphing, −3 , is 5. 5

31 3776 = 35 35

1. The number 5

21 . 8

8. To find how much more 11

290 4 6. = 22 13 13

fraction is

5 written as an improper 8

7. The fractional part of a mixed number should always be a proper fraction.

17 3923 4. 186 = 21 21

3. To estimate operations on mixed numbers, we round mixed numbers to the nearest whole number.

6. To multiply mixed numbers, we first write them as equivalent improper fractions and then multiply as we multiply for fractions.

Calculator Explorations 1. 25

ISM: Prealgebra

3 8

ISM: Prealgebra

Chapter 4: Fractions and Mixed Numbers

3 rounds to 20. 14 8 4 rounds to 5. 11 20 ÷ 5 = 4 The best estimate is c.

10.

4 11 2 +3 11 6 10 11 4 2 Estimate: 7 rounds to 7, 3 rounds to 3. 11 11 7 + 3 = 10 so the answer is reasonable.

28. Exact:

8. 20

5 1 5 21 5 ⋅ 3 ⋅ 7 7 1 ⋅4 = ⋅ = = or 2 9 5 9 5 3⋅3⋅5 3 3

2 9 5 9 3 9 ⋅ 3 27 2 12. 9 ÷ 1 = ÷ = ⋅ = = or 5 3 1 3 1 5 1⋅ 5 5 5

1 6 10 2 ⋅ 3 ⋅10 20 18. 6 ⋅ 3 = ⋅ = = = 20 3 1 3 1⋅ 3 1

5 1 rounds to 12, 4 rounds to 4. 6 12 12 + 4 = 16 so the answer is reasonable.

Estimate: 12

32.

34.

2 13 7 +8 26

4 26 7 +8 26 11 14 26

2 9

8 32

4 5 14 21 2 ⋅ 7 ⋅ 21 147 7 20. 2 ⋅ 2 = ⋅ = = or 7 5 8 5 8 5⋅2⋅ 4 20 20

22.

5 12 2 +4 12 7 16 12

5 12 1 +4 6

30. Exact: 12

1 1 9 57 513 1 14. Exact: 2 ⋅ 7 = ⋅ = or 16 4 8 4 8 32 32 1 1 Estimate: 2 rounds to 2, 7 rounds to 7. 8 4 2 ⋅ 7 = 14

16. Exact: 5 3 35 38 7 ⋅ 5 ⋅ 2 ⋅19 133 1 = 5 ⋅7 = ⋅ = or 44 6 5 6 5 2 ⋅3⋅5 3 3 5 3 Estimate: 5 rounds to 6, 7 rounds to 8. 6 5 6 ⋅ 8 = 48

8 32

10 21

30 63 44 49 63

+ 9

5 1 5 21 5 5 5⋅5 25 ÷4 = ÷ = ⋅ = = 9 5 9 5 9 21 9 ⋅ 21 189 7 rounds to 4. 8 1 2 rounds to 2. 5 4−2=2 The best estimate is d.

36.

1 26. 8 rounds to 8. 3 1 1 rounds to 2. 2 8−2=6 The best estimate is c.

38.

24. 3

3 5 8 + 8 15 23

1 3 2 9 5 1 +3 6 4

14 63

9 15 8 + 8 15 17 2 2 31 = 31 + 1 = 32 15 15 15 23

10 30 12 9 30 5 +3 30 27 9 16 = 16 30 10 4

137

Chapter 4: Fractions and Mixed Numbers

4 9 2 −3 9 2 4 9 4 2 Estimate: 7 rounds to 7, 3 rounds to 3. 9 9 7 − 3 = 4 so the answer is reasonable.

40. Exact:

42. Exact:

5 12 1 − 4 6

5 12 2 − 4 12

3 1 =8 12 4 5 1 Estimate: 12 rounds to 12, 4 rounds to 4. 6 12 12 − 4 = 8 so the answer is reasonable. 8

44.

46.

2 13 7 −4 26 5

50.

138

30 26 7 −4 26 23 26 4

10 10 7 −1 10 3 6 10

52.

7 10

2 15 3 − 27 10 86

4 30 9 − 27 30

5 8 3 +2 8 8 7 = 7 +1 = 8 8

8 15 9 − 8 15 23

23 15 9 − 8 15 14 14 15 22

1 1 26 13 ÷ 56. 5 ÷ 3 = 5 4 5 4 26 4 = ⋅ 5 13 2 ⋅13 ⋅ 4 = 5 ⋅13 2⋅4 = 5 8 = 5 3 =1 5

58. 15

3 2 3 10 13 + 5 = 15 + 5 = 20 25 5 25 25 25

3 3 6 3 3 60. 5 − 2 = 5 − 2 = 3 8 16 16 16 16

62.

34 30 9 − 27 30 25 5 58 = 58 30 6

8 15 3 − 8 5 23

1 9 3 27 1 54. 4 ⋅ 3 = ⋅ = = 13 2 2 1 2 2

8 −1

48.

4 26 7 −4 26 5

ISM: Prealgebra

7 16 1 6 2 3 +9 8 3

64.

5 12 19 − 23 24 47

7 16 8 6 16 6 +9 16 21 5 5 18 = 18 + 1 = 19 16 16 16 3

10 24 19 − 23 24 47

34 24 19 − 23 24 15 5 23 = 23 24 8 46

2 3 20 11 20 ⋅11 4 ⋅ 5 ⋅11 55 1 66. 6 ⋅ 2 = ⋅ = = = = 18 3 4 3 4 3⋅ 4 3⋅ 4 3 3

ISM: Prealgebra

68.

7 15

Chapter 4: Fractions and Mixed Numbers

3 7

7 15

7 14 13 42 14

1 + 20 2

70. The sum of 8

6 14

+ 20

3 and a number translates as 4

3 8 + x. 4

72. Divide a number by −6

1 1 9 19 3 ⋅ 3 ⋅19 57 1 82. 4 ⋅ 6 = ⋅ = or 28 = 2 3 2 3 2⋅3 2 2 57 1 The area is or 28 square yards. 2 2

1 13 5 65 1 = 16 84. 3 ⋅ 5 = ⋅ = 4 4 1 4 4 1 The perimeter is 16 yards. 4

86. 1 translates as 11

 1 x ÷  −6  .  11  3 1 111 1 111 4 111 ÷ = ⋅ = = 111 74. 27 ÷ = 4 4 4 4 4 1 1 This will make 111 quarter-pound hamburgers.

76. Subtract the standard track gauge in Portugal from the standard gauge in Spain. 9 18 65 65 10 20 11 11 − 65 − 65 20 20 7 20 7 inch wider Spain’s standard track gauge is 20 than that of Portugal. 78.

8 8 5 5 − 41 − 41 8 8 3 1 8 The short crutch should be lengthened 3 1 inches. 8 43

8 8 3 3 −3 −3 8 8 5 2 8 5 The remaining piece is 2 feet. 8 5

1 47 4 47 1 47 ⋅1 47 7 88. 23 ÷ 4 = ÷ = ⋅ = = =5 2 2 1 2 4 2⋅4 8 8 7 The length of each side is 5 feet. 8

90.

92.

7 15 1 +2 15 8 6 15 The total duration for the eclipses in even8 numbered years is 6 minutes. 15 4

7 15 37 −2 60 4

It will be 1

28 60 37 −2 60 4

88 60 37 −2 60 51 17 1 =1 60 20 3

17 minutes longer. 20

1 5 7 5 ⋅ 7 35 1 = or 17 80. 5 ⋅ 3 = ⋅ = 2 1 2 1⋅ 2 2 2 35 1 The area is or 17 square inches. 2 2

139

Chapter 4: Fractions and Mixed Numbers

5  2  23 11 94. −3 ÷  −3  = ÷ 6  3 6 3 23 3 = ⋅ 6 11 23 ⋅ 3 = 2 ⋅ 3 ⋅11 23 = 22 1 =1 22

ISM: Prealgebra

1 3 17 19 104. −4 ÷ 2 = − ÷ 4 8 4 8 17 8 =− ⋅ 4 19 17 ⋅ 4 ⋅ 2 =− 4 ⋅19 34 =− 19 15 = −1 19

5  2 5 2 96. 17 +  −14  = 17 − 14 9  3 9 3 5 6 = 17 − 14 9 9 14 6 = 16 − 14 9 9 8 =2 9

106.

108. −

3  1 7  7 100. 1 ÷  −3  = ÷  −  4  2 4  2 7  2 = ⋅−  4  7 7⋅2 =− 4⋅7 2 =− 4 1 =− 2

9  10   9 10   − m  =  ⋅  m = 1⋅ m = m 10  9   10 9 

110. a.

7 3 3 = 6 +1 = 7 4 4 4

11 3 3 = 5+ 2 = 7 4 4 4

12 3 =7 16 4

7  5 7 5 98. −31 −  −26  = −31 + 26 8  12  8 12 21 10 = −31 + 26 24 24 11 = −5 24

The answer is d, all of them. 112. Incorrect; to multiply mixed numbers, first write each mixed number as improper fraction. 114. answers may vary 19 1 + 9 ≈ 12 + 9 = 21 20 10 The best sum estimate is d.

116. 11

118. answers may vary

2  3 4  3 7 102. −20 +  −30  = −20 +  −30  = −50 5  10  10  10  10

140

1 1  (5 y ) =  ⋅ 5  y = 1⋅ y = y 5 5 

Section 4.8 Practice Exercises 1.

2 5 = 3 12 2 2 5 2 y− + = + 3 3 12 3 5 2⋅4 y= + 12 3 ⋅ 4 5 8 y= + 12 12 13 y= 12 y−

ISM: Prealgebra

Chapter 4: Fractions and Mixed Numbers

2 5 = 3 12 13 2 5 − 0 12 3 12 13 2 ⋅ 4 5 − 0 12 3 ⋅ 4 12 13 8 5 − 0 12 12 12 5 5 True = 12 12 13 The solution is . 12

Check:

1 y=2 5 1 5⋅ y = 5⋅2 5 y = 10 1 y=2 Check: 5 1 ⋅10 0 2 5 2 = 2 True The solution is 10.

5 b = 25 7 7 5 7 ⋅ b = ⋅ 25 5 7 5 7 ⋅ 25 1b = 5 b = 35 5 b = 25 Check: 7 5 ⋅ 35 0 25 7 25 = 25 True The solution is 35.

7 2 x= 10 5 10 7 10 2 − ⋅− x = − ⋅ 7 10 7 5 10 ⋅ 2 x=− 7 ⋅5 4 x=− 7 −

7 2 x= 10 5 7 4 2 − ⋅− 0 10 7 5 28 2 0 70 5 2 2 = True 5 5 4 The solution is − . 7

y−

Check:

5x = −

−

3 4

1 1 3 ⋅ 5x = ⋅ − 5 5 4 1⋅ 3 x=− 5⋅4 3 x=− 20 3 4 3 3 5⋅− 0 − 20 4 15 3 − 0− 20 4 3 3 − = − True 4 4 3 The solution is − . 20 Check:

5x = −

6. The LCD is 15. 11 3 x=− 15 5 11 3 15 ⋅ x = 15 ⋅ − 15 5 11x = −9 11x −9 = 11 11 9 x=− 11

The solution is −

9 . 11

141

Chapter 4: Fractions and Mixed Numbers 7. Solve by multiplying by the LCD: The LCD is 8. y 3 + =2 8 4  y 3 8  +  = 8(2)  8 4 y 3 8 ⋅ + 8 ⋅ = 8(2) 8 4 y + 6 = 16 y + 6 − 6 = 16 − 6 y = 10 Solve with fractions: y 3 + =2 8 4 3 y 3 3 + − = 2− 8 4 4 4 y 8 3 = − 8 4 4 y 5 = 8 4 5 y 8⋅ = 8⋅ 8 4 1 2⋅4⋅5 8⋅ ⋅ y = 8 1⋅ 4 y = 10 y 3 + =2 8 4 10 3 + 02 8 4 10 6 + 02 8 8 16 02 8 2 = 2 True The solution is 10.

ISM: Prealgebra

1 in the original 4 equation to see that a true statement results. The 1 solution is − . 4

To check, replace x with −

9. The LCD is 10. y y 3 = + 2 5 2  y  y 3 10   = 10  +  2  5 2 y    y 3 10   = 10   + 10   2 5 2 5 y = 2 y + 15 5 y − 2 y = 2 y − 2 y + 15 3 y = 15 3 y 15 = 3 3 y=5

y y 3 = + 2 5 2 5 5 3 0 + 2 5 2 1 1 2 0 1+1 2 2 1 1 2 =2 True 2 2 The solution is 5.

Check:

142

1 x −x= 5 5 x  1 5 − x  = 5  5  5 x 1 5   − 5( x) = 5   5   5 x − 5x = 1 −4 x = 1 −4 x 1 = −4 −4 1 x=− 4

10.

9 y 9 3 y 10 − = ⋅ − ⋅ 10 3 10 3 3 10 9 ⋅ 3 y ⋅10 = + 10 ⋅ 3 3 ⋅10 27 10 y = − 30 30 27 − 10 y = 30

Vocabulary, Readiness & Video Check 4.8 1. The LCD of 3 and 6 is 6. 2. The LCD of 21 and 7 is 21. 3. The LCD of 5 and 3 is 15. 4. The LCD of 11 and 2 is 22. 5. addition property of equality

ISM: Prealgebra

Chapter 4: Fractions and Mixed Numbers

6. We want z with an understood coefficient of 1. 4 The coefficient of z is − . Two numbers are 9 reciprocals if their product is 1, so we use the multiplication property of equality to multiply 9 both sides of the equation by − , the reciprocal 4 4 of − . 9 7. We multiply by 12 because it is the LCD of all fractions in the equation. The equation no longer contains fractions.

5 4 = 14 14 9 5 4 − 0 14 14 14 9−5 4 0 14 14 4 4 True = 14 14 9 The solution is . 14 Check:

1 2 5 − 4x = − 11 11 11 1 3 x+ = − 11 11 1 1 3 1 x+ − = − − 11 11 11 11 4 x=− 11

6. 5 x +

8. Multiplying through by the LCD is a step in solving an equation, but we don’t have an equation⎯we have an expression. Exercise Set 4.8 2.

1 7 =− 9 9 1 1 7 1 x+ − = − − 9 9 9 9 8 x=− 9 1 7 Check: x + = − 9 9 8 1 7 − + 0− 9 9 9 −8 + 1 7 0− 9 9 7 7 − =− True 9 9 8 The solution is − . 9

1 2 5 − 4x = − 11 11 11 2 5  4 1  4 5 −  + − 4 −  0 − 11 11 11 11 11     20 1 16 2 5 − + + 0 − 11 11 11 11 11 3 3 − =− True 11 11 4 The solution is − . 11

5 4 = 14 14 5 5 4 5 z− + = + 14 14 14 14 4+5 z= 14 9 z= 14 z−

z−

Check:

5x +

8 1 = 9 3 8 8 1 8 y− + = + 9 9 3 9 1⋅ 3 8 y= + 3⋅3 9 3 8 y= + 9 9 11 y= 9 y−

143

Chapter 4: Fractions and Mixed Numbers

ISM: Prealgebra

2 11 9x − − 8x = 7 14  15  2  15  11 9  − − 8  0  14  7  14  14 135 4 120 11 − − 0 14 14 14 14 11 11 = True 14 14 15 The solution is . 14

8 1 = 9 3 11 8 1 − 0 9 9 3 3 1 0 9 3 1 1 = True 3 3 11 The solution is . 9

Check:

10.

y−

1 3 +a = − 2 8 1 1 1 3 − + +a = − − 2 2 2 8 1⋅ 4 3 a=− − 2⋅4 8 4 3 a=− − 8 8 7 a=− 8 1 3 +a = − Check: 2 8 1  7 3 +−  0 − 2  8 8 4 7 3 − 0− 8 8 8 3 3 − =− True 8 8 7 The solution is − . 8

2 11 12. 9 x − − 8 x = 7 14 2 11 x− = 7 14 2 2 11 2 x− + = + 7 7 14 7 11 2 ⋅ 2 x= + 14 7 ⋅ 2 11 4 x= + 14 14 15 x= 14

144

Check:

14.

16.

18.

1 7 = y− 4 10 1 7 7 7 − + = y− + 4 10 10 10 1⋅ 5 7 ⋅ 2 − + =y 4 ⋅ 5 10 ⋅ 2 5 14 − + =y 20 20 9 =y 20 1 7 Check: − = y− 4 10 1 9 7 − 0 − 4 20 10 1⋅ 5 9 7⋅2 0 − − 4 ⋅ 5 20 10 ⋅ 2 5 9 14 − 0 − 20 20 20 5 5 True − =− 20 20 9 The solution is . 20 −

−5 x = 4 1 1 − ⋅ (−5 x) = − ⋅ 4 5 5 4 x=− 5

1 x=6 3 1 3⋅ x = 3⋅ 6 3 x = 18

ISM: Prealgebra

20.

22.

24.

4 x = −8 7 7 4 7 ⋅ x = ⋅ −8 4 7 4 56 x=− 4 x = −14

11 2 x=− 10 7 10 11 10 2 − ⋅− x = − ⋅− 11 10 11 7 20 x= 77

Chapter 4: Fractions and Mixed Numbers

32.

−

2z = −

34.

5 12

1 1 5 ⋅ 2z = ⋅ − 2 2 12 5 z=− 24

26.

28.

30.

12 25 1 1 12 − ⋅ −4 z = − ⋅ − 4 4 25 12 z= 4 ⋅ 25 3 z= 25 −4 z = −

3 7 y=− 5 20 3   7  20  y  = 20  −  5   20  12 y = −7 12 y −7 = 12 12 7 y=− 12 x 7 −1 = 5 5 7 x  5  − 1 = 5 ⋅ 5 5   x 5 ⋅ − 5 ⋅1 = 7 5 x −5 = 7 x −5+5 = 7+5 x = 12

36.

x − x = −6 3 x  3  − x  = 3(−6) 3  x 3   − 3( x) = −18 3 x − 3 x = −18 −2 x = −18 −2 x −18 = −2 −2 x=9 2 1 x − = 3 4 12 2 1  x 12  −  = 12   3 4    12  2 1 12   − 12   = x 3 4 8−3 = x 5= x

a a 5 = + 2 7 2 a a 5 14   = 14  +  2 7 2 a 5 7a = 14 ⋅ + 14 ⋅ 7 2 7a = 2a + 35 7a − 2a = 2a − 2a + 35 5a = 35 5a 35 = 5 5 a=7

5 y 5 8 y 9 40 9 y −40 + 9 y 38. − + = − ⋅ + ⋅ = − + = 9 8 9 8 8 9 72 72 72

40. 2 +

42.

7x 2 3 7x 6 7x 6 + 7x = ⋅ + = + = 3 1 3 3 3 3 3

9 x 5 x 9 x 3 5 x 4 27 x 20 x 7 x − = ⋅ − ⋅ = − = 8 6 8 3 6 4 24 24 24

145

Chapter 4: Fractions and Mixed Numbers

44.

46.

48.

2 3 y= 5 10 5 2 5 3 ⋅ y= ⋅ 2 5 2 10 15 y= 20 3 y= 4 4 x 21 + = 5 4 20 21 4 x 20  +  = 20 ⋅ 5 4 20   4 x 20 ⋅ + 20 ⋅ = 21 5 4 16 + 5 x = 21 16 + 5 x − 16 = 21 − 16 5x = 5 5x 5 = 5 5 x =1

3 20 3 −4n = 20 1 1 3 − ⋅ −4n = − ⋅ 4 4 20 3 n=− 80

146

54.

56.

58.

60.

5 5 a= 16 6 16 5 16 5 ⋅ a= ⋅ 5 16 5 6 2⋅8⋅5 a= 5⋅ 2⋅3 8 a= 3

50. 30n − 34n =

52.

ISM: Prealgebra

y 1 −2 = 7 7 y  1 7 − 2 = 7  7  7 y 7⋅ −7⋅2 =1 7 y − 14 = 1 y − 14 + 14 = 1 + 14 y = 15

62.

8 1 8 2 1 11 16 11 5 − = ⋅ − ⋅ = − = 11 2 11 2 2 11 22 22 22

1 x = −2 4 1 4 ⋅ x = 4 ⋅ (−2) 4 x = −8 3 9 − x= 4 2 4 3 4 9 − ⋅− x = − ⋅ 3 4 3 2 4⋅9 x=− 3⋅ 2 2 ⋅ 2 ⋅3⋅3 x=− 3⋅ 2 x = −6 y = −4 + y 3  y 3   = 3(−4 + y ) 3 y = 3(−4) + 3 ⋅ y y = −12 + 3 y y − 3 y = −12 + 3 y − 3 y −2 y = −12 −2 y −12 = −2 −2 y=6 7 5 4 − x=− − 9 18 18 7 9 − x=− 9 18 9 7 9 9 − ⋅− x = − ⋅− 7 9 7 18 9⋅9 x= 7 ⋅9⋅ 2 9 x= 14

4 9 4 −3x = 9

64. 27 x − 30 x =

ISM: Prealgebra

66.

68.

70.

Chapter 4: Fractions and Mixed Numbers

5 1 7 y= − 4 2 10 5  1 7  20  y  = 20  −  4   2 10  1 7 25 y = 20 ⋅ − 20 ⋅ 2 10 25 y = 10 − 14 25 y = −4 25 y −4 = 25 25 4 y=− 25

76. a.

a a 1 = + 6 3 2 a a 1 6⋅ = 6 +  6 3 2 a 1 a = 6⋅ + 6⋅ 3 2 a = 2a + 3 a − 2a = 2 a + 3 − 2 a −a = 3 −a 3 = −1 −1 a = −3

78.

y y −2 = −4 5 3 y  y  15  − 2  = 15  − 4  5  3  y y 15 ⋅ − 15 ⋅ 2 = 15 ⋅ − 15 ⋅ 4 5 3 3 y − 30 = 5 y − 60 3 y − 5 y − 30 = 5 y − 5 y − 60 −2 y − 30 = −60 −2 y − 30 + 30 = −60 + 30 −2 y = −30 −2 y −30 = −2 −2 y = 15

80.

x 5 − =2 6 3  x 5 6  −  = 6(2)  6 3  x 5 6   − 6   = 12  6 3 x − 10 = 12 x − 10 + 10 = 12 + 10 x = 22 x 5 x 5 2 x 10 x − 10 − = − ⋅ = − = 6 3 6 3 2 6 6 6

19 353x 23 = + 53 1431 27 19  353x 23  1431 ⋅ = 1431 +  53  1431 27  353x 23 27 ⋅19 = 1431 ⋅ + 1431 ⋅ 1431 27 513 = 353x + 1219 513 − 1219 = 353x + 1219 − 1219 −706 = 353x −706 353x = 353 353 −2 = x 5 3 5 4 5⋅ 4 5⋅4 5 ÷ = ⋅ = = = 12 4 12 3 12 ⋅ 3 4 ⋅ 3 ⋅ 3 9 5 The length is inch. 9

Chapter 4 Vocabulary Check 1. Two numbers are reciprocals of each other if their product is 1. 2. A composite number is a natural number greater than 1 that is not prime. 3. Fractions that represent the same portion of a whole are called equivalent fractions.

72. To round 576 to the nearest hundred, observe that the digit in the tens place is 7. Since this digit is at least 5, we add 1 to the digit in the hundreds place. 576 rounded to the nearest hundred is 600. 74. To round 2333 to the nearest ten, observe that the digit in the ones place is 3. Since this digit is less than 5, we do not add 1 to the digit in the tens place. 2333 rounded to the nearest ten is 2330.

4. An improper fraction is a fraction whose numerator is greater than or equal to its denominator. 5. A prime number is a natural number greater than 1 whose only factors are 1 and itself. 6. A fraction is in simplest form when the numerator and the denominator have no factors in common other than 1.

147

Chapter 4: Fractions and Mixed Numbers

ISM: Prealgebra

7. A proper fraction is one whose numerator is less than its denominator. 8. A mixed number contains a whole number part and a fraction part. 9. In the fraction

7 , the 7 is called the numerator and the 9 is called the denominator. 9

10. The prime factorization of a number is the factorization in which all the factors are prime numbers. 11. The fraction

3 is undefined. 0

12. The fraction

0 = 0. 5

13. Fractions that have the same denominator are called like fractions. 14. The LCM of the denominators in a list of fractions is called the least common denominator. 15. A fraction whose numerator or denominator or both numerator and denominator contain fractions is called a complex fraction. 16. In

a c = , a ⋅ d and b ⋅ c are called cross products. b d

Chapter 4 Review 1. 2 out of 6 equal parts are shaded:

2 6

2. 4 out of 7 equal parts are shaded:

4 7

3. Each part is

1 7 1 of a whole and there are 7 parts shaded, or 2 wholes and 1 more part: or 2 3 3 3

4. Each part is

1 13 1 or 3 of a whole and there are 13 parts shaded, or 3 wholes and 1 more part: 4 4 4

5. successful free throws → 11 total free throws →12 11 of the free throws were made. 12

6. a. b.

131 − 23 = 108 108 cars are not blue.

not blue →108 total → 131 108 of the cars on the lot are not blue. 131

148

ISM: Prealgebra

Chapter 4: Fractions and Mixed Numbers

3 7. − = −1 3

−20 =1 −20

0 =0 −1

10.

4 is undefined 0

11.

12.

13.

14. 3 15. 4 15 12 3

15 3 =3 4 4 3 16. 13 39 39 0 39 =3 13 1 5 ⋅ 2 + 1 10 + 1 11 17. 2 = = = 5 5 5 5 8 9 ⋅ 3 + 8 27 + 8 35 18. 3 = = = 9 9 9 9

19.

12 4 ⋅ 3 3 = = 28 7 ⋅ 4 7

20.

15 3 ⋅ 5 5 = = 27 3 ⋅ 9 9

149

Chapter 4: Fractions and Mixed Numbers

21. −

22. −

25 x 75 x

=−

1 ⋅ 25 ⋅ x 1 =− 3 ⋅ 25 ⋅ x ⋅ x 3x

29ab 29 ⋅ a ⋅ b 29 = = 32abc 32 ⋅ a ⋅ b ⋅ c 32c

24.

18 xyz 18 ⋅ x ⋅ y ⋅ z 18 z = = 23 xy 23 ⋅ x ⋅ y 23

26.

27.

33. −

36 y 3 1 ⋅ 36 ⋅ y ⋅ y ⋅ y y2 =− =− 72 y 2 ⋅ 36 ⋅ y 2

23.

25.

45 x 2 y 27 xy 3

42ab 2 c 30abc3

34.

24 x  15  24 x ⋅15 ⋅ − = 5  8 x3  5 ⋅ 8 x 3 8⋅3⋅ x ⋅5⋅3 = 5⋅8⋅ x ⋅ x ⋅ x 3⋅3 = x⋅ x 9 = x2

27 y 3 7 27 y 3 ⋅ 7 3 ⋅ 9 ⋅ y ⋅ y ⋅ y ⋅ 7 y ⋅ = = = 21 18 y 2 21 ⋅18 y 2 3 ⋅ 7 ⋅ 9 ⋅ 2 ⋅ y ⋅ y 2 3

9⋅5⋅ x ⋅ x ⋅ y 5x = 9 ⋅ 3 ⋅ x ⋅ y ⋅ y ⋅ y 3y2

1 ⋅1 ⋅ 1 1  1   1  1  1  =− 35.  −  =  −  −  −  = − 3⋅3⋅3 27  3   3  3  3 

6⋅7 ⋅ a ⋅b ⋅b ⋅c 7b = 6 ⋅ 5 ⋅ a ⋅ b ⋅ c ⋅ c ⋅ c 5c 2

5⋅5 25  5  5  5  = 36.  −  =  −  −  = 12 12 12 12 12 144 ⋅     

8 inches 8 4⋅2 2 = = = 12 inches 12 4 ⋅ 3 3 2 8 inches represents of a foot. 3

3 3 3 8 3⋅8 3⋅ 4 ⋅ 2 2 37. − ÷ = − ⋅ = − =− = − = −2 4 8 4 3 4⋅3 1⋅ 4 ⋅ 3 1

38.

28. 15 − 6 = 9 cars are not white. 9 non-white cars 9 3 ⋅ 3 3 = = = 15 total cars 15 3 ⋅ 5 5 3 of the cars are not white. 5 29. Not equivalent, since the cross products are not equal: 34 ⋅ 4 = 136 and 10 ⋅ 14 = 140 30. Equivalent, since the cross products are equal: 30 ⋅ 15 = 450 and 50 ⋅ 9 = 450 31.

ISM: Prealgebra

3 1 3 ⋅1 3 ⋅ = = 5 2 5 ⋅ 2 10

21a 7 a 21a 5 ÷ = ⋅ 4 5 4 7a 21a ⋅ 5 = 4 ⋅ 7a 7 ⋅3⋅ a ⋅5 = 4⋅7⋅a 3⋅5 = 4 15 = 4

9 1 9  3  9 ⋅ 3 27 = 39. − ÷ − = − ⋅  −  = 2 3 2  1  2 ⋅1 2 5 5 2y 5 1 5 ⋅1 5 =− ⋅ =− =− 40. − ÷ 2 y = − ÷ 3 3 1 3 2y 3⋅ 2y 6y

6 5 6⋅5 6⋅5 5 5 32. − ⋅ = − =− =− =− 7 12 7 ⋅12 7⋅6⋅2 7⋅2 14

150

ISM: Prealgebra

Chapter 4: Fractions and Mixed Numbers

52. 4 = 2 ⋅ 2 8=2⋅2⋅2 12 = 2 ⋅ 2 ⋅ 3 LCD = 2 ⋅ 2 ⋅ 2 ⋅ 3 = 24

9 3 ÷ 7 4 9 4 = ⋅ 7 3 9⋅4 = 7 ⋅3 3⋅3⋅ 4 = 7 ⋅3 3⋅ 4 = 7 12 = 7

41. x ÷ y =

42. ab = −7 ⋅

9 7 9 7 ⋅9 63 =− ⋅ =− =− 10 1 10 1 ⋅10 10

43. area = length ⋅ width =

The area is

11 7 11 ⋅ 7 77 ⋅ = = 6 8 6 ⋅ 8 48

77 square feet. 48

2 2 2⋅2 4 44. area = side ⋅ side = ⋅ = = 3 3 3⋅3 9 4 The area is square meter. 9

45.

53.

2 2 10 2 ⋅10 20 = ⋅ = = 3 3 10 3 ⋅10 30

54.

5 5 7 5 ⋅ 7 35 = ⋅ = = 8 8 7 8 ⋅ 7 56

55.

7 a 7 a 7 7 a ⋅ 7 49a = ⋅ = = 6 6 7 6⋅7 42

56.

9b 9b 5 9b ⋅ 5 45b = ⋅ = = 4 4 5 4⋅5 20

57.

4 4 10 4 ⋅10 40 = ⋅ = = 5 x 5 x 10 5 x ⋅10 50 x

58.

5 5 2 5⋅2 10 = ⋅ = = 9 y 9 y 2 9 y ⋅ 2 18 y

59.

7 3 7 + 3 10 + = = 11 11 11 11

60.

4 2 4 + 2 6 2⋅3 2 46. + = = = = 9 9 9 9 3⋅3 3

47.

1 5 1 − 5 −4 1⋅ 4 1 − = = =− =− 12 12 12 12 3⋅ 4 3

48.

11x x 11x + x 12 x 3 ⋅ 4 ⋅ x 4 x + = = = = 15 15 15 15 3⋅5 5

49.

4y 3 4y − 3 − = 21 21 21

50.

4 3 2 4 − 3 − 2 −1 1 − − = = =− 15 15 15 15 15 15

51. 3 = 3 x=x LCD = 3 ⋅ x = 3x

3 2 1 3 + 2 +1 6 2 ⋅3 3 + + = = = = 8 8 8 8 8 2⋅4 4 3 He did of his homework that evening. 4

9 3 9 3 9+3+9+3 + + + = 16 16 16 16 16 24 = 16 3⋅8 = 2 ⋅8 3 = 2 3 The perimeter is miles. 2

61. The LCD is 18. 7 2 7 2 ⋅ 2 7 4 11 + = + = + = 18 9 18 9 ⋅ 2 18 18 18 62. The LCD is 26. 4 1 4⋅2 1 8 1 7 − = − = − = 13 26 13 ⋅ 2 26 26 26 26 63. The LCD is 12. 1 1 1⋅ 4 1⋅ 3 4 3 1 − + =− + =− + =− 3 4 3⋅ 4 4 ⋅3 12 12 12

151

Chapter 4: Fractions and Mixed Numbers 64. The LCD is 12. 2 1 2 ⋅ 4 1⋅ 3 8 3 5 − + =− + =− + =− 3 4 3⋅ 4 4 ⋅3 12 12 12 65. The LCD is 55. 5 x 2 5 x ⋅ 5 2 25 x 2 25 x + 2 + = + = + = 11 55 11 ⋅ 5 55 55 55 55 66. The LCD is 15. 4 b 4 b ⋅ 3 4 3b 4 + 3b + = + = + = 15 5 15 5 ⋅ 3 15 15 15 67. The LCD is 36. 5 y 2 y 5 y ⋅ 3 2 y ⋅ 4 15 y 8 y 7 y − = − = − = 12 9 12 ⋅ 3 9 ⋅ 4 36 36 36 68. The LCD is 18. 7 x 2 x 7 x 2 x ⋅ 2 7 x 4 x 11x + = + = + = 18 9 18 9 ⋅ 2 18 18 18 69. The LCD is 9y. 4 5 4 ⋅ y 5 ⋅ 9 4 y 45 4 y + 45 + = + = + = 9 y 9⋅ y y ⋅9 9y 9y 9y 70. The LCD is 11y. 9 3 9 ⋅ y 3 ⋅11 9 y 33 9 y − 33 − = − = − = 11 y 11 ⋅ y y ⋅11 11 y 11 y 11 y 71. The LCD is 150. 4 23 7 4 ⋅ 6 23 ⋅ 2 7 ⋅ 3 + + = + + 25 75 50 25 ⋅ 6 75 ⋅ 2 50 ⋅ 3 24 46 21 = + + 150 150 150 91 = 150 72. The LCD is 18. 2 2 1 2 ⋅ 6 2 ⋅ 2 1⋅ 3 − − = − − 3 9 6 3⋅ 6 9 ⋅ 2 6 ⋅3 12 4 3 = − − 18 18 18 5 = 18

152

ISM: Prealgebra 73. The LCD is 18. 2 5 2 5 2⋅ 2 5⋅3 2⋅ 2 5⋅3 + + + = + + + 9 6 9 6 9⋅ 2 6⋅3 9⋅ 2 6⋅3 4 15 4 15 = + + + 18 18 18 18 38 = 18 2 ⋅19 = 2⋅9 19 = 9 19 meters. The perimeter is 9 74. The LCD is 10. 1 3 7 1⋅ 2 3 ⋅ 2 7 + + = + + 5 5 10 5 ⋅ 2 5 ⋅ 2 10 2 6 7 = + + 10 10 10 15 = 10 3⋅5 = 2⋅5 3 = 2 3 The perimeter is feet. 2 75. The LCD is 50. 9 3 9⋅2 3 18 3 21 + = + = + = 25 50 25 ⋅ 2 50 50 50 50 21 of the donors have type A blood. 50 76. The LCD is 12. 2 5 2⋅4 5 8 5 3 1⋅ 3 1 − = − = − = = = 3 12 3 ⋅ 4 12 12 12 12 4 ⋅ 3 4 1 The difference in length is yard. 4

ISM: Prealgebra

77.

Chapter 4: Fractions and Mixed Numbers

2x 2x 7 5 = ÷ 7 5 10 10

81.

2 x 10 = ⋅ 5 7 2 x ⋅10 = 5⋅7 2⋅ x ⋅5⋅ 2 = 5⋅7 4x = 7

78.

79.

= =

(

)

( )

30 − 23 + 30 ⋅ 54

) )

)

12 y ⋅ 56 − 12 y ⋅ 14

( 2)

82.

x+ y 2 + −3 = 4 z = =

(

30 12 − 23

)

30 ⋅ 54 30 ⋅ 12 − 30 ⋅ 23

6⋅4 15 − 20 = 24 5 =− 24

8 − 10 = 15 − 14 −2 = 1 = −2

80.

(

30 − 23 + 54

15 −20 + 24 15 = 4

2−1 20 52 − 12 5 2 = 3− 7 7 20 34 − 10 4 10 20 ⋅ 52 − 20 ⋅ 12 = 7 20 ⋅ 34 − 20 ⋅ 10

5−1 12 y 56 − 14 6 4 = − 121 y 12 y − 1 12 y

30 ⋅ 12

3y 3 y 11 3 y 7 3 y ⋅ 7 3 y 7 = ÷ = ⋅ = = 11 7 7 7 11 7 ⋅11 11 7

( (

x 2 = y+z −2+ 4

83.

5 1 4 5 2 4 5⋅ 2⋅4 8 ÷ ⋅ = ⋅ ⋅ = = 13 2 5 13 1 5 13 ⋅1 ⋅ 5 13 2

84.

−1 10 y − 3 y = −1 7y = −1 = −7 y

85.

2 1 2 1 −  = − 27  3  27 9 2 1⋅ 3 = − 27 9 ⋅ 3 2 3 = − 27 27 1 =− 27 9 1 2 1 9 ⋅1 2 ⋅1 ⋅ − ⋅ = − 10 3 5 11 10 ⋅ 3 5 ⋅11 3 2 = − 10 55 3 ⋅11 2 ⋅ 2 = − 10 ⋅11 55 ⋅ 2 33 4 = − 110 110 29 = 110

153

Chapter 4: Fractions and Mixed Numbers

1 22 5 22 ⋅ 5 110 2 92. 7 ⋅ 5 = ⋅ = = = 36 3 3 1 3 ⋅1 3 3 110 2 There are or 36 grams of fat in a 5-ounce 3 3 hamburger patty.

2 1 3  2  1⋅ 2 3  86. − ⋅  +  = − ⋅  +  7  5 10  7  5 ⋅ 2 10  2  2 3 = − ⋅ +  7  10 10  2 5 =− ⋅ 7 10 2⋅5 =− 7 ⋅ 2⋅5 1 =− 7

87.

3 8 5 9 6 1 +3 12 7

9 24 20 9 24 2 +3 24 31 7 7 19 = 19 + 1 = 20 24 24 24 7

1 rounds to 8. 5 3 5 rounds to 5. 11 An estimate is 8 − 5 = 3. 1 11 8 8 5 55 3 15 −5 −5 11 55

88. 8

66 55 15 −5 55 51 2 55 7

5 rounds to 2. 8 1 3 rounds to 3. 5 An estimate is 2 ⋅ 3 = 6. 5 1 13 16 13 ⋅16 13 ⋅ 8 ⋅ 2 26 1 1 ⋅3 = ⋅ = = = =5 8 5 8 5 8⋅5 8⋅5 5 5

89. 1

3 2 27 9 27 7 9 ⋅ 3 ⋅ 7 21 1 90. 6 ÷ 1 = ÷ = ⋅ = = =5 4 7 4 7 4 9 4⋅9 4 4 1 341 31 341 2 11 ⋅ 31 ⋅ 2 = ÷ = ⋅ = = 22 2 1 2 1 31 1 ⋅ 31 We would expect 22 miles on one gallon.

91. 341 ÷ 15

154

ISM: Prealgebra

7 3 4 1 93. 18 − 10 = 8 = 8 8 8 8 2 1 17 2 17 1 17 1 8 ÷2 = ÷ = ⋅ = =4 2 2 1 2 2 4 4 1 Each measurement is 4 inches. 4 3 3 13 3 13 3 ⋅ 2 13 6 7 − = − = − = − = 10 5 10 5 10 5 ⋅ 2 10 10 10 7 yard. The unknown measurement is 10

94. 1

1  3 2  3 5 95. −12 +  −15  = −12 +  −15  = −27 7  14  14  14  14 7 7 7 7  96. 23 − 24 = −  24 − 23  8 10 8  10 7 28 68 24 24 23 10 40 40 7 35 35 − 23 − 23 − 23 8 40 40 33 40 7 7 33 23 − 24 = − 8 10 40 1  7 16  27  97. −3 ÷  −2  = − ÷  −  5  10  5  10  16  10  = − ⋅ −  5  27  16 ⋅ 5 ⋅ 2 = 5 ⋅ 27 32 = 27 5 =1 27 1 3 9 7 63 15 98. −2 ⋅1 = − ⋅ = − = −3 4 4 4 4 16 16

ISM: Prealgebra

99.

2 1 = 3 6 2 2 1 2 a− + = + 3 3 6 3 1 2⋅2 a= + 6 3⋅ 2 1 4 a= + 6 6 5 a= 6

a−

1 7 100. 9 x + − 8 x = − 5 10 1 7 x+ = − 5 10 1 1 7 1 x+ − = − − 5 5 10 5 7 2 x=− − 10 10 9 x=− 10

101.

102.

103.

3 − x=6 5 5 3 5 − ⋅− x = − ⋅6 3 5 3 x = −10

Chapter 4: Fractions and Mixed Numbers

104.

105.

106.

2 4 y=− 9 3 9 2 9 4 ⋅ y = ⋅− 2 9 2 3 9⋅4 y=− 2⋅3 3⋅3⋅ 2 ⋅ 2 y=− 2⋅3 6 y=− 1 y = −6

x 6 −3 = − 7 7 x   6 7  − 3 = 7  −  7   7 x 7 ⋅ − 7 ⋅ 3 = −6 7 x − 21 = −6 x − 21 + 21 = −6 + 21 x = 15

107.

y 11 +2= 5 5 y   11  5 + 2  = 5  5  5 y 5 ⋅ + 5 ⋅ 2 = 11 5 y + 10 = 11 y + 10 − 10 = 11 − 10 y =1 1 x 17 + = 6 4 12 17 1 x 12  +  = 12 ⋅ 6 4 12   x 1 12 ⋅ + 12 ⋅ = 17 6 4 2 + 3 x = 17 2 + 3 x − 2 = 17 − 2 3 x = 15 3 x 15 = 3 3 x=5

x 5 x 1 − = − 5 4 2 20  x 5 x 1  20  −  = 20  −  5 4  2 20  x x 5 1 20 ⋅ − 20 ⋅ = 20 ⋅ − 20 ⋅ 5 4 2 20 4 x − 25 = 10 x − 1 4 x − 25 − 10 x = 10 x − 1 − 10 x −25 − 6 x = −1 −25 − 6 x + 25 = −1 + 25 −6 x = 24 −6 x 24 = −6 −6 x = −4 6 5 6⋅5 3⋅ 2 ⋅5 1 ⋅ = = = 15 8 15 ⋅ 8 5 ⋅ 3 ⋅ 2 ⋅ 4 4

155

Chapter 4: Fractions and Mixed Numbers

108.

5 x 2 10 x3 5 x 2 y3 ÷ = ⋅ y y 10 x3 y3 =

ISM: Prealgebra

1 rounds to 12. 7 3 9 rounds to 10. 5 An estimate is 12 − 10 = 2. 1 5 12 12 7 35 3 21 − 9 − 9 5 35

115. 12

5x2 ⋅ y3

y ⋅10 x3 5⋅ x ⋅ x ⋅ y ⋅ y ⋅ y = y ⋅5⋅2⋅ x ⋅ x ⋅ x = 109.

y2 2x

3 1 3 − 1 2 1⋅ 2 1 − = = = = 10 10 10 10 5 ⋅ 2 5

7 2 7⋅2 7⋅2 7 110. ⋅− = − =− =− 8x 3 8x ⋅ 3 2⋅4⋅ x ⋅3 12 x

5 2 5 5 2 25 2 23 + =− ⋅ + =− + =− 11 55 11 5 55 55 55 55

7 rounds to 3. 8 1 9 rounds to 10. 2 An estimate is 3 + 10 = 13. 7 7 2 2 8 8 1 4 +9 +9 2 8 11 3 3 11 = 11 + 1 = 12 8 8 8

156

( ) 8 (1 − 18 )

8 2 + 34

8 ⋅ 2 + 8 ⋅ 34

8 ⋅1 − 8 ⋅ 18 16 + 6 = 8 −1 22 1 or 3 = 7 7

3 1 8 1 113. −1 ÷ = − ÷ 5 4 5 4 8 4 =− ⋅ 5 1 8⋅ 4 =− 5 ⋅1 32 2 or − 6 =− 5 5 114. 2

1 − 18

2x x 2x 4 x 3 + = ⋅ + ⋅ 111. 3 4 3 4 4 3 8 x 3x = + 12 12 8 x + 3x = 12 11x = 12 112. −

116.

2 + 34

3 2 4 3  2⋅3 4  117. − ⋅  −  = −  −  8 3 9 8  3⋅3 9  3 6 4 =−  −  89 9 3 2 =−   89 3⋅ 2 =− 8⋅9 3⋅ 2 =− 2 ⋅ 4 ⋅3⋅3 1 =− 12 2 13 118. 11x − − 10 x = − 7 14 2 13 x− = − 7 14 2 2 13 2 x− + = − + 7 7 14 7 13 2 ⋅ 2 x=− + 14 7 ⋅ 2 13 4 x=− + 14 14 9 x=− 14

40 35 21 − 9 35 19 2 35 11

ISM: Prealgebra

119.

Chapter 4: Fractions and Mixed Numbers

3 4 − x= 5 15 5 3 5 4 − ⋅− x = − ⋅ 3 5 3 15 5⋅ 4 x=− 3 ⋅15 5⋅ 4 x=− 3⋅5⋅3 4 x=− 9

3 5 ⋅ 4 + 3 20 + 3 23 5. 4 = = = ; C 5 5 5 5

6. 8

8 2 8⋅2 16 ⋅ = = , the operation is 11 11 11 ⋅11 121 multiplication; C.

7. Since

8 2 8 11 8 ⋅11 8 2 ⋅ 4 ÷ = ⋅ = = = = 4, the 11 11 11 2 11 ⋅ 2 2 2 operation is division; D.

8. Since

8 2 8−2 6 − = = , the operation is 11 11 11 11 subtraction; B.

1 50 11 − 121. 50 − 5 = 2 1 2 50 ⋅ 2 11 = − 1⋅ 2 2 100 11 = − 2 2 89 = 2 1 = 44 2

9. Since

8 2 8 + 2 10 + = = , the operation is 11 11 11 11 addition; A.

10. Since

The length of the remaining piece is 44 4 1 81 11 81 ⋅11 81 1 ⋅5 = ⋅ = = or 40 11 2 11 2 11 ⋅ 2 2 2 81 1 or 40 square feet. The area is 2 2

122. 7

Chapter 4 Getting Ready for the Test −2 = 1; A −2

−2 −1 ⋅ 2 = = −1; B 2 2

2 is undefined; D 0

2 23 −16 7

23 7 =2 ; B 8 8

x 5 3 120. + =− 12 6 4  x 5  3 12  +  = 12  −   12 6   4 x 5 12 ⋅ + 12 ⋅ = −9 12 6 x + 10 = −9 x + 10 − 10 = −9 − 10 x = −19

0 = 0; C −2

1 yards. 2

11.

5 1 5 +1 6 + = = ; F 7 7 7 7

12.

5 1 5 ⋅1 5 ⋅ = = ; H 7 7 7 ⋅ 7 49

13.

5 1 5 7 5⋅7 5 ÷ = ⋅ = = = 5; B 7 7 7 1 7 ⋅1 1

14.

5 1 5 −1 4 − = = ; D 7 7 7 7

15. Since multiplication and division are performed before addition and subtraction, the first operation that should be performed to simplify 1 1 1 1 1 + ⋅ is the multiplication of and ; C. 2 5 4 5 4

157

Chapter 4: Fractions and Mixed Numbers 16. Since operations in parentheses (grouping symbols) are performed first, the first operation 1 1 1 that should be performed to simplify  +  ⋅ 2 5 4 1 1 is the addition of and ; A. 5 2

ISM: Prealgebra

18 3. 4 75 4 35 32 3 75 3 = 18 4 4

17. Since multiplication and division are performed in order from left to right, the first operation that 1 1 1 should be performed to simplify ÷ ⋅ is the 2 5 4 1 1 division of by ; D. 5 2

18. Since operations in parentheses (grouping symbols) are performed first, the first operation 1 1 1 that should be performed to simplify ÷  ⋅  2 5 4 1 1 is the multiplication of and ; C. 5 4

6. Check the cross products. 5 ⋅ 11 = 55 7 ⋅ 8 = 56 Since 55 ≠ 56, the fractions are not equivalent.

19.

x 3 + =2 10 5  x 3 10  +  = 10 ⋅ 2  10 5   x 3 10   + 10   = 10 ⋅ 2  10  5 x + 6 = 20 The correct equivalent equation is choice D. 2 1 − does not contain an equal sign, it is 3 9 an expression; A.

20. Since

2 1 x − = contains an equal sign, it is an 3 9 9 equation; B.

21. Since

Chapter 4 Test 1. 7 of the 16 equal parts are shaded:

2. 7

2 3 ⋅ 7 + 2 21 + 2 23 = = = 3 3 3 3

7 . 16

5. −

42 x 3 ⋅14 ⋅ x 3x =− =− 70 5 ⋅14 5

7. Check the cross products. 6 ⋅ 63 = 378 27 ⋅ 14 = 378 Since 378 = 378, the fractions are equivalent. 8. 84 = 2 ⋅ 42 = 2 ⋅ 2 ⋅ 21 = 2 ⋅ 2 ⋅ 3 ⋅ 7 = 2 2 ⋅ 3 ⋅ 7 9. 495 = 3 ⋅ 165 = 3 ⋅ 3 ⋅ 55 = 3 ⋅ 3 ⋅ 5 ⋅ 11 = 32 ⋅ 5 ⋅11 10.

4 3 4 4 4⋅4 4 ÷ = ⋅ = = 4 4 4 3 4⋅3 3

4 4 4⋅4 4 11. − ⋅ = − =− 3 4 3⋅ 4 3

12.

7 x x 7 x + x 8x + = = 9 9 9 9

13. The LCD is 7x. 1 3 1 x 3 7 x 21 x − 21 − = ⋅ − ⋅ = − = 7 x 7 x x 7 7x 7x 7x 14.

xy 3 z xy 3 ⋅ z x ⋅ y ⋅ y ⋅ y ⋅ z y ⋅ y ⋅ = = = = y2 z xy z ⋅ xy x⋅ y⋅z 1

2 8 2 ⋅ 8 16 15. − ⋅ − = = 3 15 3 ⋅15 45

16. 158

24 6⋅4 4 = = 210 6 ⋅ 7 ⋅ 5 35

9 a 2 9 a 2 ⋅ 2 9 a 4 9a + 4 + = + = + = 10 5 10 5 ⋅ 2 10 10 10

ISM: Prealgebra

17. −

18.

19.

20.

21.

Chapter 4: Fractions and Mixed Numbers

8 2 −8 − 2 −10 2⋅5 2 − = = =− =− 15 y 15 y 15 y 15 y 3⋅5⋅ y 3y

3a 16 3a ⋅16 3⋅ a ⋅8⋅ 2 1 1 ⋅ = = = = 8 6a 3 8 ⋅ 6a 3 8 ⋅ 2 ⋅ 3 ⋅ a ⋅ a ⋅ a a ⋅ a a 2 11 3 5 11 ⋅ 2 3 ⋅ 3 5 − + = − + 12 8 24 12 ⋅ 2 8 ⋅ 3 24 22 9 5 = − + 24 24 24 22 − 9 + 5 = 24 18 = 24 3⋅6 = 4⋅6 3 = 4 7 8 2 7 5 3 +2 4 3

35 40 16 7 40 30 +2 40 81 1 1 12 = 12 + 2 = 14 40 40 40 3

19 3 −2 11

22. −

11 11 3 −2 11 8 16 11 18

1 3 10 27 23. 3 ⋅ 6 = ⋅ 3 4 3 4 10 ⋅ 27 = 3⋅ 4 2 ⋅5⋅3⋅9 = 3⋅ 2 ⋅ 2 5⋅9 = 2 45 1 = or 22 2 2 2  1 2 6 1 24. − ⋅  6 −  = − ⋅  −  7  6 7 1 6 2  6⋅6 1  = − ⋅ −  7  1⋅ 6 6  2  36 1  = − ⋅ −  7  6 6 2 35 =− ⋅ 7 6 2 ⋅ 35 =− 7⋅6 2⋅7⋅5 =− 7 ⋅2⋅3 5 2 = − or − 1 3 3

25.

1 2 3 1 3 3 1⋅ 3 ⋅ 3 9 ÷ ⋅ = ⋅ ⋅ = = 2 3 4 2 2 4 2 ⋅ 2 ⋅ 4 16 2

 3 2 5  3 2 2 5 26.  −  ÷  +  =  −  ÷  ⋅ +   4 3 6  4 3 2 6 2  3 4 5 = −  ÷ +   4 6 6 2

 3 9 = −  ÷  4 6 9 9 = ÷ 16 6 9 6 = ⋅ 16 9 9⋅ 2⋅3 = 2⋅8⋅9 3 = 8

16 3 16 12 ÷− = − ⋅− 3 12 3 3 16 ⋅12 = 3⋅3 16 ⋅ 3 ⋅ 4 = 3⋅3 64 1 or 21 = 3 3

159

Chapter 4: Fractions and Mixed Numbers

5 4 7   5⋅ 2 4⋅4 7  + +  ÷3 27.  + +  ÷ 3 =  6 3 12    6 ⋅ 2 3 ⋅ 4 12   10 16 7  =  + +  ÷3  12 12 12  33 3 = ÷ 12 1 33 1 = ⋅ 12 3 33 ⋅1 = 12 ⋅ 3 11 ⋅ 3 ⋅1 = 12 ⋅ 3 11 = 12

28.

5x 5 x 20 x 2 7 = ÷ 2 20 x 7 21 21

5 x 21 ⋅ 7 20 x 2 5⋅ x ⋅3⋅ 7 = 7 ⋅4⋅5⋅ x ⋅ x 3 = 4x =

29.

( ) = 14 ⋅ 5 + 14 ⋅ 73 = 70 + 6 = 76 2 − 12 14 ( 2 − 1 ) 14 ⋅ 2 − 14 ⋅ 12 28 − 7 21 2

5 + 73

or 3

30.

160

14 5 + 73

13 21

3 3 − x= 8 4 8 3 8 3 − ⋅− x = − ⋅ 3 8 3 4 8⋅3 x=− 3⋅ 4 4⋅ 2⋅3 x=− 3⋅ 4 x = −2

ISM: Prealgebra

31.

32.

x 24 +x=− 5 5 x   24  5 + x  = 5 −  5   5  x 5 ⋅ + 5 ⋅ x = −24 5 x + 5 x = −24 6 x = −24 6 x −24 = 6 6 x = −4 2 x 5 x + = + 3 4 12 2 2 x  5 x 12  +  = 12  +  3 4  12 2  x x 2 5 12 ⋅ + 12 ⋅ = 12 ⋅ + 12 ⋅ 3 4 12 2 8 + 3x = 5 + 6 x 8 + 3x − 6 x = 5 + 6 x − 6 x 8 − 3x = 5 8 − 8 − 3x = 5 − 8 −3 x = −3 −3 x −3 = −3 −3 x =1

1  1  5 1 5 ⋅1 5 = =2 33. −5 x = −5  −  = ⋅ = 2  2  1 2 1⋅ 2 2 1 7 ÷3 2 8 1 31 = ÷ 2 8 1 8 = ⋅ 2 31 1⋅ 8 = 2 ⋅ 31 1⋅ 2 ⋅ 4 = 2 ⋅ 31 4 = 31

34. x ÷ y =

ISM: Prealgebra

35.

1 2 3 −2 4

Chapter 4: Fractions and Mixed Numbers

2 4 3 −2 4

The remaining piece is 3

36. Housing:

6 4 3 −2 4 3 3 4 5

2 2 +1+ 3 3 3 2 3 2 = + + + 3 3 3 3 10 = 3 1 =3 3 2 2 area = length ⋅ width = 1 ⋅ = 3 3 1 The perimeter is 3 feet and the area is 3 2 square foot. 3

39. perimeter = 1 +

3 feet. 4

17 50

3 25 17 3 17 3 ⋅ 2 17 6 23 + = + = + = 50 25 50 25 ⋅ 2 50 50 50 23 of spending goes for housing and food 50 combined.

Food:

37. Education:

Cumulative Review Chapters 1−4

1 50

4 Transportation: 25 3 Clothing: 100 1 4 3 1⋅ 2 4⋅4 3 + + = + + 50 25 100 50 ⋅ 2 25 ⋅ 4 100 2 16 3 = + + 100 100 100 21 = 100 21 of spending goes for education, 100 transportation, and clothing.

38.

3 258 43 258 4 43 ⋅ 6 ⋅ 4 = ÷ = ⋅ = = 24 4 1 4 1 43 1 ⋅ 43 Expect to travel 24 miles on 1 gallon of gas.

40. 258 ÷ 10

2 2 47, 000 ⋅ 47, 000 = ⋅ 25 25 1 2 ⋅ 25 ⋅1880 = 25 ⋅1 2 ⋅1880 = 1 = 3760 Expect $3760 was spent on health care.

1. 546 in words is five hundred forty-six. 2. 115 in words is one hundred fifteen. 3. 27,034 in words is twenty-seven thousand, thirty-four. 4. 6573 in words is six thousand, five hundred seventy-three. 5.

46 + 713 759

587 + 44 631

543 − 29 514 Check:

514 + 29 543

995 − 62 933

161

Chapter 4: Fractions and Mixed Numbers

Check:

933 + 62 995

15. 7 ⋅ 7 ⋅ 7 = 73 16. 7 ⋅ 7 = 7 2

9. To round 278,362 to the nearest thousand, observe that the digit in the hundreds place is 3. Since this digit is less than 5, we do not add 1 to the digit in the thousands place. 278,362 rounded to the nearest thousand is 278,000. 10. To round 1436 to the nearest ten, observe that the digit in the ones place is 6. Since this digit is at least 5, we add 1 to the digit in the tens place. 1436 rounded to the nearest ten is 1440. 11.

12.

96 × 12 192 960 1152 Twelve boxes of single-serve coffee maker pods hold 1152 pods. 435 × 3 1305 The family travels 1305 miles in 3 days.

7089 13. 8 56, 717 56 07 0 71 64 77 72 5

17. 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 9 ⋅ 9 ⋅ 9 = 34 ⋅ 93 18. 9 ⋅ 9 ⋅ 9 ⋅ 9 ⋅ 5 ⋅ 5 = 94 ⋅ 52 19. 2(x − y) = 2(6 − 3) = 2(3) = 6 20. 8a + 3(b − 5) = 8 ⋅ 5 + 3(9 − 5) = 8⋅5 + 3⋅ 4 = 40 + 12 = 52 21. Let 0 represent the surface of Earth. Then 7257 feet below the surface is represented as −7257. 22. Let 0 represent a temperature of 0°F. Then 21°F below zero is represented as −21. 23. −7 + 3 = −4 24. −3 + 8 = 5 25. 7 − 8 − (−5) − 1 = 7 − 8 + 5 − 1 = 7 + (−8) + 5 + (−1) = −1 + 5 + (−1) = 4 + (−1) =3 26. 6 + (−8) − (−9) + 3 = 6 + (−8) + 9 + 3 = −2 + 9 + 3 = 7+3 = 10 27. (−5)2 = (−5)(−5) = 25

56,717 ÷ 8 = 7089 R 5 Check: 7089 × 8 + 5 = 56,712 + 5 = 56,717 379 14. 12 4558 36 95 84 118 108 10 4558 ÷ 12 = 379 R 10 Check: 379 × 12 + 10 = 4548 + 10 = 4558 162

ISM: Prealgebra

28. −24 = −2 ⋅ 2 ⋅ 2 ⋅ 2 = −16 29. 3(4 − 7) + (−2) − 5 = 3(−3) + (−2) − 5 = −9 + (−2) + (−5) = −11 + (−5) = −16 30. (20 − 52 ) 2 = (20 − 25)2 = (−5)2 = 25 31. 2 y − 6 + 4 y + 8 = (2 y + 4 y ) + (−6 + 8) = 6 y + 2 32. 5x − 1 + x + 10 = (5x + x) + (−1 + 10) = 6x + 9

ISM: Prealgebra

Chapter 4: Fractions and Mixed Numbers

33. 5 x + 2 − 4 x = 7 − 19 x + 2 = −12 x + 2 − 2 = −12 − 2 x = −14

7 40. 5 39 −35 4 39 4 =7 5 5

34. 9 y + 1 − 8 y = 3 − 20 y + 1 = −17 y + 1 − 1 = −17 − 1 y = −18 35.

36.

17 − 7 x + 3 = −3x + 21 − 3x 20 − 7 x = 21 − 6 x 20 − 7 x + 7 x = 21 − 6 x + 7 x 20 = 21 + x 20 − 21 = 21 − 21 + x −1 = x or x = −1

41.

42 x 6 ⋅ 7 ⋅ x 7 x = = 66 6 ⋅11 11

42.

70 35 ⋅ 2 2 = = 105 y 35 ⋅ 3 ⋅ y 3 y

1 7 10 7 43. 3 ⋅ = ⋅ 3 8 3 8 10 ⋅ 7 = 3⋅8 2⋅5⋅7 = 3⋅ 2 ⋅ 4 5⋅7 = 3⋅ 4 35 11 = or 2 12 12

9 x − 2 = 7 x − 24 9 x − 7 x − 2 = 7 x − 7 x − 24 2 x − 2 = −24 2 x − 2 + 2 = −24 + 2 2 x = −22 2 x −22 = 2 2 x = −11

2 37. Two of five equal parts are shaded: 5

44.

2 2 4 2⋅4 8 2 ⋅4 = ⋅ = = or 2 3 3 1 3 ⋅1 3 3

38. 156 = 2 ⋅ 78 ↓ ↓ 2 ⋅ 2 ⋅ 39 ↓ ↓ ↓ 2 ⋅ 2 ⋅ 3 ⋅ 13

45.

5 3 5 4 5⋅4 5⋅ 4 5 5 ÷ = ⋅ = = = = 16 4 16 3 16 ⋅ 3 4 ⋅ 4 ⋅ 3 4 ⋅ 3 12

46. 1

1 3 11 28 11 5 11 ⋅ 5 11 ÷5 = ÷ = ⋅ = = 10 5 10 5 10 28 5 ⋅ 2 ⋅ 28 56

156 = 22 ⋅ 3 ⋅13

39. a.

2 9 ⋅ 4 + 2 36 + 2 38 = = = 9 9 9 9

8 11 ⋅1 + 8 11 + 8 19 = = = 11 11 11 11

163

Chapter 5 Section 5.1 Practice Exercises 1. a.

986 1000 2 ⋅ 493 = −209 2 ⋅ 500 493 = −209 500

11. −209.986 = −209

0.06 in words is six hundredths.

b. −200.073 in words is negative two hundred and seventy-three thousandths. c.

0.0829 in words in eight hundred twentynine ten-thousandths.

2. 87.31 in words is eighty-seven and thirty-one hundredths. 3. 52.1085 in words is fifty-two and one thousand eighty-five ten-thousandths. 4. The check should be paid to “CLECO,” for the amount of “207.40,” which is written in words as 40 .” “Two hundred seven and 100

5. Five hundred and ninety-six hundredths is 500.96. 6. Thirty-nine and forty-two thousandths is 39.042. 7. 0.051 =

51 1000

8. 29.97 = 29 9. 0.12 =

97 100

12 3⋅ 4 3 = = 100 25 ⋅ 4 25

10. 64.8 = 64

8 2⋅4 4 = 64 = 64 10 2⋅5 5

12. 29.208 26.28 ↑ ↑ 0 < 8 so 26.208 < 26.28 13. 0.12 0.026 ↑ ↑ 1 > 0 so 0.12 > 0.026 14. 0.039 0.0309 ↑ ↑ 9 > 0 so 0.039 > 0.0309 Thus, −0.039 < −0.0309. 15. To round 482.7817 to the nearest thousandth, observe that the digit in the ten-thousandths place is 7. Since this digit is at least 5, we add 1 to the digit in the thousandths place. The number 482.7817 rounded to the nearest thousandth is 482.782. 16. To round −0.032 to the nearest hundredth, observe that the digit in the thousandths place is 2. Since this digit is less than 5, we do not add 1 to the digit in the hundredths place. The number −0.032 rounded to the nearest hundredth is −0.03. 17. To round 3.14159265 to the nearest tenthousandth, observe that the digit in the hundredthousandths place is 9. Since this digit is at least 5, we add 1 to the digit in the ten-thousandths place. The number 3.14159265 rounded to the nearest the ten-thousandth is 3.1416, or π ≈ 3.1416. 18. $24.62 rounded to the nearest dollar is $25, since 6 ≥ 5.

164

ISM: Prealgebra

Chapter 5: Decimals

Vocabulary, Readiness & Video Check 5.1 1. The number “twenty and eight hundredths” is written in words and “20.08” is written in standard form. 2. Another name for the distance around a circle is its circumference. 3. Like fractions, decimals are used to denote part of a whole. 4. When writing a decimal number in words, the decimal point is written as and. 5. The place value tenths is to the right of the decimal point while tens is to the left of the decimal point. 6. The decimal point in a whole number is after the last digit. 7. as “and” 8. 9.8 is nine and eight tenths⎯the 8 should be in the hundredths place; 9.08 9. Reading a decimal correctly gives you the correct place value, which tells you the denominator of your equivalent fraction. 10. left to right 11. When rounding, we look to the digit to the right of the place value we’re rounding to. In this case we look to the hundredths-place digit, which is 7. Exercise Set 5.1 2. 9.57 in words is nine and fifty-seven hundredths. 4. 47.65 in words is forty-seven and sixty-five hundredths. 6. −0.495 in words is negative four hundred ninety-five thousandths. 8. 233.056 in words is two hundred thirty-three and fifty-six thousandths. 10. 5000.02 in words is five thousand and two hundredths. 12. 410.3 in words is four hundred ten and three tenths. 14. 31.04 in words is thirty-one and four hundredths. 16. The check should be paid to “Amani Dupre,” for the amount “513.70,” which is written in words as “Five hundred 70 thirteen and .” 100

165

Chapter 5: Decimals

ISM: Prealgebra

18. The check should be paid to “Kroger,” for the amount of “387.49,” which is written in words as “Three hundred 49 eighty-seven and .” 100

20. Five and one tenth is 5.1. 22. Twelve and six hundredths is 12.06. 24. Negative eight hundred four and three hundred ninety-nine thousandths is −804.399. 26. Eighty-three ten-thousandths is 0.0083. 28. 0.9 =

9 10

30. 0.39 = 32. 0.8 =

39 100

8 2⋅4 4 = = 10 2 ⋅ 5 5

34. 5.4 = 5

4 2⋅2 2 =5 =5 10 2⋅5 5

36. −0.024 = − 38. 9.005 = 9

24 3⋅8 3 =− =− 1000 125 ⋅ 8 125

5 1⋅ 5 1 =9 =9 1000 200 ⋅ 5 200

40. 11.406 = 11 42. 0.2006 =

406 2 ⋅ 203 203 = 11 = 11 1000 2 ⋅ 500 500

2006 2 ⋅1003 1003 = = 10, 000 2 ⋅ 5000 5000

44. Five tenths is 0.5 and as a fraction is

5 1 = . 10 2

46. In words, 0.019 is nineteen thousandths. As a fraction, 0.019 =

166

19 . 1000

ISM: Prealgebra

Chapter 5: Decimals

48. 0.12 0.15 ↑ ↑ 2 < 5 so 0.12 < 0.15

68. To round −0.892 to the nearest hundredth, observe that the digit in the thousandths place is 2. Since this digit is less than 5, we do not add 1 to the digit in the hundredths place. The number −0.892 rounded to the nearest hundredth is −0.89.

50. 0.59 0.52 ↑ ↑ 9 > 2 so 0.59 > 0.52 Thus, −0.59 < −0.52.

70. To round 63.4523 to the nearest thousandth, observe that the digit in the ten-thousandths place is 3. Since this digit is less than 5, we do not add 1 to the digit in the thousandths place. The number 63.4523 rounded to the nearest thousandth is 63.452.

52. 0.0756 0.2 ↑ ↑ 0 < 2 so 0.0756 < 0.2

72. To round π ≈ 3.14159265 to the nearest one, observe that the digit in the tenths place is 1. Since this digit is less than 5, we do not add 1 to the digit in the ones place. The number π ≈ 3.14159265 rounded to the nearest ones is 3.

54. 0.98400 0.984 ↑ ↑ 4 = 4 so 0.98400 = 0.984

74. To round π ≈ 3.14159265 to the nearest hundredthousandth, observe that the digit in the millionths place is 2. Since this digit is less than 5, we do not add 1 to the digit in the hundredthousandths place. The number π ≈ 3.14159265 rounded to the nearest hundred-thousandth is 3.14159.

56. 519.3405 519.3054 ↑ ↑ 4 > 0 so 519.3405 > 519.3054 58. 18.1 18.01 ↑ ↑ 1 > 0 so 18.1 > 18.01 Thus, −18.1 < −18.01. 60.

76. To round 14,769.52 to the nearest one, observe that the digit in the tenths place is 5. Since this digit is at least 5, we add 1 to the digit in the ones place. The number 14,769.52 rounded to the nearest one is 14,770. The amount is $14,770.

0.01 −0.1 ↑ ↑ + > − so 0.01 > −0.1

78. To round 0.7633 to the nearest hundredth, observe that the digit in the thousandths place is 3. Since this digit is less than 5, we do not add 1 to the digit in the hundredths place. The number 0.7633 rounded to the nearest hundredth is 0.76. The amount is $0.76.

62. 0.562 0.652 ↑ ↑ 5 < 6 so 0.562 < 0.652 Thus, −0.562 > −0.652. 64. To round 0.64 to the nearest tenth, observe that the digit in the hundredths place is 4. Since this digit is less than 5, we do not add 1 to the digit in the tenths place. The number 0.64 rounded to the nearest tenth is 0.6. 66. To round 68,934.543 to the nearest ten, observe that the digit in the ones place is 4. Since this digit is less than 5, we do not add 1 to the digit in the tens place. The number 68,934.543 rounded to the nearest ten is 68,930.

80. To round 0.112 to the nearest hundredth, observe that the digit in the thousandths place is 2. Since this digit is less than 5, we do not add 1 to the digit in the hundredths place. The number 0.112 rounded to the nearest hundredth is 0.11. Norway won 0.11 of the medals awarded. 82. To round 3.36 to the nearest tenth, observe that the digit in the hundredths place is 6. Since this digit is at least 5, we add 1 to the digit in the tenths place. The number 3.36 rounded to the nearest tenth is 3.4. The speed is 3.4 mph.

167

Chapter 5: Decimals

ISM: Prealgebra

84. To round 89.61 to the nearest one, observe that the digit in the tenths place is 6. Since this digit is at least 5, we add 1 to the digit in the ones place. The number 89.61 rounded to the nearest one is 90. The amount is $90 billion.

110. a. b.

90.

8 945 + 4 536 13, 481

1. a.

19.520 + 5.371 24.891

40.080 + 17.612 57.692

0.125 + 422.800 422.925

2. a.

34.5670 129.4300 + 2.8903 166.8873

11.210 46.013 + 362.526 419.749

92. To round 146.059 to the nearest ten, observe that the digit in the ones place is 6. Since this digit is at least 5, we add 1 to the digit in the tens place. The number 146.059 rounded to the nearest ten is 150, which is choice d. 94. To round 146.059 to the nearest tenth, observe that the digit in the hundredths place is 5. Since this digit is at least 5, we add 1 to the digit in the tenths place. The number 146.059 rounded to the nearest tenth is 146.1, which is choice b. 96. answers may vary 268 = 17.268 1000

100. 0.00026849576 in words is twenty-six million, eight hundred forty-nine thousand, five hundred seventy-six hundred-billionths. 3.

102. answers may vary 104. answers may vary 106. 0.0612 and 0.0586 rounded to the nearest hundredth are 0.06. 0.066 rounds to 0.07. 0.0506 rounds to 0.05. 108. From smallest to largest, 0.01, 0.0839, 0.09, 0.1

168

Moving the decimal point in 100,800. two places to the left gives 1008. The approximate number of games is 1008 thousand.

Section 5.2 Practice Exercises

4002 − 3897 105

98. 17

4800 21 4 800 96 000 100,800 4800 thousand × 21 = 100,800 thousand ×

86. To round 24.6229 to the nearest thousandth, observe that the digit in the ten-thousandths place is 9. Since this digit is at least 5, we add 1 to the digit in the thousandths place. The number 24.6229 rounded to the nearest thousandth is 24.623. The day length is 24.623 hours. 88.

4820 thousand rounded to the nearest hundred thousand is 4800 thousand.

19.000 + 26.072 45.072

4. 7.12 + (−9.92) Subtract the absolute values. 9.92 − 7.12 2.80 Attach the sign of the larger absolute value. 7.12 + (−9.92) = −2.8

ISM: Prealgebra

Chapter 5: Decimals

5. a.

6.70 − 3.92 2.78

Check:

2.78 + 3.92 6.70

9.720 − 4.068 5.652

Check:

5.652 + 4.068 9.720

6. a.

73.00 − 29.31 43.69

Check:

43.69 + 29.31 73.00

210.00 − 68.22 141.78

Check:

141.78 + 68.22 210.00

25.91 − 19.00 6.91

721.38 148.29 + 152.50 1022.17 The total monthly cost is $1022.17.

15.

71.3 − 70.8 0.5 The average height in Germany is 0.5 inch greater than the average height in Luxembourg.

1. 315.782 + 12.96 = 328.742 2. 29.68 + 85.902 = 115.582 3. 6.249 − 1.0076 = 5.2414 4. 5.238 − 0.682 = 4.556

9. −1.05 − (−7.23) = −1.05 + 7.23 Subtract the absolute values. 7.23 − 1.05 6.18 Attach the sign of the larger absolute value. −1.05 − (−7.23) = 6.18

Exact 58.10 + 326.97 385.07

Estimate 1 60 + 300 360

Estimate 2 60 + 330 390

b. Exact 16.080 − 0.925 15.155

Estimate 1 16 − 1 15

Estimate 2 20 − 1 19

11. y − z = 11.6 − 10.8 = 0.8 12.

14.

Calculator Explorations

8. −5.4 − 9.6 = −5.4 + (−9.6) Add the absolute values. 5.4 + 9.6 15.0 Attach the common sign. −5.4 − 9.6 = −15

10. a.

13. −4.3 y + 7.8 − 20.1y + 14.6 = −4.3 y − 20.1 y + 7.8 + 14.6 = (−4.3 − 20.1) y + (7.8 + 14.6) = −24.4 y + 22.4

y − 4.3 = 7.8 12.1 − 4.3 0 7.8 7.8 = 7.8 True Yes, 12.1 is a solution.

12.555 224.987 5.2 + 622.65 865.392

47.006 0.17 313.259 + 139.088 499.523

Vocabulary, Readiness & Video Check 5.2 1. The decimal point in a whole number is positioned after the last digit. 2. In 89.2 − 14.9 = 74.3, the number 74.3 is called the difference, 89.2 is the minuend, and 14.9 is the subtrahend. 3. To simplify an expression, we combine any like terms. 4. To add or subtract decimals, we line up the decimal points vertically.

169

Chapter 5: Decimals

ISM: Prealgebra

5. True or false: If we replace x with 11.2 and y with −8.6 in the expression x − y, we have 11.2 − 8.6. false

12. Exact:

734.89 + 640.56 1375.45 Estimate: 730 + 640 1370

6. True or false: If we replace x with −9.8 and y with −3.7 in the expression x + y we have −9.8 + 3.7. false 7. Lining up the decimal points also lines up place values, so we only add or subtract digits in the same place values.

14. Exact:

200.89 7.49 + 62.8 271.18 Estimate: 200 7 + 60 267

8. check subtraction by addition 9. so the subtraction can be written vertically with decimal points lined up 10. Two: There are 2 x-terms and there are 2 constants.

16.

65.0000 5.0903 + 6.9003 76.9906

18.

Check:

3.6 + 4.1 7.7

8.9 − 3.1 5.8

5.8 + 3.1 8.9

20.

Check:

5.17 + 3.70 8.87

28.0 − 3.3 24.7

24.7 + 3.3 28.0

22.

863.230 − 39.453 823.777

Check:

823.777 + 39.453 863.230

11. To calculate the amount of border material needed, we are actually calculating the perimeter of the triangle. Exercise Set 5.2 2.

32.4000 1.5800 + 0.0934 34.0734

24. Exact: 6.4 − 3.04 = 3.36 Estimate: 6 − 3 = 3 Check: 3.36 + 3.04 = 6.40

8. −18.2 + (−10.8) Add the absolute values. 18.2 + 10.8 29.0 Attach the common sign. −18.2 + (−10.8) = −29

Check:

28.

Check: 791.1 + 8.9 800.0

2000.00 − 327.47 1672.53 Estimate: 2000 − 300 1700

10. 4.38 + (−6.05) Subtract the absolute values. 6.05 − 4.38 1.67 Attach the sign of the larger absolute value. 4.38 + (−6.05) = −1.67 170

26. Exact:

−

800.0 8.9 791.1

1672.53 + 327.47 2000.00

ISM: Prealgebra

Chapter 5: Decimals

30. −8.63 − 5.6 = −8.63 + (−5.6) Add the absolute values. 8.63 Check: 14.23 + 5.60 − 5.60 14.23 8.63 Attach the common sign. −8.63 − 5.6 = −14.23

46. 40.3 − 700 = 40.3 + (−700) Subtract the absolute values. 700.0 − 40.3 659.7 Attach the sign of the larger absolute value. 40.3 − 700 = −659.7

32. 8.53 − 17.84 = 8.53 + (−17.84) Subtract the absolute values. 17.84 Check: 9.31 − 8.53 + 8.53 9.31 17.84 Attach the sign of the larger absolute value. 8.53 − 17.84 = −9.31

48.

845.9 − 45.8 800.1

50.

100.007 2.080 + 5.014 107.101

34. −9.4 − (−10.4) = −9.4 + 10.4 Subtract the absolute values. 10.4 Check: 1.0 − 9.4 + 9.4 1.0 10.4 Attach the sign of the larger absolute value. −9.4 − (−10.4) = 1 7.000 − 0.097 6.903

Check:

38.

45.0 − 9.2 35.8

Check:

40.

0.7 + 3.4 4.1

36.

6.903 + 0.097 7.000 35.8 + 9.2 45.0

42. −5.05 + 0.88 Subtract the absolute values. 5.05 − 0.88 4.17 Attach the sign of the larger absolute value. −5.05 + 0.88 = −4.17 44.

600.47 − 254.68 345.79

52. −0.004 + 0.085 Subtract the absolute values. 0.085 − 0.004 0.081 Attach the sign of the larger absolute value. −0.004 + 0.085 = 0.081 54. −36.2 − 10.02 = −36.2 + (−10.02) Add the absolute values. 36.20 + 10.02 46.22 Attach the common sign. −36.2 − 10.02 = −46.22 56. −6.5 − (−3.3) = −6.5 + 3.3 Subtract the absolute values. 6.5 − 3.3 3.2 Attach the sign of the larger absolute value. −6.5 − (−3.3) = −3.2 58. y + x = 5 + 3.6 = 8.6 60. y − z = 5 − 0.21 = 4.79 62. x + y + z = 3.6 + 5 + 0.21 = 8.6 + 0.21 = 8.81 64.

x + 5.9 = 8.6 3.7 + 5.9 0 8.6 9.6 = 8.6 False No, 3.7 is not a solution.

171

Chapter 5: Decimals 66.

ISM: Prealgebra

45.9 + z = 23 45.9 + ( −22.9) 0 23 23 = 23 True

86. To find the late fee, add. 23.15 + 6.85 30.00 The average late fee was $30.00.

Yes, −22.9 is a solution. 68.

1.9 − x = x + 0.1 1.9 − 0.9 0 0.9 + 0.1 1 = 1 True Yes, 0.9 is a solution.

88. To find the amount of rainfall in Yuma, add 3.006 to the amount in the Atacama Desert. 3.006 + 0.004 3.010 Yuma receives an average of 3.01 inches of rainfall each year.

70. 14.2 z + 11.9 − 9.6 z − 15.2 = 14.2 z − 9.6 z + 11.9 − 15.2 = (14.2 − 9.6) z + (11.9 − 15.2) = 4.6 z − 3.3 72. −8.96 x − 2.31 − 4.08 x + 9.68 = −8.96 x − 4.08 x − 2.31 + 9.68 = (−8.96 − 4.08) x + (−2.31 + 9.68) = −13.04 x + 7.37 74. Change = 20 − 18.26 20.00 − 18.26 1.74 If Bai paid with two $10 bills, his change was $1.74. 76. Subtract the cost of the frames from the cost of the glasses. 347.89 − 97.23 250.66 The lenses cost $250.66. 78. Perimeter = 4.2 + 5.78 + 7.8 = 17.78 The perimeter is 17.78 inches. 80. Perimeter = 6.43 + 3.07 + 6.43 + 3.07 = 19.0 The perimeter is 19.0 inches. 82. The phrase “how much more” indicates that we should subtract the average rainfall in Omaha from that in New Orleans. 64.16 − 30.08 34.08 Therefore, on average New Orleans receives 34.08 inches more rain than Omaha. 84. To find the increase, subtract. 6.97 − 6.77 0.20 The increase was 0.20 hour per day. 172

90. Add the lengths of the sides to get the perimeter. 15.7 10.6 15.7 + 10.6 52.6 Therefore, 52.6 feet of railing was purchased. 92.

145.290 − 144.416 0.874 The difference is 0.874 mile per hour.

94. The shortest bar indicates the least consumption per person, so Sweden has the least chocolate consumption per person. 96.

17.8 − 17.4 0.4 The difference in consumption is 0.4 pound per year.

98. 23 ⋅ 2 = 46 100. 39 ⋅ 3 = 117 3

1 1 1 1 ⋅1 ⋅ 1 1 1 = 102.   = ⋅ ⋅ = 5 5 5 5 ⋅ 5 ⋅ 5 125 5 104. It is incorrect. After borrowing, there should be 10 tenths. 8 9 910

900.0 − 96.4 803.6

106. 17.67 − (5.26 + 7.82) = 17.67 − 13.08 = 4.59 The unknown length is 4.59 meters.

ISM: Prealgebra

Chapter 5: Decimals

108. 2 quarters, 3 dimes, 4 nickels, and 2 pennies: 0.25 + 0.25 + 0.10 + 0.10 + 0.10 + 0.05 + 0.05 + 0.05 + 0.05 + 0.01 + 0.01 = 1.02 The value of the coins shown is $1.02. 110. 5 nickels: 0.05 + 0.05 + 0.05 + 0.05 + 0.05 = 0.25 1 dime and 3 nickels: 0.10 + 0.05 + 0.05 + 0.05 = 0.25 2 dimes and 1 nickel: 0.10 + 0.10 + 0.05 = 0.25 1 quarter: 0.25 112.

256, 436.012 − 256, 435.235 0.777 The difference in measurements is 0.777 mile.

6. 203.004 × 100 = 20,300.4 7. (−2.33)(1000) = −2330 8. 6.94 × 0.1 = 0.694 9. 3.9 × 0.01 = 0.039 10. (−7682)(−0.001) = 7.682 11. 61.45 million = 61.45 × 1 million = 61.45 × 1, 000, 000 = 61, 450, 000 12. 7y = 7(−0.028) = −0.196 13.

114. no; answers may vary 116. 14.271 − 8.968x + 1.333 − 201.815x + 101.239x = −8.968x − 201.815x + 101.239x + 14.271 + 1.333 = (−8.968 − 201.815 + 101.239)x + (14.271 + 1.333) = −109.544x + 15.604 Section 5.3 Practice Exercises 1.

34.8 × 0.62 696 20 880 21.576

2. 0.0641 × 27 4487 1 2820 1.7307

1 decimal place 2 decimal places

Yes, −5.5 is a solution. 14. C = 2πr = 2π ⋅ 11 = 22π ≈ 22(3.14) = 69.08 The circumference is 22π meters ≈ 69.08 meters. 15.

60.5 × 5.6 36 30 302 50 338.80 338.8 ounces of fertilizer are needed.

Vocabulary, Readiness & Video Check 5.3 1 + 2 = 3 decimal places 4 decimal places 0 decimal places

4 + 0 = 4 decimal places

30.26 × 2.89 2 7234 24 2080 60 5200 87.4514

5. 46.8 × 10 = 468

1. When multiplying decimals, the number of decimal places in the product is equal to the sum of the number of decimal place in the factors. 2. In 8.6 × 5 = 43, the number 43 is called the product while 8.6 and 5 are each called a factor.

3. (7.3)(−0.9) = −6.57 (Be sure to include the negative sign.) 4. Exact:

−6 x = 33 −6(−5.5) 0 33 33 = 33 True

Estimate:

30 × 3 90

3. When multiplying a decimal number by powers of 10 such as 10, 100, 1000, and so on, we move the decimal point in the number to the right the same number of places as there are zeros in the power of 10. 4. When multiplying a decimal number by powers of 10 such as 0.1, 0.01, and so on, we move the decimal point in the number to the left the same number of places as there are decimal places in the power of 10. 5. The distance around a circle is called its circumference.

173

Chapter 5: Decimals

ISM: Prealgebra

6. When multiplying decimals, we need to learn where to place the decimal point in the product.

14. Exact:

7. knowing whether we placed the decimal point correctly in our product 8. We just need to know how to move the decimal point. 100 has two zeros, so we move the decimal point two places to the right.

16.

9. 3(5.7) − (−0.2) 10. We used an approximation for π. The exact answer is 10π cm. 11. This is an application problem and needs units attached. The complete answer is 24.8 grams. Exercise Set 5.3 2.

2 decimal places 0 decimal places 2 + 0 = 2 decimal places

6.8 × 0.3 2.04

1 decimal place 1 decimal place 1 + 1 = 2 decimal places

174

1 decimal place 3 decimal places

1 + 3 = 4 decimal places

18. 7.2 × 100 = 720 20. 23.4 × 0.1 = 2.34

26. (−4.72)(−0.01) = 0.0472 28. 36.41 × 0.001 = 0.03641 30.

8. The product (−7.84)(−3.5) is positive. 7.84 2 decimal places × 3.5 1 decimal place 3 920 23 520 27.440 2 + 1 = 3 decimal places 8.3 × 2.7 5 81 16 60 22.41

0.864 × 0.4 0.3456

2 × 6 12

24. 0.5 × 100 = 50

6. The product (4.7)(−9.02) is negative. 9.02 2 decimal places × 4.7 1 decimal place 6 314 36 080 −42.394 2 + 1 = 3 decimal places and include the negative sign.

12.

300.9 × 0.032 6018 9 0270 9.6288

Estimate:

22. (−1.123)(1000) = −1123

0.23 × 9 2.07

10. Exact:

2.0005 × 5.5 1 00025 10 00250 11.00275

Estimate:

3 decimal places 1 decimal place 3 + 1 = 4 decimal places

8 ×3 24

0.216 × 0.3 0.0648

32. (345.2)(100) = 34,520 34.

0.42 × 5.7 294 2 100 2.394

36. (−0.001)(562.01) = −0.56201 38. 993.5 × 0.001 = 0.9935 40.

9.21 × 3.8 7 368 27 630 34.998

42. 48.3 million = 48.3 × 1 million = 48.3 × 1, 000, 000 = 48,300, 000 About 48,300,000 American households own at least one dog.

ISM: Prealgebra

Chapter 5: Decimals

44. 343.8 billion = 343.8 ×1 billion = 343.8 × 1, 000, 000, 000 = 343,800, 000, 000 The restaurant industry had projected online delivery sales of $343,800,000,000 in 2022. 46. yz = (−0.2)(5.7) = −1.14 48. −5y + z = −5(−0.2) + 5.7 = 1.0 + 5.7 = 6.7 50.

52.

100 z = 14.14 100(1414) 0 14.14 141, 400 = 14.14 False No, 1414 is not a solution.

64. Circumference = 2 ⋅ π ⋅ radius C = 2 ⋅ π ⋅ 3950 = 7900π 7900 × 3.14 316 00 790 00 23700 00 24806.00 The circumference is 7900π miles, which is approximately 24,806 miles. 66. C = π ⋅ d C = π ⋅ 80 = 80π 3.14 × 80 251.20 The circumference is 80π meters or approximately 251.2 meters.

0.7 x = −2.52 0.7(−3.6) 0 − 2.52 −2.52 = −2.52 True

Yes, −3.6 is a solution. 54. C = πd is π(22 inches) = 22π inches C ≈ 22(3.14) inches = 69.08 inches 56. C = 2πr is 2π ⋅ 5.9 kilometers = 11.8π kilometers C ≈ 11.8π = (11.8)(3.14) kilometers = 37.052 kilometers

68. Volume = length ⋅ width ⋅ height = 16 ⋅ 7.6 ⋅ 2 = 243.2 The volume of the box is 243.2 cubic inches. 70. a.

58. pay before taxes = 19.52 × 20 19.52 × 20 390.40 The worker’s pay for last week was $390.40. 60. Multiply the number of servings by the number of grams of saturated fat in 3.5 ounces. 3 × 0.1 = 0.3 There is 0.3 gram of saturated fat in 3 servings. 62. Area = length ⋅ width 6.43 × 3.1 643 19 290 19.933 The area is 19.933 square inches.

Circumference = π ⋅ diameter Smaller circle: C = π ⋅ 16 = 16π C ≈ 16(3.14) = 50.24 The circumference of the smaller circle is approximately 50.24 inches. Larger circle: C = π ⋅ 32 = 32π C ≈ 32(3.14) = 100.48 The circumference of the larger circle is approximately 100.48 inches.

b. Yes, the circumference gets doubled when the diameter is doubled. 72. 6.7775 × 10,000 = 67,775 They would pay $67,775 for 10,000 bushels of corn. 74.

6.7918 × 300 2037.5400 You would receive 2037.54 Chinese yuan for 300 American dollars.

175

Chapter 5: Decimals 76.

ISM: Prealgebra 92. answers may vary

1.0722 130 32 1660 107 2200 139.3860 The tourist spent 139.39 euros. ×

Section 5.4 Practice Exercises 1. 8

78. Multiply the doorway height in meters by the number of inches in 1 meter. 39.37 × 2.15 1 9685 3 9370 78 7400 84.6455 The doorway is approximately 84.6455 inches.

2. 48

8 80. 365 2920 −2920 0

82.

84.

3. a.

162 9 162 75 ÷− = ⋅− 25 75 25 9 9 ⋅18 ⋅ 25 ⋅ 3 =− 25 ⋅ 9 18 ⋅ 3 =− 1 = −54 7.20 0.14 + 98.60 105.94

86.

100.0 − 48.6 51.4

88.

−3.6 × 0.04 −0.144

0.71 34.08 −33 6 48 −48 0 1.135 14 15.890 −14 18 −1 4 49 −42 70 −70 0

Check: 46.3 × 8 370.4

Check:

0.71 × 48 5 68 28 40 34.08

Check: 1.135 × 14 4 540 11 350 15.890

Thus, −15.89 ÷ 14 = −1.135. b.

90. 8.3 minutes = 8.3 × 60 seconds = 498 seconds 498 × 1.86 29 88 398 40 498 00 926.28 926.28 × 100,000 = 92,628,000 The radio wave travels 92,628,000 miles. 176

46.3 370.4 −32 50 −48 24 −2 4 0

0.027 104 2.808 −2 08 728 −728 0

Check: 0.027 × 104 108 2 700 2.808 Thus, −2.808 ÷ (−104) = 0.027

29.8 4. 5.6 166.88 becomes 56 1668.8 −112 548 −504 44 8 −44 8 0

ISM: Prealgebra

Chapter 5: Decimals

12.35 5. 0.16 1.976 becomes 16 197.60 −16 37 −32 56 −48 80 −80 0 41.052 ≈ 41.05 6. 0.57 23.4 becomes 57 2340.000 −228 60 −57 3 00 −2 85 150 −114 36 7.9 7. 91.5 722.85 becomes 915 7228.5 −6405 823 5 −823 5 0

8 Estimate: 90 720 8.

11.84 12. 1250 14800.00 −1250 2300 −1250 1050 0 −1000 0 50 00 −50 00 0 He needs 11.84 bags or 12 whole bags.

Calculator Explorations 1. 102.62 × 41.8 ≈ 100 × 40 = 4000 Since 4000 is not close to 428.9516, it is not reasonable. 2. 174.835 ÷ 47.9 ≈ 200 ÷ 50 = 4 Since 4 is close to 3.65, it is reasonable. 3. 1025.68 − 125.42 ≈ 1000 − 100 = 900 Since 900 is close to 900.26, it is reasonable.

Vocabulary, Readiness & Video Check 5.4 1. In 6.5 ÷ 5 = 1.3, the number 1.3 is called the quotient, 5 is the divisor, and 6.5 is the dividend.

0.49 = −0.049 10

10. x ÷ y = 0.035 ÷ 0.02 0.02 0.035 becomes 2

x = 3.9 100 39 0 3.9 100 0.39 = 3.9 False No, 39 is not a solution.

4. 562.781 + 2.96 ≈ 563 + 3 = 566 Since 566 is not close to 858.781, it is not reasonable.

362.1 = 0.3621 1000

9. −

11.

1.75 3.50 −2 15 −1 4 10 −10 0

2. To check a division exercise, we can perform the following multiplication: quotient ⋅ divisor = dividend. 3. To divide a decimal number by a power of 10 such as 10, 100, 1000, or so on, we move the decimal point in the number to the left the same number of places as there are zeros in the power of 10. 4. True or false: If we replace x with −12.6 and y with 0.3 in the expression y ÷ x, we have 0.3 ÷ (−12.6) true

177

Chapter 5: Decimals

ISM: Prealgebra

5. a whole number 6. deciding if our decimal point is in the correct place in the quotient 7. We just need to know how to move the decimal point. 1000 has three zeros, so we move the decimal point in the decimal number three places to the left. 8. This actually means 4 divides into 1 zero times, so we place a zero in the quotient. 9. We want the answer rounded to the nearest tenth, so we go to one extra place value, to the hundredths place, in order to round. Exercise Set 5.4 5.9 2. 4 23.6 −20 36 −3 6 0

11 Estimate: 2 22 0.519 12. 16 8.304 −8 0 30 −16 144 −144 0

14. A positive number divided by a negative number is a negative number. 900 0.04 36 becomes 4 3600 −36 000

0.085 4. 6 0.510 −48 30 −30 0

36 ÷ (−0.04) = −900 16. A negative number divided by a negative number is a positive number. 4 0.9 3.6 becomes 9 36 −36 0

500 6. 0.04 20 becomes 4 2000 −20 000

(−3.6) ÷ (−0.9) = 4

4.9 8. 0.36 1.764 becomes 36 176.4 −144 32 4 −22 4 0

178

9.9 10. Exact: 2.2 21.78 becomes 22 217.8 −198 19 8 −19 8 0

6.4 18. 0.34 2.176 becomes 34 217.6 −204 13 6 −13 6 0 800 20. 0.03 24 becomes 3 2400 −24 000

ISM: Prealgebra

Chapter 5: Decimals

3.6 22. 0.92 3.312 becomes 92 331.2 −276 55 2 −55 2 0

24. A negative number divided by a negative number is a positive number. 2.2 9.9 21.78 becomes 99 217.8 −198 19 8 −19 8 0

−21.78 ÷ (−9.9) = 2.2 8.6 26. Exact: 6.3 54.18 becomes 63 541.8 −504 37 8 −37 8 0

9 Estimate: 6 54 4.4 28. 7.7 33.88 becomes 77 338.8 −308 30 8 −30 8 0 63 30. 0.051 3.213 becomes 51 3213 −306 153 −153 0 3.213 = 63 0.051

66.488 32. 0.75 49.866 becomes 75 4986.600 −450 486 −450 36 6 −30 0 6 60 −6 00 600 −600 0 0.0045 34. 2.96 0.01332 becomes 296 1.3320 −1 184 1480 −1480 0 0.0462 ≈ 0.046 36. 0.98 0.0453 becomes 98 4.5300 −3 92 610 −588 220 −196 24 28.23 ≈ 28.2 38. 3.5 98.83 becomes 35 988.30 −70 288 −280 83 −7 0 1 30 −1 05 25

40.

64.423 = 0.64423 100

42.

13.49 = 1.349 10

44. 13.49 ÷ (−10,000) = −0.001349

179

Chapter 5: Decimals

ISM: Prealgebra

50, 000 62. 0.0007 35 becomes 7 350, 000 −35 00 000

14.5 46. 9 130.5 −9 40 −36 45 −4 5 0

48.

17.7 = 1.77 10

50.

986.11 = 0.098611 10, 000

35 ÷ (−0.0007) = −50,000 2.2 64. 0.98 2.156 becomes 98 215.6 −196 19 6 −19 6 0

−2.156 ÷ 0.98 = −2.2 66. −86.79 ÷ (−1000) =

0.045 52. 12 0.540 −48 60 −60 0

680 68. 0.035 23.8 becomes 35 23800 −210 280 −280 00

7 49 −49 0

54. 0.7 4.9 becomes 7

23.8 = 680 0.035

70. z ÷ x = 4.52 ÷ 5.65

4.9 ÷ (−0.7) = −7 3.2 56. 0.42 1.344 becomes 42 134.4 −126 84 −8 4 0

−1.344 ÷ 0.42 = −3.2 900 58. 0.03 27 becomes 3 2700 −27 000

27 ÷ 0.03 = 900 60. 0.4 20 becomes 4

5.65 4.52 becomes 565

z ÷ x = 4.52 ÷ 5.65 = 0.8 72. y ÷ 2 = −0.8 ÷ 2 0.4 2 0.8 −8 0

y ÷ 2 = −0.8 ÷ 2 = −0.4 74.

50 200 −20 00

y = 0.89 8 7.12 0 0.89 8 0.89 = 0.89 True Yes, 7.12 is a solution.

20 ÷ 0.4 = 50

180

−86.79 = 0.08679 −1000

0.8 452.0 −452 0 0

ISM: Prealgebra

76.

Chapter 5: Decimals

x = 0.23 10 23 0 0.23 10 2.3 = 0.23 False No, 23 is not a solution.

78. Number of boxes = 486 ÷ 36 13.5 36 486.0 −36 126 −108 18 0 −18 0 0 Since whole boxes must be used, 14 boxes are needed. 19.68 ≈ 19.7 80. 2.54 50 becomes 254 5000.00 −254 2460 −2286 174 0 −152 4 21 60 −20 32 1 28 There are approximately 19.7 inches in 50 centimeters.

82. Divide the price per hundred pounds by 100. 121.20 = 1.2120 ≈ 1.21 100 The average price per pound was $1.21. 84. Each dose is 0.5 teaspoon. 48 0.5 24 becomes 5 240 −20 40 −40 0 There are 48 doses in the bottle.

86. From Exercise 83, we know there are 24 teaspoons in 4 fluid ounces. Thus, there are 48 half teaspoons (0.5 tsp) or doses in 4 fluid ounces. To see how long the medicine will last, if a dose is taken every 6 hours, there are 24 ÷ 6 = 4 doses taken per day. 48 (doses) ÷ 4 (per day) = 12 days. The medicine will last 12 days. 21.90 ≈ 21.9 88. 19.35 423.8 becomes 1935 42380.00 −3870 3680 −1935 17450 −17415 350 Drake’s car averages about 21.9 miles per gallon. 345.5 90. 24 8292.0 −72 109 −96 132 −120 12 0 −12 0 0 There were 345.5 thousand books sold each hour.

92.

3 7 3 10 3 ⋅10 3 ⋅ 5 ⋅ 2 6 ÷ = ⋅ = = = 5 10 5 7 5⋅7 5⋅7 7

3 1 3 7 1 2 94. − − = − ⋅ − ⋅ 4 14 4 7 14 2 3 ⋅ 7 1⋅ 2 =− − 4 ⋅ 7 14 ⋅ 2 21 2 =− − 28 28 23 =− 28 96.

1.278 × 0.3 0.3834

3 decimal places 1 decimal place 3 + 1 = 4 decimal places

181

Chapter 5: Decimals 98.

1.278 − 0.300 0.978

100.

7.20 0.05 + 49.10 56.35

102.

ISM: Prealgebra

134.78 ≈ 134.8 1.15 155 becomes 115 15500.00 −115 400 −345 550 −460 90 0 −805 950 −9 20 30

87.2 = −0.00872 −10, 000

104. 1.437 + 20.69 is approximately 1 + 21 = 22, which is choice b. 106. 302.729 − 28.697 is approximately 300 − 30 = 270, which is choice a. 108.

56 + 75 + 80 211 = ≈ 70.3 3 3

45.2 180.8 −16 20 −20 08 −8 0 The length of a side is 45.2 centimeters.

110. 4

112. answers may vary 113.91 ≈ 113.9 114. 1.15 131 becomes 115 13100.00 −115 160 −115 450 −345 1050 −1035 150 −115 35

182

The range of wind speeds is 113.9 − 134.8 knots. 13.6 116. 185.35 2523.86 becomes 18535 252386.0 −18535 67036 −55605 11431 0 −11121 0 310 0 No; it will take the student more than 13 months to pay it off at the minimum rate.

Mid-Chapter Review 1.

1.60 + 0.97 2.57

3.20 + 0.85 4.05

9.8 − 0.9 8.9

10.2 − 6.7 3.5

0.8 × 0.2 0.16

ISM: Prealgebra 6.

Chapter 5: Decimals

0.6 × 0.4 0.24

8.6 15. 3.4 29.24 becomes 34 292.4 −272 20 4 −20 4 0

0.27 7. 8 2.16 −1 6 56 −56 0

8. 6

10.

11.

12.

29.24 ÷ (−3.4) = −8.6 5.4 16. 1.9 10.26 becomes 19 102.6 −95 76 −7 6 0

0.52 3.12 −3 0 12 −12 0

−10.26 ÷ (−1.9) = 5.4 17. −2.8 × 100 = −280 18. 1.6 × 1000 = 1600

9.6 × 0.5 4.80 (9.6)(−0.5) = −4.8 8.7 × 0.7 6.09 (−8.7)(−0.7) = 6.09 123.60 − 48.04 75.56

19.

96.210 7.028 + 121.700 224.938

20.

0.268 1.930 + 142.881 145.079

21. 46

325.20 − 36.08 289.12

13. Subtract absolute values. 25.000 − 0.026 24.974 Attach the sign of the larger absolute value. −25 + 0.026 = −24.974 14. Subtract absolute values. 44.000 − 0.125 43.875 Attach the sign of the larger absolute value. 0.125 + (−44) = −43.875

0.56 25.76 −23 0 2 76 −2 76 0

−25.76 ÷ (−46) = 0.56 0.63 22. 43 27.09 −25 8 1 29 −1 29 0

−27.09 ÷ 43 = −0.63

183

Chapter 5: Decimals

23.

12.004 2.3 3 6012 24 0080 27.6092

3 decimal places 1 decimal place

28.006 5.2 5 6012 140 0300 145.6312

3 decimal places 1 decimal place

24.

3 + 1 = 4 decimal places

25.

ISM: Prealgebra

28 30. 0.061 1.708 becomes 61 1708 −122 488 −488 0 −1.708 = −28 0.061

3 + 1 = 4 decimal places

10.0 − 4.6 5.4

26. Subtract absolute values. 18.00 − 0.26 17.74 Attach the sign of the greater absolute value. 0.26 − 18 = −17.74 27. −268.19 − 146.25 = −268.19 + (−146.25) Add absolute values. 268.19 + 146.25 414.44 Attach the common sign. −268.19 − 146.25 = −414.44

31.

160.00 − 43.19 116.81

32.

120.00 − 101.21 18.79

33. 15.62 × 10 = 156.2 34. 15.62 ÷ 10 = 1.562 35.

15.62 + 10.00 25.62

36.

15.62 − 10.00 5.62

37.

53.7 79.2 + 71.2 204.1 The exact distance is 204.1 miles. 50 80 + 70 200 The estimated distance is 200 miles.

38.

16.75 − 13.85 2.90 It costs $2.90 more to send the package as Priority Mail.

39.

18.8 + 21.8 40.6 The total number of vinyl albums shipped was 40.6 million.

28. −860.18 − 434.85 = −860.18 + (−434.85) Add absolute values. 860.18 + 434.85 1295.03 Attach the common sign. −860.18 − 434.85 = −1295.03 34 29. 0.087 2.958 becomes 87 2958 −261 348 −348 0 2.958 = −34 −0.087

184

ISM: Prealgebra

Chapter 5: Decimals

Section 5.5 Practice Exercises 1. a.

0.4 5 2.0 −2.0 0

0.225 40 9.000 −8 0 1 00 −80 200 −200 0

0.375 2. 8 3.000 −2 4 60 −56 40 −40 0

3. a.

0.833... 6 5.000 −4 8 20 −18 20 −18 2 0.22... 9 2.00 −1 8 20 −18 2

2 = 0.4 5

9 = 0.225 40

2.1538 ≈ 2.154 4. 13 28.0000 −26 20 −1 3 70 −65 50 −39 110 −104 6 5 53 = 16 16 3.3125 16 53.0000 −48 50 −4 8 20 −16 40 −32 80 −80 0

5. 3 3 − = −0.375 8

5 = 0.83 6

Thus, 3

2 = 0.2 9

5 = 3.3125. 16

3 3 2 6 = ⋅ = = 0.6 5 5 2 10

3 3 2 6 = ⋅ = = 0.06 50 50 2 100

0.2 8. 5 1.0 −1 0 0

Since 0.2 < 0.25, then

1 < 0.25. 5

185

Chapter 5: Decimals

9. a.

ISM: Prealgebra

1 1 = 0.5 and 0.5 < 0.54, so < 0.54. 2 2 9

0.55... 5.00 −4 5 50 −45 5

20.06 − (1.2)2 ÷ 10 20.06 − 1.44 ÷ 10 = 0.02 0.02 20.06 − 0.144 = 0.02 19.916 = 0.02 = 995.8 1 ⋅ base ⋅ height 2 1 = ⋅ 7 ⋅ 2.1 2 = 0.5 ⋅ 7 ⋅ 2.1 = 7.35 The area of the triangle is 7.35 square meters.

15. Area =

0.714 7 5.000 −4 9 10 −7 30 −28 2

0.714 < 0.72, so

13. (−0.7)2 + 2.1 = 0.49 + 2.10 = 2.59 14.

5 0.5 = 0.55..., so 0.5 = . 9

12. −8.69(3.2 − 1.8) = −8.69(1.4) = −12.166

16. 1.7y − 2 = 1.7(2.3) − 2 = 3.91 − 2 = 1.91 5 < 0.72. 7

Vocabulary, Readiness & Video Check 5.5 1. The number 0.5 means 0.555. false

10. a.

1 = 0.333... 3 0.302 = 0.302 3 = 0.375 8 1 3 0.302, , 3 8

9 as a decimal, perform the division 19 19 9. true

2. To write

3. (−1.2)2 means (−1.2)(−1.2) or −1.44. false 4. To simplify 8.6(4.8 − 9.6), we first subtract. true

b. 1.26 = 1.26 1 1 = 1.25 4 2 1 = 1.40 5 1 2 1 , 1.26, 1 5 4 c.

5. We place a bar over just the repeating digits and only 6 repeats in our decimal answer. 6. It is easier to compare decimal numbers. 7. The fraction bar serves as a grouping symbol. 8. A = l ⋅ w; 0.248 sq yd

0.4 = 0.40 0.41 = 0.41 3 ≈ 0.43 7 3 0.4, 0.41, 7

9. 4(0.3) − (−2.4) Exercise Set 5.5

11. 897.8 ÷ 100 × 10 = 8.978 × 10 = 89.78

186

0.05 2. 20 1.00 −1 00 0

1 = 0.05 20

ISM: Prealgebra

4. 25

0.52 13.00 −12 5 50 −50 0

0.125 6. 8 1.000 −8 20 −16 40 −40 0

8. 25

0.12 3.00 −2 5 50 −50 0

1.6 10. 5 8.0 −5 30 −3 0 0 0.4166... 12. 12 5.0000 −4 8 20 −12 80 −72 80 −72 8

14. 25

0.76 19.00 −17 5 1 50 −1 50 0

Chapter 5: Decimals

13 = 0.52 25

1 = 0.125 8

−

3 = −0.12 25

8 = 1.6 5

5 = 0.416 12

16. 40

0.775 31.000 − 28 0 3 00 −2 80 200 −200 0

0.777... 18. 9 7.000 −6 3 70 −63 70 −63 7 0.5625 20. 16 9.0000 −8 0 1 00 −96 40 −32 80 −80 0 0.8181... 22. 11 9.0000 −8 8 20 −11 90 −88 20 −11 9

31 = 0.775 40

−

7 = −0.7 9

9 = 0.5625 16

9 = 0.81 11

19 = 0.76 25

187

Chapter 5: Decimals

0.875 24. 8 7.000 −6 4 60 −56 40 −40 0 0.424 26. 375 159.000 −150 0 9 00 −7 50 1 500 −1 500 0

28. −

7 = −0.777 ≈ −0.78 9

30.

9 = 0.5625 ≈ 0.56 16

32.

9 = 0.8181 ≈ 0.8 11

0.45 34. 20 9.00 −8 0 1 00 −1 00 0 0.054 36. 125 8.000 −7 50 500 −500 0

188

ISM: Prealgebra

7 = 4.875 8

159 = 0.424 375

0.00000125 38. 800, 000 1.00000000 −800000 2000000 −1600000 4000000 −4000000 0

40. 0.983 0.988 ↑ ↑ 3 < 8 so 0.983 < 0.988 0.725 42. 40 29.000 −28 0 1 00 −80 200 −200 0 29 = 0.725 40

44. 0.00563 0.00536 ↑ ↑ 6 > 3 so, 0.00563 > 0.00536 Thus, −0.00563 < −0.00536. 0.117 ≈ 0.12 46. 17 2.000 −1 7 30 −17 130 −119 11 2 2 ≈ 0.12 and 0. 1 < 0.12, so 0. 1 < . 17 17

ISM: Prealgebra

Chapter 5: Decimals

0.5454... 48. 11 6.0000 −5 5 50 −44 60 −55 50 −44 6

54. 7

6 6 = 0.54 and 0.583 > 0.54, so 0.583 > . 11 11 0.555... 50. 9 5.000 −4 5 50 −45 50 −45 5

12.7142 ≈ 12.714 89.0000 −7 19 −14 50 −4 9 10 −7 30 −28 20 −14 6

89 ≈ 12.714 and 12.713 < 12.714, so 7 89 12.713 < . 7

56. 0.40, 0.42, 0.47

5 5 = 0.555 and 0.555 < 0.557, so < 0.557. 9 9

0.37288 ≈ 0.3729 52. 59 22.00000 −17 7 4 30 −4 13 170 −118 520 −472 480 −472 8

58. 0.72 = 0.720 0.72, 0.727, 0.728 60. 7.56 = 7.560 67 = 7.444... 9 7.562 = 7.562 67 , 7.56, 7.562 9 62.

5 = 0.833 6 5 0.821, , 0.849 6

64. (−2.5)(3) − 4.7 = −7.5 − 4.7 = −12.2

22 ≈ 0.3729 and 0.372 < 0.3729, so 59 22 0.372 < . 59

66. (−0.05) 2 + 3.13 = 0.0025 + 3.13 = 3.1325 68. (8.2)(100) − (8.2)(10) = 820 − 82 = 738 70.

0.222 − 2.13 −1.908 = = −0.159 12 12

72. 0.9(5.6 − 6.5) = 0.9(−0.9) = −0.81 74. (1.5) 2 + 0.5 = 2.25 + 0.5 = 2.75

189

Chapter 5: Decimals

ISM: Prealgebra

76.

3 3 − (9.6)(5) = − 48 = 0.75 − 48 = −47.25 4 4

78.

3 3 (4.7 − 5.9) = (−1.2) = 0.375(−1.2) = −0.45 8 8

1 ⋅b⋅h 2 1 = ⋅17 ⋅ 4.4 2 = 0.5 ⋅17 ⋅ 4.4 = 37.4 square feet

80. Area =

0.650 ≈ 0.65 102. 15,371 10000.000 −9222 6 777 40 − 768 55 8 550 The top six formats accounted for about 0.65 of radio stations.

82. Area = l ⋅ w 7 = (1.2)   8 = (1.2)(0.875) = 1.05 square miles

84. y 2 = (0.3)2 = 0.09

104. answers may vary

86. x − z = 6 − (−2.4) = 6 + 2.4 = 8.4 88.

x 6 + 2z = + 2(−2.4) = 20 − 4.8 = 15.2 y 0.3

90.

4 19 4 2 19 8 19 11 1 − = ⋅ − = − =− =− 11 22 11 2 22 22 22 22 2 2

94. 1.0000 = 1 96. 101 > 99, so

190

99 =1 99

Section 5.6 Practice Exercises 1.

2 3  2 2  3 3 3  92.     =  ⋅   ⋅ ⋅  3 2      3 3  2 2 2  2 ⋅ 2 ⋅3⋅3⋅3 = 3⋅3⋅ 2 ⋅ 2 ⋅ 2 3 = 2

98.

0.0817 ≈ 0.082 100. 15,371 1257.0000 −1229 68 27 320 −15 371 11 9490 −10 7597 1 1893 Approximately 0.08 of radio stations had a Spanish format.

101 >1 99

z + 0.9 = 1.3 z + 0.9 − 0.9 = 1.3 − 0.9 z = 0.4 Check: z + 0.9 = 1.3 0.4 + 0.9 0 1.3 1.3 = 1.3 True The solution is 0.4.

2. 0.17 x = −0.34 0.17 x −0.34 = 0.17 0.17 x = −2 Check: 0.17 x = −0.34 0.17( −2) 0 − 0.34 −0.34 = −0.34 True

The solution is −2. 3.

2.9 = 1.7 + 0.3x 2.9 − 1.7 = 1.7 + 0.3x − 1.7 1.2 = 0.3 x 1.2 0.3 x = 0.3 0.3 4=x The solution is 4.

ISM: Prealgebra 4.

Chapter 5: Decimals

8 x + 4.2 = 10 x + 11.6 8 x + 4.2 − 4.2 = 10 x + 11.6 − 4.2 8 x = 10 x + 7.4 8 x − 10 x = 10 x − 10 x + 7.4 −2 x = 7.4 −2 x 7.4 = −2 −2 x = −3.7 The solution is −3.7.

8. 3 y = −25.8 3 y −25.8 = 3 3 y = −8.6

6.3 − 5 x = 3( x + 2.9) 6.3 − 5 x = 3 x + 8.7 6.3 − 5 x − 6.3 = 3 x + 8.7 − 6.3 −5 x = 3 x + 2.4 −5 x − 3x = 3 x + 2.4 − 3 x −8 x = 2.4 −8 x 2.4 = −8 −8 x = −0.3 The solution is −0.3.

0.2 y + 2.6 = 4 10(0.2 y + 2.6) = 10(4) 10(0.2 y ) + 10(2.6) = 10(4) 2 y + 26 = 40 2 y + 26 − 26 = 40 − 26 2 y = 14 2 y 14 = 2 2 y=7 The solution is 7.

Vocabulary, Readiness & Video Check 5.6

10.

7.1 − 0.2 x = 6.1 7.1 − 0.2 x − 7.1 = 6.1 − 7.1 −0.2 x = −1 −0.2 x −1 = −0.2 −0.2 x=5

12.

7 x − 9.64 = 5 x + 2.32 7 x − 9.64 + 9.64 = 5 x + 2.32 + 9.64 7 x = 5 x + 11.96 7 x − 5 x = 5 x − 5 x + 11.96 2 x = 11.96 2 x 11.96 = 2 2 x = 5.98

14.

5( x + 2.3) = 19.5 5 x + 11.5 = 19.5 5 x + 11.5 − 11.5 = 19.5 − 11.5 5x = 8 5x 8 = 5 5 x = 1.6

16.

0.7 x + 0.1 = 1.5 10(0.7 x + 0.1) = 10(1.5) 7 x + 1 = 15 7 x + 1 − 1 = 15 − 1 7 x = 14 7 x 14 = 7 7 x=2

18.

3 y = 7 y + 24.4 10 ⋅ 3 y = 10(7 y + 24.4) 30 y = 70 y + 244 30 y − 70 y = 70 y − 70 y + 244 −40 y = 244 −40 y 244 = −40 −40 y = −6.1

1. so that we are no longer working with decimals 2. We would have avoided working with a negative coefficient⎯subtracting 2x would have given us a positive coefficient for x. Exercise Set 5.6 2.

y − 0.5 = 9 y − 0.5 + 0.5 = 9 + 0.5 y = 9.5

4. −0.4 x = 50 −0.4 x 50 = −0.4 −0.4 x = −125

9.7 = x + 11.6 9.7 − 11.6 = x + 11.6 − 11.6 −1.9 = x

191

Chapter 5: Decimals 20.

1.5 x + 2 − 1.2 x = 12.2 10(1.5 x + 2 − 1.2 x) = 10 ⋅12.2 15 x + 20 − 12 x = 122 3 x + 20 = 122 3 x + 20 − 20 = 122 − 20 3 x = 102 3 x 102 = 3 3 x = 34

22.

x + 5.7 = 8.4 x + 5.7 − 5.7 = 8.4 − 5.7 x = 2.7

ISM: Prealgebra

24. −9 y = −0.162 −9 y −0.162 = −9 −9 y = 0.018 26.

2 x − 4.2 = 8.6 2 x − 4.2 + 4.2 = 8.6 + 4.2 2 x = 12.8 2 x 12.8 = 2 2 x = 6.4

28.

9 y − 6.9 = 6 y − 11.1 9 y − 6.9 + 6.9 = 6 y − 11.1 + 6.9 9 y = 6 y − 4.2 9 y − 6 y = 6 y − 6 y − 4.2 3 y = −4.2 3 y −4.2 = 3 3 y = −1.4

30.

32.

192

2.3 + 500 x = 600 x − 0.2 2.3 − 2.3 + 500 x = 600 x − 0.2 − 2.3 500 x = 600 x − 2.5 500 x − 600 x = 600 x − 600 x − 2.5 −100 x = −2.5 −100 x −2.5 = −100 −100 x = 0.025 7( x + 8.6) = 6 x 7 x + 60.2 = 6 x 7 x − 7 x + 60.2 = 6 x − 7 x 60.2 = − x 60.2 − x = −1 −1 −60.2 = x

34.

24 y − 10 = 20 y − 17 24 y − 10 + 10 = 20 y − 17 + 10 24 y = 20 y − 7 24 y − 20 y = 20 y − 20 y − 7 4 y = −7 4 y −7 = 4 4 y = −1.75

36.

1.5 = 0.4 x + 0.5 1.5 − 0.5 = 0.4 x + 0.5 − 0.5 1 = 0.4 x 1 0.4 x = 0.4 0.4 2.5 = x

38.

−50 x + 0.81 = −40 x − 0.48 100(−50 x + 0.81) = 100(−40 x − 0.48) −5000 x + 81 = −4000 x − 48 −5000 x + 81 + 48 = −4000 x − 48 + 48 −5000 x + 129 = −4000 x −5000 x + 5000 x + 129 = −4000 x + 5000 x 129 = 1000 x 129 1000 x = 1000 1000 0.129 = x

40.

4(2 x − 1.6) = 5 x − 6.4 8 x − 6.4 = 5 x − 6.4 8 x − 6.4 + 6.4 = 5 x − 6.4 + 6.4 8x = 5x 8x − 5x = 5x − 5x 3x = 0 3x 0 = 3 3 x=0

42.

y − 5 = 0.3 y + 4.1 10( y − 5) = 10(0.3 y + 4.1) 10 y − 50 = 3 y + 41 10 y − 50 + 50 = 3 y + 41 + 50 10 y = 3 y + 91 10 y − 3 y = 3 y − 3 y + 91 7 y = 91 7 y 91 = 7 7 y = 13

44. x + 14 − 5x − 17 = (x − 5x) + (14 − 17) = −4x − 3

ISM: Prealgebra

Chapter 5: Decimals

1 1 16 55 16 6 16 ⋅ 2 ⋅ 3 32 46. 5 ÷ 9 = ÷ = ⋅ = = 3 6 3 6 3 55 3 ⋅ 55 55

48. 50 − 14 50.

9 13 9 4 = 49 − 14 = 35 13 13 13 13

y − 15.9 = −3.8 y − 15.9 + 15.9 = −3.8 + 15.9 y = 12.1

68. 100, although answers may vary 70. answers may vary 72. 7.68 y = −114.98496 7.68 y −114.98496 = 7.68 7.68 y = −14.972 74.

52. − x − 4.1 − x − 4.02 = − x − x − 4.1 − 4.02 = −2 x − 8.12 54. 9a − 5.6 − 3a + 6 = 9a − 3a − 5.6 + 6 = 6a + 0.4 56.

58.

9.7 = x + 4.3 9.7 − 4.3 = x + 4.3 − 4.3 5.4 = x

Section 5.7 Practice Exercises

2.8 x + 3.4 = −13.4 2.8 x + 3.4 − 3.4 = −13.4 − 3.4 2.8 x = −16.8 2.8 x −16.8 = 2.8 2.8 x = −6

1. Mean =

60. 9.21 + x − 4 x + 11.33 = x − 4 x + 9.21 + 11.33 = −3x + 20.54 62.

64.

6( x + 1.43) = 5 x 6 x + 8.58 = 5 x 6 x − 6 x + 8.58 = 5 x − 6 x 8.58 = − x 8.58 − x = −1 −1 −8.58 = x 8.4 z − 2.6 = 5.4 z + 10.3 10(8.4 z − 2.6) = 10(5.4 z + 10.3) 84 z − 26 = 54 z + 103 84 z − 26 + 26 = 54 z + 103 + 26 84 z = 54 z + 129 84 z − 54 z = 54 z − 54 z + 129 30 z = 129 30 z 129 = 30 30 z = 4.3

66. −3.2 x + 12.6 − 8.9 x − 15.2 = −3.2 x − 8.9 x + 12.6 − 15.2 = −12.1x − 2.6

6.11x + 4.683 = 7.51x + 18.235 6.11x + 4.683 − 4.683 = 7.51x + 18.235 − 4.683 6.11x = 7.51x + 13.552 6.11x − 7.51x = 7.51x − 7.51x + 13.552 −1.4 x = 13.552 −1.4 x 13.552 = −1.4 −1.4 x = −9.68

87 + 75 + 96 + 91 + 78 427 = = 85.4 5 5

32 + 34 + 46 + 38 + 42 + 95 + 50 + 42 + 41 420 = 9 9 ≈ 46.67 The mean salary of all staff members except G is about $46.67 thousand or $46,670.

3. gpa =

4 ⋅ 2 + 3 ⋅ 4 + 2 ⋅ 5 + 1 ⋅ 2 + 4 ⋅ 2 40 = ≈ 2.67 2+ 4+5+ 2+ 2 15

4. Because the numbers are in numerical order, and there are an odd number of items, the median is the middle number, 24. 5. Write the numbers in numerical order: 36, 65, 71, 78, 88, 91, 95, 95 Since there are an even number of scores, the median is the mean of the two middle numbers. 78 + 88 median = = 83 2 6. Mode: 15 because it occurs most often, 3 times. 7. Median: Write the numbers in order. 15, 15, 15, 16, 18, 26, 26, 30, 31, 35 Median is mean of middle two numbers, 18 + 26 = 22. 2 Mode: 15 because it occurs most often, 3 times.

193

Chapter 5: Decimals

ISM: Prealgebra

Vocabulary, Readiness & Video Check 5.7 1. Another word for “mean” is average. 2. The number that occurs most often in a set of numbers is called the mode. 3. The mean (or average) of a set of number items sum of items is . number of items 4. The median of a set of numbers is the middle number. If the number of numbers is even, it is the mean (or average) of the two middle numbers. 5. An example of weighted mean is a calculation of grade point average. 6. The answer is not exact; it’s an approximation since we rounded to the nearest tenth. 7. Place the data numbers in numerical order (or verify that they already are). 8. All occurrences of a number will be grouped together, making it easier to locate and count numbers that occur more than once.

Median: Write the numbers in order: 0.1, 0.3, 0.4, 0.5, 0.6, 0.6, 0.7, 0.8, 0.8 The middle number is 0.6. Mode: Since 0.6 and 0.8 occur twice, there are two modes, 0.6 and 0.8. 8. Mean: 451 + 356 + 478 + 776 + 892 + 500 + 467 + 780 8 4700 = 8 = 587.5 Median: Write the numbers in order: 356, 451, 467, 478, 500, 776, 780, 892 478 + 500 The mean of the middle two: = 489 2 Mode: There is no mode, since each number occurs once. 10. Because the numbers are in numerical order, the median height is the middle number (of the top 5), 1972 feet. 12. Mean:

Exercise Set 5.7 45 + 36 + 28 + 46 + 52 207 = = 41.4 5 5 Median: Write the numbers in order: 28, 36, 45, 46, 52 The middle number is 45. Mode: There is no mode, since each number occurs once.

2. Mean:

4. Mean: 4.9 + 7.1 + 6.8 + 6.8 + 5.3 + 4.9 35.8 = ≈ 6.0 6 6 Median: Write the numbers in order: 4.9, 4.9, 5.3, 6.8, 6.8, 7.1 5.3 + 6.8 Median is mean of middle two: = 6.05 2 Mode: Since 6.8 and 4.9 occur twice, there are two modes, 6.8 and 4.9. 6. Mean: 0.6 + 0.6 + 0.8 + 0.4 + 0.5 + 0.3 + 0.7 + 0.8 + 0.1 9 4.8 = 9 ≈ 0.5 194

2717 + 2073 + 1972 + 1965 + 1819 + 1776 6 12, 322 = 6 ≈ 2053.7 feet

14. answer may vary 16. gpa =

1 ⋅1 + 0 ⋅1 + 2 ⋅ 4 + 3 ⋅ 5 24 = ≈ 2.18 1+1+ 4 + 5 11

18. gpa =

3 ⋅ 2 + 3 ⋅ 2 + 2 ⋅ 3 + 4 ⋅ 3 + 3 ⋅ 3 39 = = 3.0 2+ 2+3+3+3 13

20. Median: Write the numbers in order. 4.7, 6.9, 6.9, 7.0, 7.5, 7.8 6.9 + 7.0 The mean of the middle two: = 6.95 2 22. Mean:

93 + 85 + 89 + 79 + 88 + 91 525 = = 87.5 6 6

24. Mode: There is no mode, since each number occurs only once. 26. Median: Write the numbers in order. 64, 65, 66, 68, 70, 70, 71, 71, 72, 75, 77, 78, 80, 82, 86 The middle number is 71.

ISM: Prealgebra

Chapter 5: Decimals

28. The mean is 73 (from Exercise 25). There were 6 pulse rates higher than the mean. They are 75, 77, 78, 80, 82, and 86. 30. answers may vary 32. Mean: 4160 + 4000 + 3915 12, 075 = = 4025 miles 3 3 Median: 4000 miles 34.

9. The mean of a list of items with number values is sum of items . number of items 10. When 2 million is written as 2,000,000, we say it is written in standard form. Chapter 5 Review 1. In 23.45, the 4 is in the tenths place.

4x 4 ⋅ x x = = 36 4 ⋅ 9 9

2. In 0.000345, the 4 is in the hundred-thousandths place.

35a 3

3. −23.45 in words is negative twenty-three and forty-five hundredths.

5 ⋅ 7 ⋅ a ⋅ a ⋅ a 7a 36. = = 2 5 ⋅ 20 ⋅ a ⋅ a 20 100a

38. Since the mode is 21, 21 must occur at least twice in the set. Since there are an odd number of numbers in the set, the median, 20 is in the set. Therefore the missing numbers are 20, 21, 21.

4. 0.00345 in words is three hundred forty-five hundred-thousandths. 5. 109.23 in words is one hundred nine and twentythree hundredths. 6. 200.000032 in words is two hundred and thirtytwo millionths.

40. answers may vary Chapter 5 Vocabulary Check

7. Eight and six hundredths is 8.06.

1. Like fractional notation, decimal notation is used to denote a part of a whole. 2. To write fractions as decimals, divide the numerator by the denominator.

8. Negative five hundred three and one hundred two thousandths is −503.102. 9. Sixteen thousand twenty-five and fourteen tenthousandths is 16,025.0014.

3. To add or subtract decimals, write the decimals so that the decimal points line up vertically.

10. Fourteen and eleven thousandths is 14.011.

4. When writing decimals in words, write “and” for the decimal point.

11. 0.16 =

5. When multiplying decimals, the decimal point in the product is placed so that the number of decimal places in the product is equal to the sum of the number of decimal places in the factors. 6. The mode of a set of numbers is the number that occurs most often.

16 4⋅4 4 = = 100 25 ⋅ 4 25

12. −12.023 = −12

13.

23 1000

231 = 0.00231 100, 000

7. The distance around a circle is called the circumference.

1 25 14. 25 = 25 = 25.25 4 100

8. The median of a set of numbers in numerical order is the middle number. If there are an even number of numbers, the mode is the mean of the two middle numbers.

15. 0.49 0.43 ↑ ↑ 9 > 3 so 0.49 > 0.43 16. 0.973 = 0.9730

195

Chapter 5: Decimals

ISM: Prealgebra

17. 38.0027 38.00056 ↑ ↑ 2 > 0 so 38.0027 > 38.00056 Thus, −38.0027 < −38.00056.

27. Add the absolute values. 6.40 + 0.88 7.28 Attach the common sign. −6.4 + (−0.88) = −7.28

18. 0.230505 0.23505 ↑ ↑ 0 < 5 so 0.230505 < 0.23505 Thus, −0.230505 > −0.23505. 19. To round 0.623 to the nearest tenth, observe that the digit in the hundredths place is 2. Since this digit is less than 5, we do not add 1 to the digit in the tenths place. The number 0.623 rounded to the nearest tenth is 0.6. 20. To round 0.9384 to the nearest hundredth, observe that the digit in the thousandths place is 8. Since this digit is at least 5, we add 1 to the digit in the hundredths place. The number 0.9384 rounded to the nearest hundredth is 0.94. 21. To round −42.895 to the nearest hundredth, observe that the digit in the thousandths place is 5. Since this digit is at least 5, we add 1 to the digit in the hundredths place. The number −42.895 rounded to the nearest hundredth is −42.90. 22. To round 16.34925 to the nearest thousandth, observe that the digit in the ten-thousandths place is 2. Since this digit is less than 5, we do not add 1 to the digit in the thousandths place. The number 16.34925 rounded to the nearest thousandth is 16.349. 23. 890.5 million = 890.5 × 1 million = 890.5 × 1, 000, 000 = 890,500, 000 24. 600 thousand = 600 × 1 thousand = 600 × 1000 = 600, 000 25.

8.6 + 9.5 18.1

26.

3.9 + 1.2 5.1

196

28. Subtract the absolute values. 19.02 − 6.98 12.04 Attach the sign of the larger absolute value. −19.02 + 6.98 = −12.04 29.

200.490 16.820 + 103.002 320.312

30.

0.00236 100.45000 + 48.29000 148.74236

31.

4.9 − 3.2 1.7

32.

5.23 − 2.74 2.49

33. −892.1 − 432.4 = −892.1 + (−432.4) Add the absolute values. 892.1 + 432.4 1324.5 Attach the common sign. −892.1 − 432.4 = −1324.5 34. 0.064 − 10.2 = 0.064 + (−10.2) Subtract the absolute values. 10.200 − 0.064 10.136 Attach the sign of the larger absolute value. 0.064 − 10.2 = −10.136 35.

100.00 − 34.98 65.02

ISM: Prealgebra 36. −

37.

Chapter 5: Decimals

200.00000 0.00198 199.99802

0.0877 47. 3 0.2631 −24 23 −21 21 −21 0

19.9 15.1 10.9 + 6.7 52.6 The total distance is 52.6 miles.

38. x − y = 1.2 − 6.9 = −5.7 39. Perimeter = 6.2 + 4.9 + 6.2 + 4.9 = 22.2 The perimeter is 22.2 inches. 40. Perimeter = 11.8 + 12.9 + 14.2 = 38.9 The perimeter is 38.9 feet. 41. 7.2 × 10 = 72 42. 9.345 × 1000 = 9345 43. A negative number multiplied by a positive number is a negative number. 34.02 2 decimal places × 2.3 1 decimal place 10 206 68 040 78.246 2 + 1 = 3 decimal places

−34.02 × 2.3 = −78.246

15.825 48. 20 316.500 −20 116 −100 16 5 −16 0 50 −40 100 −100 0

49. A negative number divided by a negative number is a positive number. 70 0.3 21 becomes 3 210 −21 00

−21 ÷ (−0.3) = 70

44. A negative number multiplied by a negative number is a positive number. 839.02 2 decimal places × 87.3 1 decimal place 251 706 5873 140 67121 600 73246.446 2 + 1 = 3 decimal places

−839.02 × (−87.3) = 73,246.446

50. A negative number divided by a positive number is a negative number. 0.21 0.03 0.0063 becomes 3 0.63 −6 03 −3 0

−0.0063 ÷ 0.03 = −0.21

45. C = 2πr = 2π ⋅ 7 = 14π meters C ≈ 14 ⋅ 3.14 = 43.96 meters 46. C = πd = π ⋅ 20 = 20π inches C ≈ 20 ⋅ 3.14 = 62.8 inches

197

Chapter 5: Decimals

ISM: Prealgebra

8.0588 ≈ 8.059 51. 0.34 2.74 becomes 34 274.0000 −272 20 −0 2 00 −1 70 300 −272 280 −272 8

52. 19.8 601.92 becomes 198

53.

23.65 = 0.02365 1000

54.

93 = −9.3 −10

30.4 6019.2 −594 79 −0 79 2 −79 2 0

7.31 ≈ 7.3 55. 3.28 24 becomes 328 2400.00 −2296 104 0 −98 4 5 60 −3 28 2 32 There are approximately 7.3 meters in 24 feet. 45 56. 69.71 3136.95 becomes 6971 313695 −27884 34855 −34855 0 The loan will be paid off in 45 months.

57. 5

0.8 4.0 −4 0 0

0.9230 ≈ 0.923 58. 13 12.0000 −11 7 30 −26 40 −39 10 −0 10 −

12 ≈ −0.923 13

0.3333... 59. 3 1.0000 −9 10 −9 10 −9 10 −9 1 1 2 = 2.3 or 2.333 3

60. 60

0.2166... 13.0000 −12 0 1 00 −60 400 −360 400 −360 40

13 = 0.216 or 0.217 16

61. 0.392 = 0.39200

198

4 = 0.8 5

ISM: Prealgebra

Chapter 5: Decimals

62. 0.0231 0.0221 ↑ ↑ 3 > 2 so 0.0231 > 0.0221, thus −0.0231 < −0.0221.

70. (−2.3) 2 − 1.4 = 5.29 − 1.4 = 3.89 71. 0.0726 ÷ 10 × 1000 = 0.00726 × 1000 = 7.26

0.571 ≈ 0.57 63. 7 4.000 −3 5 50 −49 10 −7 3

72. 0.9(6.5 − 5.6) = 0.9(0.9) = 0.81

4 4 ≈ 0.57 and 0.57 < 0.625, so < 0.625. 7 7

64. 17

69. −7.6 × 1.9 + 2.5 = −14.44 + 2.5 = −11.94

0.2941 ≈ 0.294 5.0000 −3 4 1 60 −1 53 70 −68 20 −17 3

73.

(1.5) 2 + 0.5 2.25 + 0.5 2.75 = = = 55 0.05 0.05 0.05

74.

7 + 0.74 7.74 = = −129 −0.06 −0.06

1 ⋅b⋅h 2 1 = (4.6)(3) 2 = 0.5(4.6)(3) = 6.9 square feet

75. Area =

1 ⋅b⋅h 2 1 = (5.2)(2.1) 2 = 0.5(5.2)(2.1) = 5.46 square inches

76. Area =

5 5 ≈ 0.294 and 0.293 < 0.294, so 0.293 < . 17 17

77.

x + 3.9 = 4.2 x + 3.9 − 3.9 = 4.2 − 3.9 x = 0.3

78.

70 = y − 22.81 70 + 22.81 = y − 22.81 + 22.81 92.81 = y

65. 0.832, 0.837, 0.839 66.

67.

68.

5 = 0.625 8 5 , 0.626, 0.685 8 3 ≈ 0.428 7 3 0.42, , 0.43 7

18 = 1.6363 11 1.63 = 1.63 19 = 1.583 12 19 18 , 1.63, 12 11

79. 2 x = 17.2 2 x 17.2 = 2 2 x = 8.6 80. −1.1y = 88 −1.1y 88 = −1.1 −1.1 y = −80 81.

3 x − 0.78 = 1.2 + 2 x 3 x − 0.78 + 0.78 = 1.2 + 2 x + 0.78 3x = 1.98 + 2 x 3x − 2 x = 1.98 + 2 x − 2 x x = 1.98

199

Chapter 5: Decimals

ISM: Prealgebra

82.

− x + 0.6 − 2 x = −4 x − 0.9 −3x + 0.6 = −4 x − 0.9 −3 x + 0.6 − 0.6 = −4 x − 0.9 − 0.6 −3x = −4 x − 1.5 −3 x + 4 x = −4 x + 4 x − 1.5 x = −1.5

83.

−1.3 x − 9.4 = −0.4 x + 8.6 10(−1.3 x − 9.4) = 10(−0.4 x + 8.6) −13 x − 94 = −4 x + 86 −13x − 94 + 94 = −4 x + 86 + 94 −13 x = −4 x + 180 −13 x + 4 x = −4 x + 4 x + 180 −9 x = 180 −9 x 180 = −9 −9 x = −20

84.

3( x − 1.1) = 5 x − 5.3 3 x − 3.3 = 5 x − 5.3 3x − 3.3 + 3.3 = 5 x − 5.3 + 3.3 3x = 5 x − 2 3x − 5 x = 5 x − 5 x − 2 −2 x = −2 −2 x −2 = −2 −2 x =1 13 + 23 + 33 + 14 + 6 89 = = 17.8 5 5 Median: Write the numbers in order. 6, 13, 14, 23, 33 The middle number is 14. Mode: There is no mode, since each number occurs once.

85. Mean:

45 + 86 + 21 + 60 + 86 + 64 + 45 7 407 = 7 ≈ 58.1 Median: Write the numbers in order. 21, 45, 45, 60, 64, 86, 86 The middle number is 60. Mode: There are 2 numbers that occur twice, so there are two modes, 45 and 86.

86. Mean =

14, 000 + 20, 000 + 12, 000 + 20, 000 + 36, 000 + 45, 000 147, 000 = = 24,500 6 6 Median: Write the numbers in order. 12,000, 14,000, 20,000, 20,000, 36,000, 45,000 20, 000 + 20, 000 = 20, 000 The mean of the middle two: 2 Mode: 20,000 is the mode because it occurs twice.

87. Mean =

200

ISM: Prealgebra

Chapter 5: Decimals

560 + 620 + 123 + 400 + 410 + 300 + 400 + 780 + 430 + 450 4473 = = 447.3 10 10 Median: Write the numbers in order. 123, 300, 400, 400, 410, 430, 450, 560, 620, 780 410 + 430 = 420 The mean of the two middle numbers: 2 Mode: 400 is the mode because it occurs twice.

88. Mean =

89. gpa =

4 ⋅ 3 + 4 ⋅ 3 + 2 ⋅ 2 + 3 ⋅ 3 + 2 ⋅1 39 = = 3.25 3 + 3 + 2 + 3 +1 12

90. gpa =

3 ⋅ 3 + 3 ⋅ 4 + 2 ⋅ 2 + 1 ⋅ 2 + 3 ⋅ 3 36 = ≈ 2.57 3+ 4+ 2+ 2+3 14

91. 200.0032 in words is two hundred and thirty-two ten-thousandths. 92. Negative sixteen and nine hundredths is −16.09 in standard form. 93. 0.0847 =

94.

847 10, 000

6 ≈ 0.857 7 8 ≈ 0.889 9 0.75 = 0.750 6 8 0.75, , 7 9

95. −

7 = −0.07 100

0.1125 96. 80 9.0000 −8 0 1 00 −80 200 −160 400 −400 0

9 = 0.1125 80

201

Chapter 5: Decimals

ISM: Prealgebra 104. Subtract absolute values. 4.9 − 3.2 1.7 Attach the sign of the larger absolute value. 3.2 − 4.9 = −1.7

51.0571 ≈ 51.057 97. 175 8935.0000 −875 185 −175 10 0 −0 10 00 −875 1 250 −1 225 250 −175 75

105.

106. Subtract absolute values. 102.06 − 89.30 12.76 Attach the sign of the larger absolute value. −102.06 + 89.3 = −12.76

8935 ≈ 51.057 175

98. 402.000032 402.00032 ↑ ↑ 0 < 3 so 402.000032 < 402.00032 Thus, −402.000032 > −402.00032. 6 = 0.54 99. 11 0.54 < 0.55, so

9.12 − 3.86 5.26

6 < 0.55 11

100. To round 86.905 to the nearest hundredth, observe that the digit in the thousandths place is 5. Since this digit is at least 5, we add 1 to the digit in the hundredths place. The number 86.905 rounded to the nearest hundredth is 86.91. 101. To round 3.11526 to the nearest thousandth, observe that the digit in the ten-thousandths place is 2. Since this digit is less than 5, we do not add 1 to the digit in the thousandths place. The number 3.11526 rounded to the nearest thousandth is 3.115. 102. To round 123.46 to the nearest one, observe that the digit in the tenths place is 4. Since this digit is less than 5, we do not add 1 to the digit in the ones place. The number $123.46 rounded to the nearest dollar (or one) is $123.00.

107.

−4.021 −10.830 ( +) − 0.056 −14.907

108.

2.54 × 3.2 508 7 620 8.128

2 + 1 = 3 decimal places

109. The product of a negative number and a positive number is a negative number. 3.45 2 decimal places × 2.1 1 decimal place 345 6 900 7.245 2 + 1 = 3 decimal places

The product is −7.245. 4900 110. 0.005 24.5 becomes 5 24500 −20 45 −45 000

103. To round 3645.52 to the nearest one, observe that the digit in the tenths place is 5. Since this digit is at least 5, we add 1 to the digit in the ones place. The number $3645.52 rounded to the nearest dollar (or one) is $3646.00.

202

2 decimal places 1 decimal place

ISM: Prealgebra

Chapter 5: Decimals

23.9043 ≈ 23.904 111. 2.3 54.98 becomes 23 549.8000 −46 89 −69 20 8 −20 7 10 −0 100 −92 80 −69 11

1. In the number 1026.89704, the digit 8 is in the tenths place: C. 2. In the number 1026.89704, the digit 2 is in the tens place; A. 3. In the number 1026.89704, the digit 0 to the left of the decimal point is in the hundreds place; B. 4. In the number 1026.89704, the digit 0 to the right of the decimal point is in the tenthousandths place; E.

Replace x with 3 and y with 0.2 in each expression. A. x + y = 3 + 0.2 = 3.0 + 0.2 = 3.2 B. x − y = 3 − 0.2 = 3.0 − 0.2 = 2.8

112. length = 115.9 ≈ 120 width = 77.3 ≈ 80 Area = length ⋅ width = 120 ⋅ 80 = 9600 square feet 113. $3.79 $2.49 $1.99

Chapter 5 Getting Ready for the Test

C. xy = (3)(0.2) = 0.6 D.

rounds to rounds to rounds to

$4 $2 $2 $8 Yes, the items can be purchased with a $10 bill.

x ÷ y = 3 ÷ 0.2 =

3 3 ⋅10 30 = = = 15 0.2 0.2 ⋅10 2

5. Since xy = 0.6, choice C is the correct expression. 6. Since x + y = 3.2, choice A is the correct expression. 7. Since x − y = 2.8, choice B is the correct expression.

(3.2)2 10.24 114. = = 0.1024 100 100

115. (2.6 + 1.4)(4.5 − 3.6) = (4)(0.9) = 3.6 73 + 82 + 95 + 68 + 54 372 116. Mean: = = 74.4 5 5 Median: Write the numbers in order. 54, 68, 73, 82, 95 The median is the middle value: 73. Mode: There is no mode, since each number occurs once.

117. Mean: 952 + 327 + 566 + 814 + 327 + 729 3715 = 6 6 ≈ 619.17 Median: Write the numbers in order. 327, 327, 566, 729, 814, 952 The mean of the two middle values: 566 + 729 = 647.5 2 Mode: 327, since this number appears twice.

8. since x ÷ y = 15, choice D is the correct expression. 9. 2(0.5)2 + 0.3 = 2(0.25) + 0.3 = 0.50 + 0.3 = 0.8; B 10.

5( x − 3.2) = 9.5 5 ⋅ x − 5 ⋅ 3.2 = 9.5 5 x − 16 = 9.5 Choice B is the correct equation.

11.

0.3 x + 1.8 = 2.16 100(0.3 x + 1.8) = 100(2.16) 100 ⋅ 0.3 x + 100 ⋅1.8 = 100 ⋅ 2.16 30 x + 180 = 216 Choice C is the correct equation.

203

Chapter 5: Decimals

ISM: Prealgebra

7 + 9 + 10 + 13 + 13 52 = = 10.4 5 5 Median: Since the numbers are in order, the median is the middle number, 10. Mode: 13 occurs more often than any other value.

Mean:

12. The median is 10; B. 13. The mode is 13; C.

−0.00843 ÷ (−0.23) ≈ 0.037

14. The mean is 10.4; A. Chapter 5 Test 1. 45.092 in words is forty-five and ninety-two thousandths. 2. Three thousand and fifty-nine thousandths in standard form is 3000.059. 3.

2.893 4.210 + 10.492 17.595

5. Subtract the absolute values. 30.25 − 9.83 20.42 Attach the sign of the larger absolute value. 9.83 − 30.25 = −20.42 10.2 × 4.01 102 40 800 40.902

8. To round 34.8923 to the nearest tenth, observe that the digit in the hundredths place is 9. Since this digit is at least 5, we add 1 to the digit in the tenths place. 34.8923 rounded to the nearest tenth is 34.9. 9. To round 0.8623 to the nearest thousandth, observe that the digit in the ten-thousandths place is 3. Since this digit is less than 5, we do not add 1 to the digit in the thousandths place. 0.8623 rounded to the nearest thousandth is 0.862. 10. 25.0909 25.9090 ↑ ↑ 0 < 9 so 25.0909 < 25.9090

4. Add the absolute values. 47.92 + 3.28 51.20 Attach the common sign. −47.92 − 3.28 = −51.20

0.0366 ≈ 0.037 7. 0.23 0.00843 becomes 23 0.8430 −69 153 −138 150 −138 12

1 decimal place 2 decimal places

0.444... 11. 9 4.000 −3 6 40 −36 40 −36 4 0.444 < 0.445, so

1 + 2 = 3 decimal places

12. 0.345 =

345 5 ⋅ 69 69 = = 1000 5 ⋅ 200 200

13. −24.73 = −24 14. −

204

4 < 0.445. 9

73 100

13 1 ⋅13 1 1⋅ 5 5 =− =− =− = − = −0.5 26 2 ⋅13 2 2⋅5 10

ISM: Prealgebra

Chapter 5: Decimals 22. Mean: 8 + 10 + 16 + 16 + 14 + 12 + 12 + 13 101 = = 12.625 8 8 Median: List the numbers in order. 8, 10, 12, 12, 13, 14, 16, 16 12 + 13 = 12.5 The mean of the middle two: 2 Mode: 12 and 16 each occur twice, so the modes are 12 and 16.

0.9411 ≈ 0.941 15. 17 16.0000 −15 3 70 −68 20 −17 30 −17 13

23. gpa =

16 ≈ 0.941 17 2

16. (−0.6) + 1.57 = 0.36 + 1.57 = 1.93 17.

0.23 + 1.63 1.86 = = −6.2 −0.3 −0.3

20.

1 (4.2 miles)(1.1 miles) 2 = 0.5(4.2)(1.1) square miles = 2.31 square miles

26. C = 2πr = 2π ⋅ 9 = 18π miles C ≈ 18 ⋅ 3.14 = 56.52 miles

0.2 x + 1.3 = 0.7 0.2 x + 1.3 − 1.3 = 0.7 − 1.3 0.2 x = −0.6 0.2 x −0.6 = 0.2 0.2 x = −3

27. a.

2( x + 5.7) = 6 x − 3.4 2 x + 11.4 = 6 x − 3.4 2 x + 11.4 − 11.4 = 6 x − 3.4 − 11.4 2 x = 6 x − 14.8 2 x − 6 x = 6 x − 6 x − 14.8 −4 x = −14.8 −4 x −14.8 = −4 −4 x = 3.7 26 + 32 + 42 + 43 + 49 192 = = 38.4 5 5 Median: The numbers are listed in order. The middle number is 42. Mode: There is no mode since each number occurs once.

21. Mean:

24. 4,583 million = 4583 × 1 million = 4583 × 1, 000, 000 = 4,583, 000, 000 25. Area =

18. 2.4 x − 3.6 − 1.9 x − 9.8 = (2.4 x − 1.9 x ) + (−3.6 − 9.8) = 0.5 x − 13.4 19.

4 ⋅ 3 + 3 ⋅ 3 + 2 ⋅ 3 + 3 ⋅ 4 + 4 ⋅1 43 = ≈ 3.07 3 + 3 + 3 + 4 +1 14

Area = length ⋅ width = (123.8) × (80) = 9904 The area is 9904 square feet.

b. 9904 × 0.02 = 198.08 Teru needs to purchase 198.08 ounces. 28.

14.2 16.1 + 23.7 54.0 The total distance is 54 miles.

Cumulative Review Chapters 1−5 1. 72 in words is seventy-two. 2. 107 in words is one hundred seven. 3. 546 in words is five hundred forty-six. 4. 5026 in words is five thousand twenty-six. 5.

46 + 713 759

205

Chapter 5: Decimals

ISM: Prealgebra

6. 3 + 7 + 9 = 19 The perimeter is 19 inches. 7.

543 − 29 514

14.

Check:

514 + 29 543

17. 9 2 = 9 ⋅ 9 = 81 18. 53 = 5 ⋅ 5 ⋅ 5 = 125 19. 34 = 3 ⋅ 3 ⋅ 3 ⋅ 3 = 81 20. 103 = 10 ⋅10 ⋅ 10 = 1000

9. To round 278,362 to the nearest thousand, observe that the digit in the hundreds place is 3. Since this digit is less than 5, we do not add 1 to the digit in the thousands place. The number 278,362 rounded to the nearest thousand is 278,000. 10. 30 = 2 ⋅ 15 = 2 ⋅ 3 ⋅ 5 ×

The opposite of 13 is −13.

The opposite of 0 is 0. The opposite of −7 is −(−7) = 7.

b. The opposite of 4 is −4. c.

b. 25 ÷ 1 = 25

Check: 25 × 1 = 25

6 =1 6

Check: 1 × 6 = 6

d. 12 ÷ 12 = 1

Check: 1 × 12 = 12

20 = 20 1

Check: 20 × 1 = 20

1 18 18

2a + 4 2 ⋅ 7 + 4 14 + 4 18 = = = =6 3 3 3 c

24. a.

Check: 7 × 1 = 7

22.

7 17

x − 5 y 35 − 5 ⋅ 5 35 − 25 10 = = = =2 5 5 5 y

b. The opposite of −2 is −(−2) = 2.

12. 236 × 86 × 0 = 0

21.

23. a.

236 86 1 416 18 880 20, 296

13. a.

15. 2 ⋅ 4 − 3 ÷ 3 = 8 − 3 ÷ 3 = 8 − 1 = 7 16. 77 ÷ 11 ⋅ 7 = 7 ⋅ 7 = 49

121 R 1 8. 27 3268 −27 56 −54 28 −27 1

11.

25 + 17 + 19 + 39 100 = =5 4 4 The average is 25.

The opposite of −1 is −(−1) = 1.

25. −2 + (−21) = −23 26. −7 + (−15) = −22 27. 5 ⋅ 6 2 = 5 ⋅ 36 = 180 28. 4 ⋅ 23 = 4 ⋅ 8 = 32

Check: 1 × 18 = 18

29. −7 2 = −(7 ⋅ 7) = −49 30. (−2)5 = (−2)(−2)(−2)(−2)(−2) = −32 31. (−5)2 = (−5)(−5) = 25 32. −32 = −(3 ⋅ 3) = −9

206

ISM: Prealgebra

Chapter 5: Decimals

1 of a whole. Four parts 3 are shaded, or 1 whole and 1 part. 4 1 or 1 3 3

33. Each part represents

42.

Since 0.571 > 0.556, then

1 of a whole. Seven parts 4 are shaded or 1 whole and 3 parts. 7 3 or 1 4 4 1 of a whole. Eleven parts 4 are shaded or 2 wholes and 3 parts. 11 3 or 2 4 4 1 of a whole. Fourteen 3 parts are shaded, or 4 wholes and 2 parts. 14 2 or 4 3 3

36. Each part represents

37. 252 = 2 ⋅126 ↓ ↓ 2 ⋅ 2 ⋅ 63 ↓ ↓ ↓ 2 ⋅ 2 ⋅ 3 ⋅ 21 ↓ ↓ ↓ ↓ 2 ⋅ 2 ⋅3⋅3 ⋅ 7 252 = 22 ⋅ 32 ⋅ 7

38.

87 − 25 62

72 36 ⋅ 2 36 39. − =− =− 26 13 ⋅ 2 13

40. 9

41.

7 8 ⋅ 9 + 7 72 + 7 79 = = = 8 8 8 8

16 2 ⋅ 8 2 = = 40 5 ⋅ 8 5 10 2 ⋅ 5 2 = = 25 5 ⋅ 5 5 The fractions are equivalent.

4 5 > . 7 9

or 4 9 36 5 7 35 and ⋅ = ⋅ = 7 9 63 9 7 63 36 35 4 5 Since 36 > 35, then and > . > 63 63 7 9

34. Each part represents

35. Each part represents

4 ≈ 0.571 7 5 ≈ 0.556 9

43.

2 5 2 ⋅ 5 10 ⋅ = = 3 11 3 ⋅11 33

5 4 21 4 21 ⋅ 4 7 ⋅ 3 ⋅ 4 3 1 44. 2 ⋅ = ⋅ = = = or 1 8 7 8 7 8⋅7 4⋅ 2⋅7 2 2

45.

1 1 1 ⋅1 1 ⋅ = = 4 2 4⋅2 8

46. 7 ⋅ 5

47.

2 7 37 7 ⋅ 37 37 = ⋅ = = = 37 7 1 7 1⋅ 7 1

z = 11 − 5 −4 z =6 −4 z −4 ⋅ = −4 ⋅ 6 −4 z = −24

48. 6 x − 12 − 5 x = −20 x − 12 = −20 x − 12 + 12 = −20 + 12 x = −8 49.

763.7651 22.0010 + 43.8900 829.6561

50.

89.2700 14.3610 + 127.2318 230.8628

207

Chapter 5: Decimals

51. ×

52.

208

23.6 0.78 1 888 16 520 18.408

43.8 × 0.645 2190 1 7520 26 2800 28.2510

ISM: Prealgebra

1 decimal place 2 decimal places

1 + 2 = 3 decimal places 1 decimal place 3 decimal places

1 + 3 = 4 decimal places

Chapter 6 Section 6.1 Practice Exercises 1. The ratio of 19 to 30 is

19 . 30

2. The ratio of $16 to $12 is

4. The ratio of 2

8 28 ÷ 3 15 8 15 = ⋅ 3 28 4 ⋅ 2 ⋅3⋅5 = 3⋅ 7 ⋅ 4 10 = . 7

11. unit price =

6.5 12 78.0 −72 60 −60 0

The ratio of the length of the shortest side to the length of the longest side is 6 meters 6 2⋅3 3 = = = . 10 meters 10 2 ⋅ 5 5

b. The ratio of the length of the longest side to the perimeter is 10 meters 10 5 ⋅ 2 5 = = = . 24 meters 24 12 ⋅ 2 12 $1350 $225 = 6 weeks 1 week

295 miles 59 miles = 15 gallons 3 gallons

6.5 bushels or 6.5 bushels/ tree. 1 tree

$170 $34 = or $34 per day 5 days 1 day

$3.99 $0.36 ≈ 11 ounces 1 ounce $5.99 $0.37 unit price = ≈ 16 ounces 1 ounce Since the 11-ounce bag has a cheaper price per ounce, it is the better buy.

12. unit price =

5. The ratio of work miles to total miles is 4800 miles 8 ⋅ 600 8 = = . 15,000 miles 25 ⋅ 600 25

400 feet or 400 feet/second. 1 second

78 bushels 12 trees

The unit rate is

2 13 = 2 ÷1 13 3 15 1

6. a.

10.

2 13 to 1 is 3 15

2 23

The unit rate is

$16 16 4 ⋅ 4 4 = = = . $12 12 4 ⋅ 3 3

3. The ratio of 1.68 to 4.8 is 1.68 1.68 ⋅100 168 7 ⋅ 24 7 = = = = . 4.8 4.8 ⋅100 480 20 ⋅ 24 20

400 8 3200 −32 000

3200 feet 8 seconds

Vocabulary, Readiness & Video Check 6.1 1. A rate with a denominator of 1 is called a unit rate. 2. When a rate is written as money per item, a unit rate is called a unit price. 3. The word per translates to “division.” 4. Rates are used to compare different types of quantities. 5. To write a rate as a unit rate, divide the numerator of the rate by the denominator. 6. The quotient of two quantities is called a ratio. 7. True or false: The ratio ratio

7 means the same as the 5

5 . false 7

209

Chapter 6: Ratio, Proportion, and Triangle Applications 8. The numerator and denominator were both 7 .7 multiplied by 10. Thus, is written as the 10 77 equivalent fraction . 100 9. The units are different in Video Notebook Number 5 (shrubs and feet); they were the same in Video Notebook Number 4 (days). 10. We want a unit rate, which is a rate with a denominator of 1. A unit rate tells us how much of the first quantity ($) will occur in 1 of the second quantity (years). 11. when shopping for the best buy

14. The ratio of 3 3 13 4 16

25 1 ⋅ 25 1 2. The ratio of 25 to 150 is = = . 150 6 ⋅ 25 6

1 1 = 3 ÷4 3 6 10 25 ÷ 3 6 10 6 = ⋅ 3 25 5⋅ 2 ⋅3⋅ 2 = 3⋅5⋅5 4 = . 5 =

25 12 days 2 56 days

6. The ratio of 9.61 to 7.62 is 9.61 9.61 ⋅100 961 . = = 7.62 7.62 ⋅100 762 8. The ratio of 35 meters to 100 meters is 35 meters 7 ⋅5 7 = = . 100 meters 20 ⋅ 5 20 $46 23 ⋅ 2 23 = = . $102 51 ⋅ 2 51

12. The ratio of 120 miles to 80 miles is 120 miles 3 ⋅ 40 3 = = . 80 miles 2 ⋅ 40 2

1 5 days to 2 days is 2 6

1 5 = 25 ÷ 2 2 6 51 17 ÷ 2 6 51 6 = ⋅ 2 17 3 ⋅17 ⋅ 2 ⋅ 3 = 2 ⋅17 ⋅1 9 = 1 =

8.1 8.1 ⋅10 81 . = = 10 10 ⋅10 100

10. The ratio of $46 to $102 is

1 1 to 4 is 3 6

16. The ratio of 25

Exercise Set 6.1

4. The ratio of 8.1 to 10 is

ISM: Prealgebra

18. The ratio of the land area of Tuvalu to the land 10 sq mi 2 ⋅ 5 5 area of San Marino is = = . 24 sq mi 2 ⋅12 12 20. Perimeter = 94 + 50 + 94 + 50 = 288 feet The ratio of width to perimeter is 50 feet 2 ⋅ 25 25 = = . 288 feet 2 ⋅144 144 22. The ratio of the parents to the total is 100 parents 100 4 ⋅ 25 4 = = = . 125 students + 100 parents 225 9 ⋅ 25 9 24. The ratio of white blood cells to red blood cells 1 cell 1 is = . 600 cells 600 26. The ratio of the miniDVD diameter to the standard DVD diameter is 8 cm 8 2⋅4 2 = = = . 12 cm 12 3 ⋅ 4 3

210

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications

28. The ratio of the SSD width to the HDD width is 2 12 in. 2 12 = 3 12 in. 3 12 1 1 = 2 ÷3 2 2 5 7 = ÷ 2 2 5 2 = ⋅ 2 7 5⋅ 2 = 2⋅7 5 = . 7 30. Perimeter = 5 + 12 + 13 = 30 feet The ratio of the base to the perimeter is 5 feet 1 ⋅ 5 1 = = . 30 feet 5 ⋅ 6 6 32. The ratio of the average number of servings of Coca-Cola consumed by citizens of the United States to the average number of servings consumed by citizens of Canada is 400 8-oz servings 400 20 ⋅ 20 20 = = = . 260 8-oz servings 260 13 ⋅ 20 13 34. The rate of 14 lab tables for 28 students is 14 lab tables 1 lab table = . 28 students 2 students 36. The rate of 8 phone lines for 36 employees is 8 phone lines 2 phone lines = . 36 employees 9 employees 38. The rate of 4 inches of rain in 18 hours is 4 inches 2 inches = . 18 hours 9 hours 40. The rate of 150 graduate students for 8 advisors is 150 graduate students 75 graduate students = . 8 advisors 4 advisors 25 42. 11 275 −22 55 −55 0

275 miles in 11 hours is 25 miles/hour.

25 miles or 1 hour

3 44. 6 18 −18 0

18 signs in 6 blocks is

3 signs or 3 signs/block. 1 block

20 46. 60 1200 −120 00

1200 wingbeats per 60 seconds is

20 wingbeats 1 second

or 20 wingbeats/second.

50 48. 8000 400, 000 −400 00 00 400,000 library books for 8000 students is 50 library books or 50 library books/student. 1 student 22, 084 50. 93 2, 053,812 −1 86 193 −186 7 81 −7 44 372 −372 0 2,053,812 residents for 93 counties is 22, 084 residents or 22,084 residents/county. 1 county

1, 250, 000 52. 4 5, 000, 000 −4 10 −8 20 −20 0 5,000,000 lottery tickets for 4 lottery winners is 1, 250, 000 tickets or 1,250,000 tickets/winner. 1 winner

211

Chapter 6: Ratio, Proportion, and Triangle Applications

3, 000, 000 54. 273 819, 000, 000 −819 0

$819 million for 273 shows is

ISM: Prealgebra

$3, 000, 000 or 1 show

$3,000,000/show. 56. a.

58. a.

212

15.5 713.0 −46 253 −230 2 30 −2 30 0 The unit rate for Akira is 15.5 bricks/minute. 46

13.2 30 396.0 −30 96 −90 60 −6 0 0 The unit rate for Rowen is 13.2 bricks/minute.

26.90 ≈ 26.9 5.5 148 becomes 55 1480.00 −110 380 −330 50 0 −49 5 50 The unit rate for Senegal is ≈26.9 items/minute.

Milan is the faster scanner, since 28.6 > 26.9.

0.29 60. 3 0.87 −6 27 −27 0 The unit price is $0.29 per apple. 12.25 73.50 −6 13 −12 15 −1 2 30 −30 0 The unit price is $12.25 per lawn chair.

62. 6

Akira is the faster bricklayer, since 15.5 > 13.2. 28.57 ≈ 28.6 3.5 100 becomes 35 1000.00 −70 300 −280 200 −175 250 −245 5 The unit rate for Milan is ≈28.6 items/minute.

0.2079 ≈ 0.208 64. 24 4.9900 −4 8 190 −168 220 −216 4 The 24-ounce size costs $0.208 per ounce.

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications

0.1404 ≈ 0.140 8.9900 −6 4 2 59 −2 56 300 −256 44 The 64-ounce size costs $0.140 per ounce. The 64-ounce size is the better buy. 64

0.3491 ≈ 0.349 66. 12 4.1900 −3 6 59 −48 110 −108 20 −12 8 The dozen costs $0.349 per egg. 0.3463 ≈ 0.346 30 10.3900 −9 0 1 39 −1 20 190 −180 100 −90 10 The flat costs $0.346 per egg. The flat is the better buy.

0.2120 ≈ 0.212 68. 24 5.0900 −4 8 29 −24 50 −48 20 The 24-ounce size costs $0.212 per ounce.

0.2202 ≈ 0.220 34 7.4900 −6 8 69 −68 100 −68 32 The 34-ounce size costs $0.220 per ounce. The 24-ounce size is the better buy. 0.2757 ≈ 0.276 70. 21 5.7900 −4 2 1 59 −1 47 120 −105 150 −147 3 The 21-ounce size costs $0.276 per ounce. 0.2908 ≈ 0.291 12 3.4900 −2 4 1 09 −1 08 100 −96 4 The 12-ounce size costs $0.291 per ounce. The 21-ounce size is the better buy. 8.6 72. 7 60.2 −56 42 −42 0 0.25 74. 4.6 1.15 becomes 46 11.50 −9 2 2 30 −2 30 0

76. answers may vary

213

Chapter 6: Ratio, Proportion, and Triangle Applications

78. no; 4

ISM: Prealgebra

88. 32,056 − 31,623 = 433

1 2 ⋅ 4 +1 9 = = 2 2 2

36.38 ≈ 36.4 11.9 433 becomes 119 4330.00 −357 760 −714 46 0 −35 7 10 30 −9 52 78 433 miles were driven, averaging 36.4 miles per gallon.

1.840 ≈ 1.84 80. 34.5 63.5 becomes 345 635.000 −345 290 0 −276 0 14 00 −13 80 200 The aspect ratio of the screen is closer to 1.85 : 1 than to 2.39 : 1. 2.285 ≈ 2.29 82. 14 32.000 −28 40 −2 8 1 20 −1 12 80 −70 10 The aspect ratio of the screen is closer to 2.39 : 1 than 1.85 : 1.

84.

3 bruised 1 bruised = 33 good 11 good

501.0 ≈ 501 90. 98, 700 49, 450, 000.0 −49 350 0 100 000 −98 700 1 300 0 There are 501 students per school.

92. increases; answers may vary 94. answers may vary 96. answers may vary Section 6.2 Practice Exercises

1 1 Since < , the shipment should not be 11 10 refused.

1. a.

86. 16,895 − 16,543 = 352

22.27 ≈ 22.3 15.8 352 becomes 158 3520.00 −316 360 −316 440 −316 1240 −1106 134 352 miles were driven, averaging approximately 22.3 miles per gallon.

214

cups → 24 4 ← cups = quarts → 6 1 ← quarts students → 560 112 ← students = 5 ← instructors instructors → 25

4 10 0 8 20 4 ⋅ 20 0 8 ⋅10 80 = 80 The proportion is true. 4.2 4.8 0 6 8 4.2 ⋅ 8 0 6 ⋅ 4.8 33.6 ≠ 28.8 The proportion is false.

ISM: Prealgebra

3 3 10

Chapter 6: Ratio, Proportion, and Triangle Applications

4 15

2 13 1 56 3 1 5 1 3 ⋅2 01 ⋅4 10 3 6 5 33 7 11 21 ⋅ 0 ⋅ 10 3 6 5 231 231 = 30 30 The proportion is true.

2 x 5. = 5 25 2 ⋅ 25 = 5 ⋅ x 50 = 5 x 50 5 x = 5 5 10 = x

−15 60 = x 2 −15 ⋅ x = 2 ⋅ 60 −15 x = 120 −15 x 120 = −15 −15 x = −8 7 8

10.

4 7

y 0.6 = 9 1.2 y ⋅1.2 = 9 ⋅ 0.6 1.2 y = 5.4 1.2 y 5.4 = 1.2 1.2 y = 4.5

17 8 = z 10 17 ⋅10 = z ⋅ 8 170 = 8 z 170 8 z = 8 8 85 =z 4 21.25 = z 4.5 y = 1.8 3 4.5 ⋅ 3 = 1.8 ⋅ y 13.5 = 1.8 y 13.5 1.8 y = 1.8 1.8 7.5 = y

Vocabulary, Readiness & Video Check 6.2 1.

4.2 1 7 = is called a proportion while is called 8.4 2 8 a ratio. a c = , a ⋅ d and b ⋅ c are called cross b d products.

2. In

= 3

7 4 2 ⋅ = z⋅ 8 7 3 1 2 = z⋅ 2 3 1 3 2 3 ⋅ = z⋅ ⋅ 2 2 3 2 3 =z 4

3. In a proportion, if cross products are equal, the proportion is true. 4. In a proportion, if cross products are not equal, the proportion is false. 5. equals or = 6. complex fractions 7. It is a ratio equal to a ratio. 8. Our answer is not exact; it is a rounded approximation. So the cross products should be close but not exactly equal. Exercise Set 6.2 2.

1 raisin 8 raisins = 5 cornflakes 40 cornflakes

4 hits 1 hit = 16 releases 4 releases

215

Chapter 6: Ratio, Proportion, and Triangle Applications

10.

12.

14.

16.

18.

20.

12 errors 1.5 errors = 8 pages 1 page 1 12 cups milk 10 bagels

ISM: Prealgebra

2 3 8 1 ⋅ 0 ⋅ 3 10 5 7 1 8 ≠ 5 35 The proportion is false.

3 cup milk 4

5 bagels

1 cup of instant rice 1.5 cups instant rice = 1.5 cups cooked rice 2.25 cups cooked rice

8 20 0 6 15 8 ⋅15 0 6 ⋅ 20 120 = 120 The proportion is true. 7 9 0 3 5 7 ⋅5 0 3⋅9 35 ≠ 27 The proportion is false.

0.7 0.3 0 0.4 0.1 0.7 ⋅ 0.1 0 0.4 ⋅ 0.3 0.07 ≠ 0.12 The proportion is false.

8 5.6 22. 0 10 0.7 8 ⋅ 0.7 0 10 ⋅ 5.6 5.6 ≠ 56 The proportion is false.

5 85

26.

5 3

4 12 1 15

5 1 5 1 5 ⋅1 0 ⋅ 4 8 5 3 2 45 6 5 9 ⋅ 0 ⋅ 8 5 3 2 27 15 ≠ 4 2 The proportion is false.

28.

6 7

0 7 3 5

6 10 ⋅5 0 3⋅ 7 7 30 30 = 7 7 The proportion is true.

8 3 0 24 9 8 ⋅ 9 0 24 ⋅ 3 72 = 72 The proportion is true. 30 600 0 50 1000 30 ⋅1000 0 50 ⋅ 600 30, 000 = 30, 000 The proportion is true.

2 1 3 0 7 8 3 5 10

24.

30.

6 9 0 8 12 6 ⋅12 0 8 ⋅ 9 72 = 72

The proportion

32.

5 7 0 3 5 5⋅5 0 3⋅ 7 25 ≠ 21

The proportion

34.

5 7 = is false. 3 5

1.8 4.5 0 2 5 1.8 ⋅ 5 0 2 ⋅ 4.5 9=9

The proportion

216

6 9 is true. = 8 12

1.8 4.5 is true. = 2 5

ISM: Prealgebra

36.

Chapter 6: Ratio, Proportion, and Triangle Applications

10 1 11 04 3 1 2 4

48.

10 1 3 1 ⋅ 0 ⋅ 11 2 4 4 5 3 ≠ 11 16 10

3 4

1 2

The proportion 11 = 4 is false.

38.

40.

42.

44.

46.

x 12 = 3 9 x ⋅ 9 = 3 ⋅12 9 x = 36 9 x 36 = 9 9 x=4

50.

52.

12 z = 10 16 12 ⋅16 = 10 ⋅ z 192 = 10 z 192 10 z = 10 10 19.2 = z 26 28 = x 49 26 ⋅ 49 = x ⋅ 28 1274 = 28 x 1274 28 x = 28 28 45.5 = x

7.8 n = 13 2.6 7.8 ⋅ 2.6 = 13 ⋅ n 20.28 = 13n 20.28 13n = 13 13 1.56 = n

12 3 4

48 n

3 ⋅ 48 4 12n = 36 12n 36 = 12 12 n=3

25 −7 = 100 n 25 ⋅ n = 100 ⋅ (−7) 25n = −700 25n −700 = 25 25 n = −28 16 z = 20 35 16 ⋅ 35 = 20 ⋅ z 560 = 20 z 560 20 z = 20 20 28 = z

−0.2 −8 = n 0.7 −0.2 ⋅ n = 0.7 ⋅ (−8) −0.2n = −5.6 −0.2n −5.6 = −0.2 −0.2 n = 28

12 ⋅ n =

54.

7 1 9 = 4 8 n 27

7 8 1 ⋅n = ⋅ 9 27 4 7 2 n= 9 27 9 7 9 2 ⋅ n= ⋅ 7 9 7 27 2 n= 21 8

56.

24 15 = 5 n 9

5 8 24 ⋅ = n ⋅ 9 15 40 8 = n⋅ 3 15 15 40 8 15 ⋅ = n⋅ ⋅ 8 3 15 8 25 = n

217

Chapter 6: Ratio, Proportion, and Triangle Applications

58.

60.

3 n 75 = 3 18 2 83 3 1 3 2 ⋅n = 3 ⋅7 8 8 5 19 25 38 ⋅n = ⋅ 8 8 5 19 95 ⋅n = 8 4 8 19 8 95 ⋅ ⋅n = ⋅ 19 8 19 4 n = 10

ISM: Prealgebra

68.

70. 7.26 7.026 ↑ ↑ 2 > 0 7.26 > 7.026

9 5 = n 11

72.

11 9⋅ = n⋅5 15 33 = 5n 5 1 33 1 ⋅ = 5n ⋅ 5 5 5 33 =n 25

74.

62.

64.

66.

218

1.8 2.5 = 8.4 z 1.8 ⋅ 8.4 = z ⋅ 2.5 15.12 = 2.5 z 15.12 2.5 z = 2.5 2.5 6.0 ≈ z 4.25 5 = 6.03 y 4.25 ⋅ y = 6.03 ⋅ 5 4.25 y = 30.15 4.25 y 30.15 = 4.25 4.25 y ≈ 7.09 17 9 = x 4 17 ⋅ 4 = x ⋅ 9 68 = 9 x 68 9 x = 9 9 7.556 ≈ x

x 18 = 12 7 x ⋅ 7 = 12 ⋅18 7 x = 216 7 x 216 = 7 7 x ≈ 30.86

76.

78.

80.

1 1 < 5 4 1 1 so 9 < 9 5 4

11 y 1 ⋅11 ⋅ y 1 = = 99 y 9 ⋅11 ⋅ y 9

28 y 2 42 y

2 ⋅14 ⋅ y ⋅ y 2 = 3 ⋅14 ⋅ y ⋅ y ⋅ y 3 y

1 5 = 4 20 1 4 = 5 20 20 5 = 4 1 4 20 = 1 5 2 4 = 7 14 14 4 = 7 2 2 7 = 4 14 7 14 = 2 4

82. answers may vary 84. answers may vary

ISM: Prealgebra

86.

88.

90.

Chapter 6: Ratio, Proportion, and Triangle Applications

0 y = 2 3.5 0 ⋅ 3.5 = 2 ⋅ y 0 = 2y 0 2y = 2 2 0= y

7. The ratio of 7 2

7 to 13 is 2

7 7 1 7 ⋅1 7 ÷ 13 = ⋅ = = . 2 2 13 2 ⋅13 26

8. The ratio of 1 1 23

585 117 = x 474 585 ⋅ 474 = x ⋅117 277, 290 = 117 x 277, 290 117 x = 117 117 2370 = x

2 34

3 2 to 2 is 3 4

2 3 5 11 5 4 5 ⋅ 4 20 =1 ÷2 = ÷ = ⋅ = = . 3 4 3 4 3 11 3 ⋅11 33

9. The ratio of 16 inches to 24 inches is 16 inches 2 ⋅ 8 2 = = . 24 inches 3 ⋅ 8 3 10. The ratio of 5 hours to 40 hours is 5 hours 5 ⋅1 1 = = . 40 hours 5 ⋅ 8 8

1425 z = 1062 177 1425 ⋅177 = 1062 ⋅ z 252, 225 = 1062 z 252, 225 1062 z = 1062 1062 237.5 = z

11.

12 inches 6 ⋅ 2 2 = = 18 inches 6 ⋅ 3 3

12. a.

Mid-Chapter Review 1. The ratio of 27 to 30 is

27 9 ⋅ 3 9 = = . 30 10 ⋅ 3 10

2. The ratio of 18 to 50 is

18 9 ⋅ 2 9 = = . 50 25 ⋅ 2 25

3. The ratio of 9.4 to 10 is 9.4 9.4 × 10 94 47 ⋅ 2 47 = = = = . 10 10 × 10 100 50 ⋅ 2 50 4. The ratio of 3.2 to 9.2 is 3.2 3.2 × 10 32 4 ⋅ 8 8 = = = = . 9.2 9.2 × 10 92 4 ⋅ 23 23 5. The ratio of 8.65 to 6.95 is 8.65 8.65 × 100 865 173 ⋅ 5 173 . = = = = 6.95 6.95 × 100 695 139 ⋅ 5 139 6. The ratio of 3.6 to 4.2 is 3.6 3.6 × 10 36 6 ⋅ 6 6 = = = = . 4.2 4.2 × 10 42 7 ⋅ 6 7

67 films were rated R.

b. 40 + 61 + 67 + 32 = 200 61 films 61 = 200 films 200 13.

4 professors 1 professor = 20 graduate assistants 5 graduate assistants

14.

6 lights 3 lights = 20 feet 10 feet

15.

100 Senators 2 Senators = 50 states 1 state

16.

5 teachers 1 teacher = 140 students 28 students

17.

21 inches 3 inches = 7 seconds 1 second

18.

$40 $8 = 5 hours 1 hour

219

Chapter 6: Ratio, Proportion, and Triangle Applications

19.

20.

21.

76 households with computers 100 households 19 households with computers = 25 households

ISM: Prealgebra

25.

538 electoral votes 269 electoral votes = 50 states 25 states 560 feet 4 seconds

The unit rate is

140 560 −4 16 −16 00

The unit rate is

26.

140 feet or 140 feet/second 1 second

26 6 156 12 36 −36 0

156 miles 6 gallons

26 miles or 26 miles/gallon. 1 gallon

112 teachers 7 computers

16 7 112 −7 42 −42 0

16 teachers or 1 computer 16 teachers/computer.

The unit rate is 22.

195 miles 3 hours

The unit rate is

63 employees 23. 3 printers

65 3 195 −18 15 −15 0

27.

8125 books 1250 college students

65 miles or 65 miles/hour. 1 hour

6.5 books or 1 college student 6.5 books/student.

21 3 63 −6 03 −3 0

The unit rate is

28.

2310 pounds 14 adults

21 employees or 1 printer 21 employees/printer.

The unit rate is

24.

85 phone calls 5 teenagers

17 85 −5 35 −35 0

165 14 2310 −14 91 −84 70 −70 0

165 pounds or 1 adult 165 pounds/adult.

The unit rate is

17 phone calls or 1 teenager 17 phone calls/teenager.

The unit rate is

220

6.5 1250 8125.0 −7500 625 0 −625 0 0

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications

1.6633 ≈ 1.663 29. 3 4.9900 −3 19 −1 8 19 −18 10 −9 10 −9 1 The 3-pound size costs $1.663 per pound. 0.9993 ≈ 0.999 16 15.9900 −14 4 1 59 −1 44 150 −144 60 −48 12 The 16-pound size costs $0.999 per pound. The 16-pound size is the better buy.

0.0649 ≈ 0.065 30. 100 6.4900 −6 00 490 −400 900 −900 0 The 100-plate size costs $0.065 per plate. 0.0768 ≈ 0.077 48 3.6900 −3 36 330 −288 420 −384 36 The 48-plate size costs $0.077 per plate. The 100 paper plates is the better buy.

0.4575 ≈ 0.458 31. 12 5.4900 −4 8 69 −60 90 −84 60 −60 0 The 12-pack size costs $0.458 per pack. 0.4162 ≈ 0.416 24 9.9900 −9 6 39 −24 150 −144 60 −48 12 The 24-pack size costs $0.416 per pack. The 24-pack size is the better buy. 1.3725 ≈ 1.373 32. 4 5.4900 −4 14 −1 2 29 −28 10 −8 20 −20 0 The 4-pack is $1.373 per battery. 0.8431 ≈ 0.843 16 13.4900 −12 8 69 −64 50 −48 20 −16 4 The 16-pack is $0.843 per battery. The 16-pack is the better buy.

221

Chapter 6: Ratio, Proportion, and Triangle Applications

33.

7 5 0 4 3 7 ⋅3 0 4⋅5 21 ≠ 20 The proportion is false.

ISM: Prealgebra Section 6.3 Practice Exercises 1. Let x be the length of the wall. feet → 4 x ← feet = inches → 1 4 14 ← inches

1 = 1⋅ x 4 4 17 ⋅ =x 1 4 17 = x The wall is 17 feet long. 4⋅4

8.2 16.4 0 34. 2 4 8.2 ⋅ 4 0 2 ⋅16.4 32.8 = 32.8 The proportion is true. 35.

36.

37.

5 40 = 3 x 5 ⋅ x = 3 ⋅ 40 5 x = 120 5 x 120 = 5 5 x = 24

y 13 = 10 4 y ⋅ 4 = 10 ⋅13 4 y = 130 4 y 130 = 4 4 y = 32.5

6 z = 11 5 6 ⋅ 5 = 11 ⋅ z 30 = 11z 30 11z = 11 11 2.72 = z or z = 2 7

38.

21 2 = x 3

7 21 ⋅ 3 = x ⋅ 2 7 63 = ⋅ x 2 2 63 2 7 ⋅ = ⋅ ⋅x 7 1 7 2 18 = x

222

8 11

ounces → 5 8 ← ounces = gallons →16 x ← gallons 5 ⋅ x = 16 ⋅ 8 5 x = 128 5 x 128 = 5 5 3 x = 25.6 or 25 5 3 Therefore, 25.6 or 25 gallons of gas can be 5 treated with 8 ounces of alcohol.

3. Let x be the number of gallons. 270 × 11 = 2970 square feet x ← gallons gallons → 1 = square feet → 450 2970 ← square feet 1 ⋅ 2970 = 450 ⋅ x 2970 450 x = 450 450 6.6 = x 7≈x Therefore, 7 gallons are needed for the wall. Vocabulary, Readiness & Video Check 6.3 1. ones 2.

ounces ounces 5 3 .5 or = = x 72 mg cholesterol mg cholesterol

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications

Exercise Set 6.3

10. Let x be the number of students. floor space → 9 21× 15 ← floor space = x ← students students → 1 9 ⋅ x = 1 ⋅ 21 ⋅15 9 x = 315 9 x 315 = 9 9 x = 35 The room can accommodate 35 students.

2. Let x be the number of attempts. field goals → 45 225 ← field goals = x ← attempts attempts →100 45 ⋅ x = 100 ⋅ 225 45 x = 22,500 45 x 22,500 = 45 45 x = 500 The player made 500 attempts. 4. Let x be the number of pages. minutes → 30 4.5 × 60 ← minutes = x pages → 4 ← pages 30 ⋅ x = 4 ⋅ 4.5 ⋅ 60 30 x = 1080 30 x 1080 = 30 30 x = 36 The assistant can type and spell-check 36 pages in 4.5 hours. 6. Let x be the number of applications received. accepted → 2 150 ← accepted = x ← applied applied → 7 2 ⋅ x = 7 ⋅150 2 x = 1050 2 x 1050 = 2 2 x = 525 The school received 525 applications. 8. Let x be the length of the wall. inches → 1 5 14 ← inches = feet → 8 x ← feet

1 4 8 21 x= ⋅ 1 4 x = 42 The wall is 42 feet long. 1⋅ x = 8 ⋅ 5

12. Let x be the number of miles she can go. x ← miles miles → 588 = gallons →11.3 6.9 ← gallons 588 ⋅ 6.9 = 11.3 ⋅ x 4057.2 = 11.3x 4057.2 11.3x = 11.3 11.3 359 ≈ x Yes, she should be able to go 359 miles (which is greater than 290 miles). 14. Let x be the distance between the small Italian village and the Mediterranean Sea. kilometers → 30 x ← kilometers = cm on map → 1 0.4 ← cm on map 30 ⋅ 0.4 = 1 ⋅ x 12 = x The actual distance is 12 kilometers. 16. Let x be the number of bags. x bags → 1 ← bags = square feet → 5000 160 ⋅160 ← square feet 1 ⋅160 ⋅160 = 5000 ⋅ x 25, 600 = 5000 x 25, 600 5000 x = 5000 5000 5.12 ≈ x 6≈x 6 bags of fertilizer should be purchased. 18. Let x be the times at bat. hits → 3 12 ← hits = at bats → 8 x ← at bats 3 ⋅ x = 8 ⋅12 3 x = 96 3 x 96 = 3 3 x = 32 The player batted 32 times.

223

Chapter 6: Ratio, Proportion, and Triangle Applications 20. Let x be the number that prefer Pepsi. Pepsi → 1 x ← Pepsi = Total → 3 36 ← Total 1 ⋅ 36 = 3 ⋅ x 36 3 x = 3 3 12 = x 12 students are likely to prefer Pepsi. 22. Let x be the number of ounces. applications → 4 35 ← applications = ounces → 3 x ← ounces 4 ⋅ x = 3 ⋅ 35 4 x = 105 4 x 105 = 4 4 x ≈ 27 They should purchase 27 ounces. 24. Let x be the number of reams. reams → 5 x ← reams = weeks → 3 15 ← weeks 5 ⋅15 = 3 ⋅ x 75 = 3x 75 3 x = 3 3 25 = x They will need 25 reams or 25 = 3.125 ≈ 4 cases. 8

30. a.

Let x be the number of ounces. x ← ounces ounces → 3 = 150 5280 ← feet feet → 3 ⋅ 5280 = 150 ⋅ x 15,840 = 150 x 15,840 150 x = 150 150 105.6 = x 105.6 ounces of pesticide are needed.

b. Let x be the number of cups. ounces → 8 105.6 ← ounces = x ← cups cups → 1 8 ⋅ x = 1 ⋅105.6 8 x 105.6 = 8 8 x = 13.2 x ≈ 13 This is about 13 cups of pesticide. 32. Let x be the amount spent on pet food at PetSmart. x pet food → 50 ← pet food = 125 6, 676, 700, 000 total sales → ← total sales 50 ⋅ 6, 676, 700, 000 = 125 ⋅ x 333,835, 000, 000 = 125 x 333,835, 000, 000 125 x = 125 125 2, 670, 680, 000 = x We expect that $2,670,680,000 was spent on pet food.

26. Let x be the number of cups of flour. flour → 2 x ← flour = servings → 4 18 ← servings 2 ⋅18 = 4 ⋅ x 36 = 4 x 36 4 x = 4 4 9=x It will take 9 cups of flour. 28. Let x be the the time to reach the restaurant (in seconds). distance → 800 500 ← distance = x ← time time → 60 800 ⋅ x = 60 ⋅ 500 800 x = 30, 000 800 x 30, 000 = 800 800 x = 37.5 It takes 37.5 seconds to reach the restaurant. 224

ISM: Prealgebra

34. a.

Let x be the length of your index finger. x ← index finger index finger → 8 = height →111 5 ← height 8 ⋅ 5 = 111 ⋅ x 40 = 111x 40 111x = 111 111 40 x= ≈ 0.36 111 40 The proposed length is foot or about 111 0.36 foot.

b. answers may vary

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications

36. Let x be the number of milligrams. milligrams → 76 x ← milligrams = ounces → 3 8 ← ounces 76 ⋅ 8 = 3 ⋅ x 608 = 3x 608 3x = 3 3 2 202 = x 3 2 There are 202 milligrams of cholesterol in 3 8 ounces of chicken. 38. Let x be the number of men that blame. blame → 2 x ← blame = total → 5 40 ← total 2 ⋅ 40 = 5 ⋅ x 80 = 5 x 80 5 x = 5 5 16 = x You would expect 16 men to blame their not eating well on fast food. 40. Let x be the number of people who are retired. x ← retired retired →13 = not retired → 5 280 ← not retired 13 ⋅ 280 = 5 ⋅ x 3640 = 5 x 3640 5 x = 5 5 728 = x You would expect 728 people age 65 and older who are retired.

44. Let x be the cups of salt. ice → 5 18 34 ← ice = salt → 1 x ← salt 3 5 ⋅ x = 1 ⋅18 4 75 5x = 4 1 1 75 ⋅ 5x = ⋅ 5 5 4 15 x= 4 3 x=3 4 3 Mix 3 cups of salt with the ice. 4 46. a.

b. Let x be the number of quarts. gallons → 1 0.5 ← gallons = quarts → 4 x ← quarts 1 ⋅ x = 4 ⋅ 0.5 x=2 0.5 gallon is 2 quarts. 48. a.

42. Let x be the number of pounds. pounds → 1 x ← pounds = cups → 2 14 6 ← cups

1 1⋅ 6 = 2 ⋅ x 4 9 6 = ⋅x 4 4 6 4 9 ⋅ = ⋅ x 9 1 9 4 8 =x 3 2 x=2 3 2 The recipe requires 2 pounds of firmly packed 3 brown sugar.

Let x be the number of gallons of oil needed. oil → x 1 ← oil = gas →10 20 ← gas x ⋅ 20 = 10 ⋅1 20 x 10 = 20 20 x = 0.5 0.5 gallon of oil is needed.

Let x be the milligrams of medicine. milligrams → x 80 ← milligrams = pounds →190 25 ← pounds 25 ⋅ x = 190 ⋅ 80 25 x = 15, 200 25 x 15, 200 = 25 25 x = 608 The daily dose is 608 milligrams. 24 = 608 ÷ 4 = 152 6 She should be given 152 milligrams every 6 hours. 608 ÷

225

Chapter 6: Ratio, Proportion, and Triangle Applications 50. 300 = 2 ⋅150 ↓ ↓ 2 ⋅ 2 ⋅ 75 ↓ ↓ ↓ 2 ⋅ 2 ⋅ 3 ⋅ 25 ↓ ↓ ↓ ↓ 2⋅2 ⋅ 3 ⋅ 5 ⋅ 5

ISM: Prealgebra

60.

= 2 2 ⋅ 3 ⋅ 52

52. 81 = 3 ⋅ 27 ↓ ↓ 3⋅3 ⋅ 9 ↓ ↓ ↓ 3⋅3 ⋅ 3 ⋅ 3

n ← feet feet → 3 = pounds →125 400 ← pounds 3 ⋅ 400 = 125 ⋅ n 1200 = 125n 1200 125n = 125 125 3 9 =n 5 3 The distance is 9 feet. 5

Section 6.4 Practice Exercises

= 34

54. Let x be the number of ml. mg →15 33 ← mg = ml → 1 x ← ml 15 ⋅ x = 1 ⋅ 33 15 x = 33 15 x 33 = 15 15 x = 2.2 2.2 ml of the medicine should be administered.

64 = 8 because 82 = 64.

169 = 13 because 132 = 169.

0 = 0 because 02 = 0.

1 1 1 1 = because   = . 4 2 4 2

9 3 9 3 = because   = . 16 4 4 16  

58. 2 dozen cookies = 24 cookies

cups of flour, or 5 cups of flour.

100 = 10 because 10 2 = 100.

56. Let x be the number of ml. mg → 8 6 ← mg = ml → 1 x ← ml 8 ⋅ x = 1⋅ 6 8x = 6 8x 6 = 8 8 x = 0.75 0.75 ml of the medicine should be administered.

If we double 24, we have 48 ≈ 50, so double 2

1 2

7. a.

10 ≈ 3.162

62 ≈ 7.874

8. Recall that 49 = 7 and 64 = 8. Since 62 is between 49 and 64, then 62 is between 49 and 64. Thus, 62 is between 7 and 8. Since 62 is closer to 64, then 62 is closer to 64, or 8. 9. Let a = 12 and b = 16. a 2 + b2 = c2 122 + 162 = c 2 144 + 256 = c 2 400 = c 2 400 = c 20 = c The hypotenuse is 20 feet.

226

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications

10. Let a = 9 and b = 7. 2

a +b = c

Vocabulary, Readiness & Video Check 6.4

1. The square roots of 100 are 10 and −10 because 10 ⋅ 10 = 100 and (−10)(−10) = 100.

92 + 7 2 = c 2 81 + 49 = c 2 130 = c 2 130 = c 11 ≈ c The hypotenuse is approximately 11 kilometers.

11. Let a = 7 and c = 13.

100 = 10 only because 10 ⋅ 10 = 100 and 10 is positive.

3. The radical sign is used to denote the positive square root of a nonnegative number. 4. The reverse process of squaring a number is finding a square root of a number.

a 2 + b2 = c2 7 2 + b 2 = 132 49 + b 2 = 169

5. The numbers 9, 1, and

b 2 = 120 b = 120 b ≈ 10.95 The length of the hypotenuse is exactly 120 feet or approximately 10.95 feet.

squares. 6. Label the parts of the right triangle. leg

100 yd

12.

1 are called perfect 25

hypotenuse

leg

53 yd

7. In the given triangle, a 2 + c 2 = b2 .

Let a = 53 and b = 100. a 2 + b2 = c 2

8. The Pythagorean Theorem can be used for right triangles.

532 + 1002 = c 2

9. The square roots of 49 are 7 and −7 since

2809 + 10, 000 = c 2

7 2 = 49 and (−7) 2 = 49. The radical sign

12,809 = c 12,809 = c 113 ≈ c The diagonal is approximately 113 yards.

Calculator Explorations

means the positive square root only, so

49 = 7.

10. Since 38 is between 36 = 6 and 49 = 7 and 38 is much closer to 36 than to 49, we know that 38 is closer to 6 than 7. 11. The hypotenuse is the side across from the right angle.

1024 = 32

676 = 26

15 ≈ 3.873

9 = 3 because 32 = 9.

19 ≈ 4.359

144 = 12 because 12 2 = 144.

97 ≈ 9.849

56 ≈ 7.483

1 1 1 1 1 1 = because   = ⋅ = . 64 8 8 8 64 8

36 6 2 6 6 36 6 = = because   = ⋅ = . 81 9 3 9 9 81 9

Exercise Set 6.4

227

Chapter 6: Ratio, Proportion, and Triangle Applications

ISM: Prealgebra

10.

5 ≈ 2.236

36. Let a = 36 and b = 27.

12.

17 ≈ 4.123

362 + 272 = c 2

14.

85 ≈ 9.220

1296 + 729 = c 2

16.

35 ≈ 5.916

a 2 + b2 = c 2

2025 = c 2 2025 = c 45 = c The missing length is 45 kilometers.

18. Since 27 is between 25 = 5 ⋅ 5 and 36 = 6 ⋅ 6, 27 is between 5 and 6; 27 ≈ 5.20. 20. Since 85 is between 81 = 9 ⋅ 9 and 100 = 10 ⋅ 10, 85 is between 9 and 10; 85 ≈ 9.22. 22.

625 = 25 because 252 = 625.

24.

18 ≈ 4.243

26.

121 11 11 11 121  11  = because   = ⋅ = . 169 13 13 13 169  13 

28.

62 ≈ 7.874

38. 12

hypotenuse = (leg) 2 + (other leg) 2 = 92 + 122 = 81 + 144 = 225 = 15 The hypotenuse has length 15 units.

30. Let a = 36 and b = 15. a 2 + b2 = c 2 362 + 152 = c 2 1296 + 225 = c 2

40.

1521 = c 2 1521 = c 39 = c The missing length is 39 feet.

leg = (hypotenuse)2 − (other leg) 2

32. Let a = 3 and c = 9. a 2 + b2 = c2 2

3 +b = 9

= 102 − 62 = 100 − 36 = 64 =8 The leg has length 8 units.

9 + b 2 = 81 b 2 = 72 b = 72 b ≈ 8.485 The missing length is approximately 8.485 yards.

34. Let a = 34 and b = 70.

42.

a 2 + b2 = c2 342 + 702 = c 2

1156 + 4900 = c 2 6056 = c 2 6056 = c 77.820 ≈ c The missing length is approximately 77.820 miles. 228

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications

hypotenuse = (leg)2 + (other leg)2 = 22 + 162 = 4 + 256 = 260 ≈ 16.125 The hypotenuse has length of about 16.125 units.

hypotenuse = (leg) 2 + (other leg) 2 = 302 + 152 = 900 + 225 = 1125 ≈ 33.541 The hypotenuse has length of about 33.541 units.

46.

hypotenuse = (leg) 2 + (other leg) 2 = 212 + 212 = 441 + 441 = 882 ≈ 29.698 The hypotenuse has length of about 29.698 units. 8

hypotenuse = (leg) 2 + (other leg) 2 = 122 + (22.5)2

52. hypotenuse = (leg) 2 + (other leg) 2 = 12 + 12 = 1+1 = 2 ≈ 1.414 The length of the diagonal is about 1.414 miles.

= 1682 − 602 = 28, 224 − 3600 = 24, 624 ≈ 156.9 The height of the antenna is about 156.9 feet.

56. hypotenuse = (leg) 2 + (other leg) 2 = 702 + 1102 = 4900 + 12,100 = 17, 000 ≈ 130.4 The length of the diagonal is about 130.4 yards.

58. leg = (hypotenuse) 2 − (other leg) 2

leg = (hypotenuse)2 − (other leg) 2 2

54. leg = (hypotenuse) 2 − (leg)2 21

48.

22.5

= 144 + 506.25 = 650.25 = 25.5 The hypotenuse has length 25.5 units.

44.

50.

= 9 −8 = 81 − 64 = 17 ≈ 4.123 The leg has length of about 4.123 units.

= (60)2 − (29.4)2 = 3600 − 864.36 = 2735.64 ≈ 52.3 The width of the television is about 52.3 inches.

229

Chapter 6: Ratio, Proportion, and Triangle Applications

60. leg = (hypotenuse) 2 − (other leg) 2 = (55) 2 − (47.9) 2 = 3025 − 2294.41 = 730.59 ≈ 27.0 The height of the television is about 27.0 inches. 2

62. hypotenuse = (leg) + (other leg) 2

= (43.6) + (24.5)

ISM: Prealgebra

76. Recall that 81 = 9 and 100 = 10. Since 85 is between 81 and 100, then 85 is between 81 and 100. Thus, 85 is between 9 and 10. Since 85 is closer to 81, then 85 is closer to 81, or 9. Check: 85 ≈ 9.22 78. answers may vary

202 + 452 0 502 400 + 2025 0 2500 2425 ≠ 2500 No; the set does not form the lengths of the sides of a right triangle.

= 1900.96 + 600.25 = 2501.21 ≈ 50 The diagonal of the television is about 50 inches. 1.774 ≈ 1.77 64. 27 47.900 −27 20 9 −18 9 2 00 −1 89 110 −108 2 Yes, the aspect ratio of the television is approximately 1.78 : 1.

66.

10 2 ⋅ 5 2 = = 15 3 ⋅ 5 3

68.

35 5⋅7 7 = = 75 y 5 ⋅15 ⋅ y 15 y

70.

3x 5 3x − 5 − = 9 9 9

72.

7 x 8 x 7 x 11 7 ⋅ x ⋅11 7 ÷ = ⋅ = = 11 11 11 8 x 11 ⋅ 8 ⋅ x 8

5. Check:

230

Section 6.5 Practice Exercises 1. a.

The triangles are congruent by Side-AngleSide.

b. The triangles are not congruent. 2.

9 meters 9 = 13 meters 13

The ratio of corresponding sides is

74. Recall that 25 = 5 and 36 = 6. Since 27 is between 25 and 36, then 27 is between 25 and 36. Thus, 27 is between 5 and 6. Since 27 is closer to 25, then 27 is closer to 25, or

a 2 + b2 = c2

80.

9 . 13

x 6 = 5 9 x ⋅9 = 5⋅6 9 x = 30 9 x 30 = 9 9 10 1 x= or 3 units 3 3

5 8 = n 60 5 ⋅ 60 = n ⋅ 8 300 = 8n 300 8n = 8 8 37.5 = n The height of the building is approximately 37.5 feet.

27 ≈ 5.20

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications

Vocabulary, Readiness & Video Check 6.5

16.

1. Two triangles that have the same shape, but not necessarily the same size are congruent. false 2. Two triangles are congruent if they have the same shape and size. true 3. Congruent triangles are also similar. true 4. Similar triangles are also congruent. false

18.

5. For the two similar triangles, the ratio of 5 corresponding sides is . false 6 6. Since the sides of both triangles are given, and no angle measures are given, we used SSS. 7. The ratios of corresponding sides are the same. 8.

20.

12 4 = 18 n

Exercise Set 6.5 2. The triangles are congruent by Side-Side-Side. 22.

4. The triangles are not congruent. 6. The triangles are congruent by Angle-SideAngle. 8. The triangles are congruent by Side-Angle-Side. 10.

8 4 7 1 = = = 32 16 28 4

1 . 4

24.

x 60 = 3 5 x ⋅ 5 = 3 ⋅ 60 5 x = 180 5 x 180 = 5 5 x = 36

x 14 = 9 8 x ⋅ 8 = 9 ⋅14 8 x = 126 8 x 126 = 8 8 x = 15.75 33.2 y = 9.6 8.3 y ⋅ 8.3 = 9.6 ⋅ 33.2 8.3 y = 318.72 8.3 y 318.72 = 8.3 8.3 y = 38.4

z 13 2 = 6 9

1 2 27 9z = 6 ⋅ 2 9 z = 81 9 z 81 = 9 9 z=9

z ⋅ 9 = 6 ⋅13

6 8 4 = = 1 42 6 3 The ratio of corresponding sides is

14.

z 22.5 = 9 15 z ⋅15 = 9 ⋅ 22.5 15 z = 202.5 15 z 202.5 = 15 15 z = 13.5

The ratio of corresponding sides is

12.

y 14 = 4 7 y ⋅ 7 = 4 ⋅14 7 y = 56 7 y 56 = 7 7 y =8

4 . 3

26.

x 13 = 13 26 x ⋅ 26 = 13 ⋅13 26 x = 169 26 x 169 = 26 26 x = 6.5

231

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications

28.

30.

32.

34.

36.

8 n = 37.5 20 n ⋅ 20 = 37.5 ⋅ 8 20n = 300 20n 300 = 20 20 n = 15

38. hypotenuse = (leg) 2 + (other leg) 2

28 5 = x 100 x ⋅ 5 = 28 ⋅100 5 x = 2800 5 x 2800 = 5 5 x = 560 The fountain is approximately 560 feet high. x 76 = 10 2 x ⋅ 2 = 10 ⋅ 76 2 x = 760 2 x 760 = 2 2 x = 380 The tree is 380 feet tall.

= 1002 + 502 = 10, 000 + 2500 = 12,500 ≈ 112 The length of the diagonal is approximately 112 yards.

40. Subtract the absolute values. 3.60 − 0.41 3.19 Attach the sign of the larger absolute value. 0.41 − 3.6 = −3.19 0.16 42. 3 0.48 −3 18 −18 0

−0.48 ÷ 3 = −0.16

x 44 = 32 24 x ⋅ 24 = 32 ⋅ 44 24 x = 1408 24 x 1408 = 24 24 x ≈ 58.7 The shadow is approximately 58.7 feet. 3 x = 5 8 3⋅8 = 5⋅ x 24 = 5 x 24 5 x = 5 5 4.8 = x 45 − 4.8 = 40.2 The speed of the truck is 40.2 miles per hour.

44. Let x be the width of the banner. 1 11 3 = 2 x 9 1 1 ⋅ 9 = x ⋅1 3 2 3 3= x 2 2 23  (3) =  x  3 32  2=x The width of the banner is 2 feet. 46.

n 11.6 = 58.7 20.8 n ⋅ 20.8 = 58.7 ⋅11.6 20.8n = 680.92 20.8n 680.92 = 20.8 20.8 n ≈ 32.7

48. answers may vary 50.

x 1 = 3 1 4

1 x ⋅ = 3 ⋅1 4 x = 12

232

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications 15. In the right triangle,

y 1 = 4 12 14 1 1 y ⋅ = 4 ⋅1 4 2 y = 18

b 2

a + b = c 2 is called the Pythagorean theorem.

z 1 = 6 1

Chapter 6 Review

1 = 6 ⋅1 4 z = 24 The lengths of the sides of the deck are 12 feet, 18 feet, and 24 feet. z⋅

Chapter 6 Vocabulary Check 1. A ratio is the quotient of two numbers. It can be written as a fraction, using a colon, or using the word to. 2.

x 7 is an example of a proportion. = 2 16

1. The ratio of 23 to 37 is

23 . 37

2. The ratio of $121 to $143 is

$121 11 ⋅11 11 = = . $143 11 ⋅13 13

3. The ratio of 4.25 yards to 8.75 yards is 4.25 yards 4.25 × 100 425 25 ⋅17 17 = = = = . 8.75 yards 8.75 × 100 875 25 ⋅ 35 35 4. The ratio of 2 2 14

3. A unit rate is a rate with a denominator of 1.

4 83

1 3 to 4 is 4 8

1 3 9 35 9 8 9 ⋅ 4 ⋅ 2 18 = 2 ÷4 = ÷ = ⋅ = = . 4 8 4 8 4 35 4 ⋅ 35 35

4. A unit price is a “money per item” unit rate. 5. A rate is used to compare different kinds of quantities. x 7 = , x ⋅ 16 and 2 ⋅ 7 are 2 16 called cross products.

6. In the proportion

7. If cross products are equal, the proportion is true. 8. If cross products are not equal, the proportion is false.

length 4.5 meters 4.5 × 10 45 5 ⋅ 9 9 = = = = = width 2 meters 2 × 10 20 5 ⋅ 4 4

width 2 meters 2 = = perimeter (4.5 + 2 + 4.5 + 2) meters 13

6000 people 1200 ⋅ 5 people 5 people = = 2400 pets 1200 ⋅ 2 pets 2 pets

15 pages 3 ⋅ 5 pages 5 pages = = 6 minutes 3 ⋅ 2 minutes 2 minutes

468 miles 9 hours

9. Congruent triangles have the same shape and the same size. 10. Similar triangles have exactly the same shape but not necessarily the same size. 11−13. Label the sides of the right triangle. 11. leg

13. hypotenuse

The unit rate is

52 468 −45 18 −18 0

52 miles or 52 miles/hour. 1 hour

12. leg

14. A triangle with one right angle is called a right triangle.

233

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications

10.

180 feet 12 seconds

The unit rate is

11.

$6.96 4 diskettes

The unit rate is

12.

104 bushels 8 trees

The unit rate is

15 180 −12 60 −60 0

0.1495 ≈ 0.150 3.5900 −2 4 1 19 −96 230 −216 140 −120 20 The 24-ounce size costs $0.150 per ounce. The 16-ounce size is the better buy. 24

15 feet or 15 feet/second. 1 second

1.74 4 6.96 −4 29 −28 16 −16 0 $1.74 or $1.74/diskette. 1 diskette

13 8 104 −8 24 −24 0 13 bushels or 13 bushels/tree. 1 tree

0.1368 ≈ 0.137 13. 16 2.1900 −1 6 59 −48 110 −96 140 −128 12 The 16-ounce size costs $0.137 per ounce.

0.1725 ≈ 0.173 14. 16 2.7600 −1 6 1 16 −1 12 40 −32 80 −80 0 The 16-ounce size costs $0.173 per ounce. 0.1695 ≈ 0.170 40 6.7800 −4 0 2 78 −2 40 380 −360 200 −200 0 The 40-ounce size costs $0.170 per ounce. The 40-ounce size is the better buy.

15.

24 uniforms 3 uniforms = 8 players 1 player

16.

12 tires 4 tires = 3 cars 1 car

17.

234

19 14 0 8 6 19 ⋅ 6 0 8 ⋅14 114 ≠ 112 The proportion is false.

ISM: Prealgebra

18.

19.

20.

21.

22.

Chapter 6: Ratio, Proportion, and Triangle Applications

3.75 7.5 0 3 6 3.75 ⋅ 60 3 ⋅ 7.5 22.5 = 22.5 The proportion is true.

24.

x 30 = 3 18 x ⋅18 = 3 ⋅ 30 18 x = 90 18 x 90 = 18 18 x=5 x 7 = 9 3 x ⋅3 = 9⋅7 3 x = 63 3 x 63 = 3 3 x = 21

25.

−8 9 = x 5 −8 ⋅ x = 5 ⋅ 9 −8 x = 45 −8 x 45 = −8 −8 x = −5.625

−27 9 4

x −5

9 ⋅x 4 9 135 = ⋅ x 4 4 135 4 9 ⋅ = ⋅ ⋅x 9 1 9 4 60 = x

−27 ⋅ (−5) =

0.4 2 23. = x 4.7 0.4 ⋅ 4.7 = x ⋅ 2 1.88 = 2 x 1.88 2 x = 2 2 0.94 = x

26.

21 x = 10 4 12 8 52 2 1 1 x ⋅8 = 4 ⋅ 2 5 2 10 42 9 21 = ⋅ x⋅ 5 2 10 42 189 ⋅x = 5 20 5 42 5 189 ⋅ ⋅x = ⋅ 42 5 42 20 9 1 x = or 1 8 8

x 4.7 = 0.4 3 x ⋅ 3 = 0.4 ⋅ 4.7 3 x = 1.88 3 x 1.88 = 3 3 x ≈ 0.63 0.07 7.2 = n 0.3 0.07 ⋅ n = 0.3 ⋅ 7.2 0.07 n = 2.16 0.07 n 2.16 = 0.07 0.07 n ≈ 30.9

27. Let x be the number of successful blocks. successful → 3 x ← successful = attempted → 7 32 ← attempted 3 ⋅ 32 = 7 ⋅ x 96 = 7 x 96 7 x = 7 7 14 ≈ x The goalie successfully blocked 14 shots. 28. Let x be the number of attempted blocks. successful → 3 15 ← successful = attempted → 7 x ← attempted 3 ⋅ x = 7 ⋅15 3 x = 105 3 x 105 = 3 3 x = 35 The goalie attempted 35 blocks.

235

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications 29. Let x be the number of bottles. x ← bottles bottles → 1 = 20, 000 180 ⋅175 ← square feet square feet → 1 ⋅180 ⋅175 = 20, 000 ⋅ x 31,500 = 20, 000 x 31,500 20, 000 x = 20, 000 20, 000 1.575 = x Purchase 2 bottles of pest-control spray. 30. Let x be the number of bottles. x bottles → 1 ← bottles = square feet → 20, 000 250 ⋅ 250 ← square feet 1 ⋅ 250 ⋅ 250 = 20, 000 ⋅ x 62,500 = 20, 000 x 62,500 20, 000 x = 20, 000 20, 000 3.125 = x Purchase 4 bottles of pest-control spray. 31. Let x be the miles. miles → x 80 ← miles = inches → 2 0.75 ← inches x ⋅ 0.75 = 2 ⋅ 80 0.75 x = 160 0.75 x 160 = 0.75 0.75 640 1 x= or 213 3 3 1 The distance is 213 miles. 3

33.

64 = 8 because 8 = 64.

34.

144 = 12 because 12 2 = 144.

35.

12 ≈ 3.464

36.

15 ≈ 3.873

236

0 = 0 because 02 = 0.

38.

1 = 1 because 12 = 1.

39.

50 ≈ 7.071

40.

65 ≈ 8.062

41.

4 2 2 2 4 2 = because   = ⋅ = . 25 5 5 5 5 25  

42.

1 1 1 1 1 1 because   = ⋅ = = . 100 10 10 10 100  10 

43. hypotenuse = (leg) 2 + (other leg) 2 = 122 + 52 = 144 + 25 = 169 = 13 The leg has length 13 units.

44. hypotenuse = (leg) 2 + (other leg) 2 = 202 + 212 = 400 + 441 = 841 = 29 The leg has length 29 units.

32. Let x be the inches. miles →1025 80 ← miles = 0.75 ← inches inches → x 1025 ⋅ 0.75 = x ⋅ 80 768.75 = 80 x 768.75 80 x = 80 80 9.6 ≈ x The distance is about 9.6 inches. 2

37.

45. leg = (hypotenuse)2 − (other leg) 2 = 142 − 92 = 196 − 81 = 115 ≈ 10.7 The leg has length of about 10.7 units.

46. leg = (hypotenuse)2 − (other leg) 2 = 862 − 662 = 7396 − 4356 = 3040 ≈ 55.1 The leg has length of about 55.1 units.

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications

47. hypotenuse = (leg) 2 + (other leg) 2

54.

= 202 + 202 = 400 + 400 = 800 ≈ 28.28 The diagonal is about 28.28 centimeters.

48. leg = (hypotenuse)2 − (other leg) 2 = 1262 − 902 = 15,876 − 8100 = 7776 ≈ 88.2 The height is about 88.2 feet.

50. The triangles are not congruent.

52.

53.

y 2 = 26 24 y ⋅ 24 = 26 ⋅ 2 24 y = 52 24 y 52 = 24 24 13 1 y= or 2 6 6 5 The unknown lengths are x = inch and 6 1 y = 2 inches. 6

55. The ratio of 15 to 25 is

49. The triangles are congruent by Angle-SideAngle.

51.

x 2 = 10 24 x ⋅ 24 = 10 ⋅ 2 24 x = 20 24 x 20 = 24 24 5 x= 6

x 20 = 20 30 x ⋅ 30 = 20 ⋅ 20 30 x = 400 30 x 400 = 30 30 40 1 x= or 13 3 3

56. The ratio of 3 pints to 81 pints is 3 pints 3 1 = = . 81 pints 81 27 57.

2 teachers 2 ⋅1 teachers 1 teacher = = 18 students 2 ⋅ 9 students 9 students

58.

6 nurses 1 ⋅ 6 nurses 1 nurse = = 24 patients 4 ⋅ 6 patients 4 patients

59.

x 24 = 5.8 8 x ⋅ 8 = 5.8 ⋅ 24 8 x = 139.2 8 x 139.2 = 8 8 x = 17.4

15 3 ⋅ 5 3 = = . 25 5 ⋅ 5 5

60.

136 miles 4 ⋅ 34 miles 34 miles = = 4 hours 4 ⋅ 1 hour 1 hour 34 miles or 34 miles/hour. The unit rate is 1 hour 12 gallons 6 ⋅ 2 gallons 2 gallons = = 6 cows 6 ⋅1 cow 1 cow 2 gallons or 2 gallons/cow. The unit rate is 1 cow

x 42 = 5.5 7 x ⋅ 7 = 5.5 ⋅ 42 7 x = 231 7 x 231 = 7 7 x = 33 The height of the building is approximately 33 feet.

237

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications

61.

139.4 ≈ 139 11.3 1576 becomes 113 15760.0 −113 446 −339 1070 −1017 53 0 −45 2 78 She ran 139 steps/minute.

62.

1576 steps 9.5 minutes

165.8 ≈ 166 9.5 1576 becomes 95 15760.0 −95 626 −570 560 −475 850 −76 0 90 He ran 166 steps/minute.

63. 24 softgels: 0.1766 ≈ 0.177 24 4.2400 −2 4 1 84 −1 68 160 −144 160 −144 16 The 24-softgel package cost $0.177 per softgel. 48 softgels:

238

0.1610 ≈ 0.161 48 7.7300 −4 8 2 93 −2 88 50 −48 20 The 48-softgel package costs $0.161 per softgel. The 48-softgel package is the better buy.

1576 steps 11.3 minutes

64. 96 ounces: 0.0487 ≈ 0.049 96 4.6800 −3 84 840 −768 720 −672 48 The 96-ounce size costs $0.049 per ounce. 64 ounces: 0.0465 ≈ 0.047 64 2.9800 −2 56 420 −384 360 −320 40 The 64-ounce size costs $0.047 per ounce. The 64-ounce size is the better buy. 65.

2 cups cookie dough 4 cups cookie dough = 30 cookies 60 cookies

66.

5 nickels 20 nickels = 3 dollars 12 dollars

67.

3 15 = x 8 3 ⋅ 8 = x ⋅15 24 = 15 x 24 15 x = 15 15 1.6 = x

ISM: Prealgebra

68.

69.

70.

Chapter 6: Ratio, Proportion, and Triangle Applications

5 x = 4 20 5 ⋅ 20 = 4 ⋅ x 100 = 4 x 100 4 x = 4 4 25 = x

77.

x 7.5 = 3 6 x ⋅ 6 = 3 ⋅ 7.5 6 x = 22.5 6 x 22.5 = 6 6 x = 3.75 1 3

78.

x 30

25 1 ⋅ 30 = 25 ⋅ x 3 10 = 25 x 10 25 x = 25 25 2 =x 5

93 n = 8 8 23 12 12 1 2 3 n ⋅12 = 8 ⋅ 9 2 3 8 25 26 75 = ⋅ n⋅ 2 3 8 25 325 n= 2 4 2 25 2 325 ⋅ n= ⋅ 25 2 25 4 13 1 n= or 6 2 2

Chapter 6 Getting Ready for the Test 1. “6 is to 7 as 18 is to 21” translates as

6 18 = ; A 7 21

3 x = 11 5 3 ⋅ 5 = 11 ⋅ x x 11 = B: 3 5 5 ⋅ x = 3 ⋅11 3 x = C: 5 11 3 ⋅11 = 5 ⋅ x 11 5 = D: x 3 11⋅ 3 = x ⋅ 5

2. A:

71.

36 = 6 because 6 2 = 36.

72.

16 4 4 4 16 4 = because   = ⋅ = . 81 9 9 9 81 9

73.

105 ≈ 10.247

74.

32 ≈ 5.657

75. hypotenuse = (leg) 2 + (other leg) 2 = 662 + 562 = 4356 + 3136 = 7492 ≈ 86.6

3 5 = , the cross products are x 11 3 ⋅ 11 and x ⋅ 5. Since the cross products in A are not equivalent to 3 ⋅ 11 and x ⋅ 5, proportion A is 3 5 not equivalent to = . x 11

In the proportion

76. leg = (hypotenuse)2 − (other leg) 2 = 242 − 122 = 576 − 144 = 432 ≈ 20.8

n 10 = 6 5 n ⋅ 5 = 6 ⋅10 5n = 60 5n 60 = 5 5 n = 12

3. Since 82 = 64, 102 = 100, and 64 < 81 < 100,

then

81 is between 8 and 10; D. Also,

36 = 6,

25 = 5, 49 = 7, and

81 = 9.

239

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications

4. Since 62 = 36, 72 = 49, and 36 < 40 < 49, is between 6 and 7; D.

650 kilometers 8 hours

5. True; the Pythagorean theorem applies only to right triangles; A. 6. False; since a triangle has three angles which sum to 180° and 90° + 90° = 180°, a right triangle can have only one 90° angle; B. 7. False; a right triangle does have 3 sides, but two of the sides are called legs and one of the sides is called the hypotenuse; B.

81.25 kilometers or 1 hour 81.25 kilometers/hour.

The unit rate is

8. True; the hypotenuse of a right triangle is the longest side; A. Chapter 6 Test

140 students 5 teachers

1. The ratio of 4500 trees to 6500 trees is 4500 trees 9 ⋅ 500 9 = = . 6500 trees 13 ⋅ 500 13

4. The ratio of 5

7 3 47 39 47 4 47 = 5 ÷9 = ÷ = ⋅ = . 3 8 4 8 4 8 39 78 9

28 140 −10 40 −40 0

The unit rate is

960 inches 60 minutes

3 7 to 9 is 8 4

5 78

28 students or 1 teacher 28 students/teacher.

2. The ratio of 9 inches of rain in 30 days is 9 inches 3 ⋅ 3 inches 3 inches = = . 30 days 3 ⋅10 days 10 days 3. The ratio of 8.6 to 10 is 8.6 8.6 × 10 86 43 = = = . 10 10 × 10 100 50

81.25 8 650.00 −64 10 −8 20 −1 6 40 −40 0

The unit rate is

16 60 960 −60 360 −360 0 16 inches or 16 inches/minute. 1 minute

5. The ratio 590 feet to 186 feet is 590 feet 2 ⋅ 295 295 = = . 186 feet 2 ⋅ 93 93

240

0.82 9. 5 4.10 −4 0 10 −10 0 The 5-ounce size costs $0.82 per ounce.

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications

0.4446 ≈ 0.445 6.6700 −6 0 67 −60 70 −60 100 −90 0 The 15-ounce size costs $0.445 per ounce. The 15-ounce size is the better buy. 15

0.0961 ≈ 0.096 1.7300 −1 62 110 −108 20 −18 2 The 18-ounce size costs $0.096 per ounce.

13.

14.

10. 18

0.0893 ≈ 0.089 32 2.8600 −2 56 300 −288 120 −96 24 The 32-ounce size costs $0.089 per ounce. The 32-ounce size is the better buy.

11.

12.

28 14 0 16 8 28 ⋅ 8 0 16 ⋅14 224 = 224 The proportion is true. 3.6 1.9 0 2.2 1.2 3.6 ⋅1.2 0 2.2 ⋅1.9 4.32 ≠ 4.18 The proportion is false.

n 15 = 3 9 n ⋅ 9 = 3 ⋅15 9n = 45 9n 45 = 9 9 n=5

8 11 = x 6 8 ⋅ 6 = x ⋅11 48 = 11x 48 11x = 11 11 4 4 =x 11 4

15.

3 7

y 1 4

1 3 = ⋅y 4 7 3 1= y 7 7 7 3 ⋅1 = ⋅ y 3 3 7 7 =y 3

4⋅

16.

1.5 2.4 = 5 n 1.5 ⋅ n = 5 ⋅ 2.4 1.5n = 12 1.5n 12 = 1.5 1.5 n=8

17. Let x be the length of the home in feet. feet → x 9 ← feet = inches →11 2 ← inches x ⋅ 2 = 11 ⋅ 9 2 x = 99 2 x 99 = 2 2 1 x = 49 2 1 The home is 49 feet long. 2

241

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications 18. Let x be the number of hours. miles → 80 100 ← miles = x ← hours hours → 3 80 ⋅ x = 3 ⋅100 80 x = 300 80 x 300 = 80 80 3 x=3 4 3 It will take 3 hours to travel 100 miles. 4 19. Let x be the number of grams. grams →10 x ← grams = pounds →15 80 ← pounds 10 ⋅ 80 = 15 ⋅ x 800 = 15 x 800 15 x = 15 15 1 53 = x 3 The standard dose for an 80-pound dog is 1 53 grams. 3

25. Let x be the height of the tower. x 48 = 3 4 5 4

3 x ⋅ 4 = 5 ⋅ 48 4 23 4 x = ⋅ 48 4 4 x = 276 4 x 276 = 4 4 x = 69 The tower is approximately 69 feet tall.

Cumulative Review Chapters 1−6

12 − 9 = 3

Check: 3 + 9 = 12

b. 22 − 7 = 15

Check: 15 + 7 = 22

35 − 35 = 0

Check: 0 + 35 = 35

d. 70 − 0 = 70

Check: 70 + 0 = 70

1. a.

2. a.

20 ⋅ 0 = 0

b. 20 ⋅ 1 = 20

20.

49 = 7 because 7 2 = 49.

21.

157 ≈ 12.530

d. 1 ⋅ 20 = 20

22.

64 8 4 8 8 64 8 = = because   = ⋅ = . 100 10 5 10 10 100  10 

23. hypotenuse = (leg) 2 + (other leg) 2

24.

242

0 ⋅ 20 = 0

3. To round 248,982 to the nearest hundred, observe that the digit in the tens place is 8. Since this digit is at least 5, we add 1 to the digit in the hundreds place. The number 248,982 rounded to the nearest hundred is 249,000.

= 42 + 42 = 16 + 16 = 32 ≈ 5.66 The hypotenuse is 5.66 centimeters.

4. To round 248,982 to the nearest thousand, observe that the digit in the hundreds place is 9. Since this digit is at least 5, we add 1 to the digit in the thousands place. The number 248,982 rounded to the nearest thousand is 249,000.

n 5 = 12 8 n ⋅ 8 = 12 ⋅ 5 8n = 60 8n 60 = 8 8 n = 7.5

5. a.

25 × 8 200 23

246 × 5 1230

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications

373 R 24 6. 28 10, 468 −8 4 2 06 −1 96 108 −84 24

18.

7. 1 + (−10) + (−8) + 9 = −9 + (−8) + 9 = −17 + 9 = −8 8. −12(7) = −84 9. 80 = 2 ⋅ 40 ↓ ↓ = 2 ⋅ 2 ⋅ 20 ↓ ↓ ↓ = 2 ⋅ 2 ⋅ 2 ⋅10 ↓ ↓ ↓ ↓ = 2⋅ 2⋅ 2⋅ 2⋅5

19. 4 ⋅ 5 = 20, so 3 3 ⋅ 5 15 = = 4 4 ⋅ 5 20 20.

21.

= 24 ⋅ 5

10. 32 − 12 = 9 − 1 = 8 11.

12 4 ⋅ 3 3 = = 20 4 ⋅ 5 5

12. 9 2 ⋅ 3 = 81 ⋅ 3 = 243

22.

 6  26  6 ⋅ 26 6 ⋅13 ⋅ 2 2 = = 13.  −  −  =  13  30  13 ⋅ 30 13 ⋅ 6 ⋅ 5 5 3 5 27 41 9 ⋅ 3 ⋅ 41 123 3 14. 3 ⋅ 4 = ⋅ = = or 15 8 9 8 9 8⋅9 8 8

23.

7 6 3 7 + 6 + 3 16 15. + + = = =2 8 8 8 8 8

16.

7 3 4 7 −3+ 4 8 4 − + = = = 10 10 10 10 10 5

17. 7 = 7 14 = 2 ⋅ 7 LCD = 2 ⋅ 7 = 14

17 3 17 2 3 5 + = ⋅ + ⋅ 25 10 25 2 10 5 17 ⋅ 2 3 ⋅ 5 = + 25 ⋅ 2 10 ⋅ 5 34 15 = + 50 50 34 + 15 = 50 49 = 50

10 5 ⋅ 2 2 = = 55 5 ⋅11 11 6 3⋅ 2 2 = = 33 3 ⋅11 11 Yes, they are equivalent.

2 10 2 11 10 3 − = ⋅ − ⋅ 3 11 3 11 11 3 22 30 = − 33 33 22 − 30 = 33 8 =− 33

5 24 5 − 9 9 17

15 72 40 − 9 72 17

87 72 40 − 9 72 47 7 72 16

5 1 5 3 5−3 2 1 − = − = = = 12 4 12 12 12 12 6 1 There is hour remaining. 6

24. 80 ÷ 8 ⋅ 2 + 7 = 10 ⋅ 2 + 7 = 20 + 7 = 27 25.

1 3 3 +5 8 2

8 24 9 +5 24 17 7 24 2

243

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications

32. 4( y + 1) − 3 = 21 4 y + 4 − 3 = 21 4 y + 1 = 21 4 y + 1 − 1 = 21 − 1 4 y = 20 4 y 20 = 4 4 y=5

+ +  27 20 33  +  ÷3 26. 5 9 15 =  + 3  45 45 45  80 1 = ⋅ 45 3 16 1 = ⋅ 9 3 16 = 27 3 3 ⋅11 33 = = 27. 4 4 ⋅11 44 9 9 ⋅ 4 36 = = 11 11 ⋅ 4 44 Since 33 < 36,

33 36 3 9 < and < . 44 44 4 11

28. 5 y − 8 y = 24 −3 y = 24 −3 y 24 = −3 −3 y = −8 29.

30.

31.

244

y − 5 = −2 − 6 y − 5 = −8 y − 5 + 5 = −8 + 5 y = −3

3y − 6 = 7 y − 6 3y − 6 + 6 = 7 y − 6 + 6 3y = 7 y 3y − 7 y = 7 y − 7 y −4 y = 0 −4 y 0 = −4 −4 y=0 3a − 6 = a + 4 3a − 6 − a = a + 4 − a 2a − 6 = 4 2a − 6 + 6 = 4 + 6 2a = 10 2a 10 = 2 2 a=5

33. 3(2 x − 6) + 6 = 0 6 x − 18 + 6 = 0 6 x − 12 = 0 6 x − 12 + 12 = 0 + 12 6 x = 12 6 x 12 = 6 6 x=2 34. Seventy-five thousandths written in standard form is 0.075. 35. To round 736.2359 to the nearest tenth, observe that the digit in the hundredths place is 3. Since this digit is less than 5, we do not add 1 to the digit in the tenths place. The number 736.2359 rounded to the nearest tenth is 736.2. 36. To round 736.2359 to the nearest thousandth, observe that the digit in the ten-thousandths place is 9. Since this digit is at least 5, we add 1 to the digit in the thousandths place. The number 736.2359 rounded to the nearest thousandth is 736.236. 37.

23.850 + 1.604 25.454

38.

700.00 − 18.76 681.24

39. 0.0531 × 16 3186 5310 0.8496

ISM: Prealgebra

Chapter 6: Ratio, Proportion, and Triangle Applications

0.375 40. 8 3.000 −2 4 60 −56 40 −40 0

45.

3 = 0.375 8

41. 115

46.

0.052 5.980 −5 75 230 −230 0

48. The ratio of 7 to 21 is

9 10 ⋅ 7 + 9 79 = = 10 10 10

n 12 = 4 16 n ⋅16 = 4 ⋅12 16n = 48 16n 48 = 16 16 n=3

$15 15 5 ⋅ 3 3 = = = . $10 10 5 ⋅ 2 2

7 1⋅ 7 1 = = . 21 3 ⋅ 7 3

49. The ratio of 2.5 to 3.15 is 2.5 2.5 ⋅100 250 50 ⋅ 5 50 = = = = . 3.15 3.15 ⋅100 315 63 ⋅ 5 63

43. −0.5(8.6 − 1.2) = −0.5(7.4) = −3.7 44.

700 meters 5 ⋅140 meters 140 meters = = 5 seconds 5 ⋅1 second 1 second The unit rate is 140 meters/second.

47. The ratio of $15 to $10 is

−5.98 ÷ 115 = −0.052 42. 7.9 = 7

9 = 0.450 20 4 = 0.4 = 0.444... 9 0.456 = 0.456 4 9 , , 0.456 9 20

50. The ratio of 900 to 9000 is 900 900 ⋅1 1 = = . 9000 900 ⋅10 10

245

Chapter 7 Section 7.1 Practice Exercises

17.

1. Since 27 students out of 100 students in the club 27 are freshmen, the fraction is . Then 100 27 = 27%. 100 2.

18.

31 = 31% 100

3. 49% = 49(0.01) = 0.49 4. 3.1% = 3.1(0.01) = 0.031 5. 175% 175(0.01) = 1.75 6. 0.46% = 0.46(0.01) = 0.0046 7. 600% = 600(0.01) = 6.00 or 6 8. 50% = 50 ⋅

1 50 1 ⋅ 50 1 = = = 100 100 2 ⋅ 50 2

9. 2.3% = 2.3 ⋅

1 2.3 2.3 ⋅10 23 = = = 100 100 100 ⋅10 1000

10. 150% = 150 ⋅

1 150 3 ⋅ 50 3 1 = = = or 1 100 100 2 ⋅ 50 2 2

2 2 1 11. 66 % = 66 ⋅ 3 3 100 200 1 = ⋅ 3 100 2 ⋅100 ⋅1 = 3 ⋅100 2 = 3 12. 12% = 12 ⋅

1 12 3⋅ 4 3 = = = 100 100 25 ⋅ 4 25

13. 0.14 = 0.14(100%) = 14.% or 14%

3 3 3 100 300 = ⋅100% = ⋅ %= % = 12% 25 25 25 1 25

9 9 = ⋅100% 40 40 9 100 = ⋅ % 40 1 900 = % 40 20 = 22 % 40 1 = 22 % 2

1 11 19. 5 = 2 2 11 = ⋅100% 2 11 100 = ⋅ % 2 1 1100 = % 2 = 550%

20.

3 3 3 100% 300 = ⋅100% = ⋅ = % ≈ 17.65% 17 17 17 1 17 17.647 ≈ 17.65 17 300.000 −17 130 −119 11 0 −10 2 80 −68 120 −119 1

Thus,

3 is approximately 17.65%. 17

14. 1.75 = 1.75(100%) = 175.% or 175% 15. 0.057 = 0.057(100%) = 05.7% or 5.7% 16. 0.5 = 0.5(100%) = 050.% or 50%

246

ISM: Prealgebra

Chapter 7: Percent

21. As a decimal 27.5% = 27.5(0.01) = 0.275. 1 As a fraction 27.5% = 27.5 ⋅ 100 27.5 = 100 27.5 10 = ⋅ 100 10 275 = 1000 11 ⋅ 25 = 40 ⋅ 25 11 = . 40 Thus, 27.5% written as a decimal is 0.275, and 11 written as a fraction is . 40 3 7 7 7 100% 700 22. 1 = = ⋅100% = ⋅ = % = 175% 4 4 4 4 1 4 3 Thus, a “1 times” increase is the same as a 4 “175% increase.”

Vocabulary, Readiness & Video Check 7.1 1. Percent means “per hundred.”

Exercise Set 7.1 2.

81 = 81% 100 81% of this player’s attempts were made.

4. Only 3 out of 100 adults preferred volleyball. 3 = 3% 100 6. 20 of the adults preferred basketball, while 15 preferred baseball. Thus, 20 + 15 = 35 preferred basketball or baseball. 35 = 35% 100 8. 62% = 62(0.01) = 0.62 10. 3% = 3(0.01) = 0.03 12. 136% = 136(0.01) = 1.36 14. 45.7% = 45.7(0.01) = 0.457 16. 1.4% = 1.4(0.01) = 0.014 18. 0.9% = 0.9(0.01) = 0.009 20. 500% = 500(0.01) = 5.00 or 5

2. 100% = 1. 3. The % symbol is read as percent.

22. 72.18% = 72.18(0.01) = 0.7218 1 22 2 ⋅11 11 = = = 100 100 2 ⋅ 50 50

4. To write a decimal or a fraction as a percent, multiply by 1 in the form of 100%.

24. 22% = 22 ⋅

5. To write a percent as a decimal, drop the % symbol and multiply by 0.01.

26. 2% = 2 ⋅

6. To write a percent as a fraction, drop the % 1 . symbol and multiply by 100

28. 7.5% = 7.5 ⋅

7. Percent means “per 100.” 8. 100% 9. Multiplying by 100% is the same as multiplying by 1. 10. The difference is how the percent symbol is replaced⎯for a decimal, replace % with 0.01 1 and for a fraction, replace % with . 100

1 2 2 ⋅1 1 = = = 100 100 2 ⋅ 50 50

1 100

7.5 100 7.5 ⋅10 = 100 ⋅10 75 = 1000 3 ⋅ 25 = 40 ⋅ 25 3 = 40 =

30. 275% = 275 ⋅

1 275 11 ⋅ 25 11 3 = = = or 2 100 100 4 ⋅ 25 4 4 247

Chapter 7: Percent

32. 8.75% = 8.75 ⋅

ISM: Prealgebra

1 100

60.

8.75 100 8.75 ⋅100 = 100 ⋅100 875 = 10, 000 7 ⋅125 = 80 ⋅125 7 = 80 =

1 1 31 3100 62. 6 = 6 ⋅100% = ⋅100% = % = 620% 5 5 5 5

64. 2

66.

3 3 1 31 1 31 34. 7 % = 7 ⋅ = ⋅ = 4 4 100 4 100 400

7 7 1 36. 21 % = 21 ⋅ 8 8 100 175 1 = ⋅ 8 100 175 = 800 7 ⋅ 25 = 32 ⋅ 25 7 = 32

7 7 27 2700 = 2 ⋅100% = ⋅100% = = 270% 10 10 10 10

11 11 1100 = ⋅100% = % 12 12 12 91.666 ≈ 91.67 12 1100.000 −108 20 −12 80 −7 2 80 −72 80 −72 8 11 is approximately 91.67%. 12

38. 0.44 = 0.44(100%) = 44%

68.

40. 0.008 = 0.008(100%) = 0.8% 42. 2.7 = 2.7(100%) = 270% 44. 0.019 = 0.019(100%) = 1.9% 46. 0.1115 = 0.1115(100%) = 11.15% 48. 9.00 = 9.00(100%) = 900% 50. 0.8 = 0.8(100%) = 80%

10 10 1000 = ⋅100% = % 11 11 11 90.909 ≈ 90.91 11 1000.000 −99 10 0 −9 9 100 −99 1 10 is approximately 90.91%. 11

52.

9 9 900 = ⋅100% = % = 90% 10 10 10

54.

2 2 200 = ⋅100% = % = 40% 5 5 5

56.

41 41 4100 = ⋅100% = % = 82% 50 50 50

58.

5 5 500 125 1 = ⋅100% = %= % = 31 % 16 16 16 4 4

248

5 5 500 1 = ⋅100% = % = 88 % 6 6 6 3

ISM: Prealgebra

70.

72.

Chapter 7: Percent

Percent

Decimal

Fraction

52.5%

0.525

21 40

75%

0.75

3 4

66 12 %

0.665

133 200

83 13 %

0.8333

5 6

100%

82. 59.5% = 59.5(0.01) = 0.595 1 59.5% = 59.5 ⋅ 100 59.5 = 100 59.5 ⋅10 = 100 ⋅10 595 = 1000 119 = 200

14%

0.14

7 50

84. 0.1144 = 0.1144(100%) = 11.44%

Percent

Decimal

Fraction

800%

320%

3.2

3 15

608%

6.08

2 6 25

926%

9.26

9 13 50

86.

1 1 100 = ⋅100% = % = 25% 4 4 4

88. 18.6% = 18.6 ⋅

1 100

18.6 100 18.6 ⋅10 = 100 ⋅10 186 = 1000 93 = 500 =

74. 38% = 38(0.01) = 0.38 1 38 19 38% = 38 ⋅ = = 100 100 50

 2 5  2 5   7  3  90.  +   −  =    −   11 11   11 11   11   11  7 ⋅3 =− 11 ⋅11 21 =− 121

76. 70% = 70(0.01) = 0.70 1 70 7 70% = 70 ⋅ = = 100 100 10 78. 20.2% = 20.2(0.01) = 0.202 1 20.2% = 20.2 ⋅ 100 20.2 = 100 20.2 ⋅10 = 100 ⋅10 202 = 1000 101 = 500 80. 9.6% = 9.6(0.01) = 0.096 1 9.6 ⋅ 10 96 12 9.6% = 9.6 ⋅ = = = 100 100 ⋅10 1000 125

2 5 20 29 ÷ 92. 6 ÷ 4 = 3 6 3 6 20 6 = ⋅ 3 29 20 ⋅ 2 ⋅ 3 = 3 ⋅ 29 40 = 29 11 =1 29

94. a.

0.5269% rounded to the nearest tenth percent is 0.5%.

b. 0.5269% rounded to the nearest hundredth percent is 0.53%.

249

Chapter 7: Percent 96. a.

ISM: Prealgebra

0.231 = 0.231(100%) = 23.1% CORRECT

b. 5.12 = 5.12(100%) = 512% INCORRECT c.

3.2 = 3.2(100%) = 320% CORRECT

d. 0.0175 = 0.0175(100%) = 1.75% INCORRECT

a and c are correct. 98. 45% + 30% + 20% = 95% 100% − 95% = 5% Other is 5% of the top four components of bone. 100. 5 of 8 equal parts are shaded so 5 5 500 = ⋅100% = % = 62.5% 8 8 8

56 ≈ 0.5490 ≈ 0.549 102 0.549(100%) = 54.9%

106. The second longest bar corresponds to data scientist, so that is predicted to be the second fastest growing occupation. 108. The percent change for statistician is 33%, so 33% = 33(0.01) = 0.33. 110. answers may vary Section 7.2 Practice Exercises

what percent of 48? 1. 8 is    ↓↓ 8 =

↓ x

↓ ↓ ⋅ 48

↓ x

3.  What number   is 90% of 0.045? ↓ x

250

↓ ↓ ↓ ↓ = 90% ⋅ 0.045

↓ x

5. 12% of what number    is 21? ↓ ↓ 12% ⋅

↓ x

↓ ↓ = 21

percent of 95 is 76? 6. What    ↓ x

↓ ↓ ↓ ↓ ⋅ 95 = 76

↓ ↓ ↓ ↓ ↓ = 25% ⋅ 90 x x = 25% ⋅ 90 x = 0.25 ⋅ 90 x = 22.5 Then 22.5 is 25% of 90. Is this reasonable? To see, round 25% to 30%. Then 30% or 0.30(90) is 27. Our result is reasonable since 22.5 is close to 27.

8. 95% of 400 is what number   ? ↓ ↓ ↓ ↓ ↓ x 95% ⋅ 400 = 0.95 ⋅ 400 = x 380 = x Then 95% of 400 is 380. Is this result reasonable? To see, round 95% to 100%. Then 100% of 400 or 1.0(400) = 400, which is close to 380.

9. 15% of what number    is 2.4?

2. 2.6 is 40% of  what number  ? ↓ ↓ ↓ ↓ 2.6 = 40% ⋅

↓ ↓ ↓ ↓ 56% ⋅ 180 =

7. What number   is 25% of 90?

102. A decimal written as a percent is less than 100% when the decimal is less than 1. 104.

4. 56% of 180 is what number   ?

↓ ↓ ↓ ↓ ↓ = 2.4 x 15% ⋅ 0.15 ⋅ x = 2.4 0.15 x 2.4 = 0.15 x 0.15 x = 16 Then 15% of 16 is 2.4. Is this result reasonable? To see, round 15% to 20%. Then 20% of 16 or 0.20(16) = 3.2, which is close to 2.4.

ISM: Prealgebra

Chapter 7: Percent 4. 100% of a number = the number

1 10. 18 is 4 % of what number   ? 2

5. Any “percent greater than 100%” of “a number” = “a number greater than the original number.”

↓↓ ↓ ↓ ↓ 1 18 = 4 % ⋅ x 2 18 = 0.045 x 18 0.045 x = 0.045 0.045 400 = x 1 Then 18 is 4 % of 400. 2

6. Any “percent less than 100%” of “a number” = “a number less than the original number.” 7. “of” translates to multiplication; “is” (or something equivalent) translates to an equal sign; “what” or “unknown” translates to our variable. 8. It is already solved for x; we just need to simplify the left side.

percent of 90 is 27? 11. What   

Exercise Set 7.2

↓ ↓ ↓ ↓↓ x ⋅ 90 = 27 90 x = 27 90 x 27 = 90 90 3 x= 10 or x = 0.30 Since we are looking for percent, we can write 3 or 0.30 as a percent. x = 30% 10 Then 30% of 90 is 27. To check, see that 30% ⋅ 90 = 27. percent of 45? 12. 63 is what   

2. 36% of 72 is what number   ?

↓ ↓ ↓ ↓ 36% ⋅ 72 =

↓ x

4. 40% of what number    is 6?

↓ ↓ 40% ⋅

↓ x

↓↓ =6

6. 0.7 is 20% of  what number  ?

↓ ↓ ↓ ↓ 0.7 = 20% ⋅

↓ x

8. 9.2 is what percent of 92?   

↓ ↓ ↓ ↓ ↓ 63 = x ⋅ 45 63 = 45 x 63 45 x = 45 45 7 =x 5 1 .4 = x 140% = x Then 63 is 140% of 45.

↓ ↓ 9.2 =

↓ x

↓ ↓ ⋅ 92

10. What number   is 25% of 55?

↓ x

↓ ↓ ↓ ↓ = 25% ⋅ 55

12. What percent of 375 is 300?   

Vocabulary, Readiness & Video Check 7.2 1. The word is translates to “=.” 2. The word of usually translates to “multiplication.” 3. In the statement “10% of 90 is 9,” the number 9 is called the amount, 90 is called the base, and 10 is called the percent.

↓ x

↓ ↓ ↓ ↓ ⋅ 375 = 300

14. 25% ⋅ 68 = x 0.25 ⋅ 68 = x 17 = x 25% of 68 is 17.

251

Chapter 7: Percent

16. x = 18% ⋅ 425 x = 0.18 ⋅ 425 x = 76.5 76.5 is 18% of 425. 18. 0.22 = 44% ⋅ x 0.22 = 0.44 x 0.22 0.44 x = 0.44 0.44 0.5 = x 0.22 is 44% of 0.5 1 20. 4 % ⋅ x = 45 2 0.045 x = 45 0.045 x 45 = 0.045 0.045 x = 1000 1 4 % of 1000 is 45. 2

ISM: Prealgebra

30.

32.

1 7.2 = 6 % ⋅ x 4 7.2 = 0.0625 x 7.2 0.0625 x = 0.0625 0.0625 115.2 = x 1 7.2 is 6 % of 115.2. 4 2.64 = x ⋅ 25 2.64 x ⋅ 25 = 25 25 0.1056 = x 10.56% = x 2.64 is 10.56% of 25.

34. x = 36% ⋅ 80 x = 0.36 ⋅ 80 x = 28.8 28.8 is 36% of 80.

22.

x ⋅ 40 = 60 40 ⋅ x 60 = 40 40 x = 1.5 x = 150% 60 is 150% of 40.

36.

24.

48 = x ⋅ 50 48 x ⋅ 50 = 50 50 0.96 = x 96% = x 48 is 96% of 50.

38. 160% ⋅ x = 40 1.6 x = 40 1.6 ⋅ x 40 = 1.6 1.6 x = 25 160% of 25 is 40.

26.

0.5 = 5% ⋅ x 0.5 = 0.05 x 0.5 0.05 x = 0.05 0.05 10 = x 0.5 is 5% of 10.

40. 4.8% ⋅ 32 = x 0.048 ⋅ 32 = x 1.536 = x 4.8% of 32 is 1.536. 42.

x ⋅ 500 = 2 2 x ⋅ 500 = 500 500 x = 0.004 x = 0.4% 0.4% of 500 is 2.

44.

9.75 = 7.5% ⋅ x 9.75 = 0.075 x 9.75 0.075 x = 0.075 0.075 130 = x 9.75 is 7.5% of 130.

28. 300% ⋅ 56 = x 3.00 ⋅ 56 = x 168 = x 300% of 56 is 168.

252

x ⋅ 120 = 76.8 x ⋅ 120 76.8 = 120 120 x = 0.64 x = 64% 64% of 120 is 76.8.

ISM: Prealgebra

Chapter 7: Percent

46. 2520 = x ⋅ 3500 2520 x ⋅ 3500 = 3500 3500 0.72 = x 72% = x 2520 is 72% of 3500. 48.

50.

52.

54.

56.

66. Since 100% of 35 is 35, and 50 is greater than 35, x must be greater than 100%; b. 68. Since 230% is greater than 100%, which is 1, 230% of 45 is greater than 45; b. 70. Since 30% is less than 100%, which is 1, 30% of y is less than y, so y is greater than 45; b.

2 = x ⋅ 40 2 x ⋅ 40 = 40 40 0.05 = x 5% = x 2 is 5% of 40.

72. Since 180% is greater than 100%, which is 1, 180% of y is greater than y, so y is less than 45; c. 74. answers may vary 76.

35 7 = x 5 35 ⋅ 5 = x ⋅ 7 175 = 7 x 175 7 x = 7 7 25 = x

x ⋅ 75, 528 = 27, 945.36 x ⋅ 75, 528 27, 945.36 = 75, 528 75, 528 x = 0.37 x = 37% 37% of 75,528 is 27,945.36.

Section 7.3 Practice Exercises

x 6 = 3 13 x ⋅ 13 = 3 ⋅ 6 13 x = 18 13 x 18 = 13 13 5 x =1 13

27%

of what number   

↓

percent

base It appears after the word of .

54? ↓ amount It is the part compared to the whole

54 27 = b 100

20 x = 25 10

5 15 = 6 x

30 is what percent of 90?    ↓ amount 30 p = 90 100

58. 68 = 8 ⋅ n 68 8 ⋅ n = 8 8 68 =n 8 Choice b is correct.

↓ percent

↓ base

3. What number   is 25% of 116?

↓ amount a 25 = 116 100

60. n = 0.7 ⋅ 12 n = 8.4 Choice a is correct.

↓ percent

↓ base

62. answers may vary 64. Since 100% of 98 is 98, the percent is 100%; a.

253

Chapter 7: Percent

680 is

ISM: Prealgebra

65% of what number   ?

↓ ↓ amount percent 680 65 = 100 b

↓ base

10. What percent of 40 is 8?   

5. What percent of 40 is 75?   

↓ percent 75 p = 40 100 6.

↓ ↓ ↓ percent base amount 8 p 1 p = or = 40 100 5 100 1 ⋅100 = 5 ⋅ p 100 = 5 p 100 5 p = 5 5 20 = p So, 20% of 40 is 8.

↓ ↓ base amount

46% of 80 is what number   ? ↓ ↓ percent base a 46 = 80 100

↓ amount 11.

↓ ↓ ↓ amount percent base a 8 = 120 100 a ⋅100 = 120 ⋅ 8 100a = 960 100a 960 = 100 100 a = 9.6 9.6 is 8% of 120.

Vocabulary, Readiness & Video Check 7.3

65% of what number    is 52? ↓ ↓ ↓ percent base amount 52 65 = b 100 52 ⋅100 = b ⋅ 65 5200 = 65b 5200 65b = 65 65 80 = b Thus, 65% of 80 is 52.

15.4 is

5% of what number   ?

↓ ↓ amount percent 254

↓ base

414 is what percent of 180?    ↓ ↓ ↓ amount percent base 414 p = 180 100 414 ⋅ 100 = 180 ⋅ p 41, 400 = 180 p 41, 400 180 p = 180 180 230 = p Then 414 is 230% of 180.

7. What number   is 8% of 120?

15.4 5 15.4 1 = or = b 100 b 20 15.4 ⋅ 20 = 1 ⋅ b 308 = b So, 154 is 5% of 308.

1. When translating the statement “20% of 15 is 3” to a proportion, the number 3 is called the amount, 15 is the base, and 20 is the percent. 2. In the question “50% of what number is 28?,” which part of the percent proportion is unknown? base 3. In the question “What number is 25% of 200?,” which part of the percent proportion is unknown? amount 4. In the question “38 is what percent of 380?,” which part of the percent proportion is unknown? percent 5. 45 follows the word “of,” so it is the base. 6. 100

ISM: Prealgebra

Chapter 7: Percent

Exercise Set 7.3 2.

14.

92% of 30 is what number   ? ↓ ↓ percent base a 92 = 30 100

↓ amount = a

4. What number   is 7% of 175?

16.

↓ ↓ ↓ amount = a percent base a 7 = 175 100 6.

1.2 is 47% of what number   ? ↓ ↓ amount percent 1.2 47 = b 100

↓ base = b

18.

85% of what number    is 520? ↓ ↓ percent base = b 520 85 = 100 b

↓ amount 20.

10. What percent of 900 is 216?   

↓ percent = p 216 p = 900 100 12.

↓ ↓ base amount

9.6 is what percent of 96?    ↓ ↓ amount percent = p 9.6 p = 96 100

↓ base

22.

a a 1 25 = = or 84 100 84 4 a ⋅ 4 = 84 ⋅1 4a = 84 4a 84 = 4 4 a = 21 25% of 84 is 21. 60 a a 3 = or = 29 100 29 5 a ⋅ 5 = 29.3 5a = 87 5a 87 = 5 5 a = 17.4 17.4 is 60% of 29. 55 55 55 11 or = = b 100 b 20 55 ⋅ 20 = b ⋅ 11 1100 = 11b 1100 11b = 11 11 100 = b 55% of 100 is 55. 1.1 44 1.1 11 or = = 25 b 100 b 1.1 ⋅ 25 = b ⋅ 11 27.5 = 11b 27.5 11b = 11 11 2.5 = b 1.1 is 44% of 2.5.

p p 147 3 = or = 98 100 2 100 3 ⋅ 100 = 2 ⋅ p 300 = 2 p 300 2 p = 2 2 150 = p 147 is 150% of 98.

255

Chapter 7: Percent

24.

26.

28.

30.

32.

34.

256

24 p 12 p = or = 50 100 25 100 12 ⋅ 100 = 25 ⋅ p 1200 = 25 p 1200 25 p = 25 25 48 = p 24 is 48% of 50.

7.4 5 7.4 1 = or = b 100 b 20 7.4 ⋅ 20 = b ⋅ 1 148 = b 7.4 is 5% of 148.

ISM: Prealgebra

36.

38.

a 2.5 = 90 100 a ⋅100 = 90 ⋅ 2.5 100a = 225 100a 225 = 100 100 a = 2.25 2.25 is 2.5% of 90.

40.

30 6 30 3 = or = b 100 b 50 30 ⋅ 50 = b ⋅ 3 1500 = 3b 1500 3b = 3 3 500 = b 30 is 6% of 500.

42.

p 550.4 = 172 100 550.4 ⋅ 100 = 172 ⋅ p 55, 040 = 172 ⋅ p 55, 040 172 ⋅ p = 172 172 320 = p 550.4 is 320% of 172.

44.

a 53 = 130 100 a ⋅ 100 = 130 ⋅ 53 100 a = 6890 100 a 6890 = 100 100 a = 68.9 68.9 is 53% of 130.

1.6 p = 5 100 1.6 ⋅100 = 5 ⋅ p 160 = 5 p 160 5 p = 5 5 32 = p 32% of 5 is 1.6. 221 170 221 17 = or = b b 100 10 221 ⋅ 10 = b ⋅ 17 2210 = 17b 2210 17b = 17 17 130 = b 170% of 130 is 221. a 7.8 = 24 100 a ⋅100 = 24 ⋅ 7.8 100a = 187.2 100a 187.2 = 100 100 a = 1.872 7.8% of 24 is 1.872.

3 p = 500 100 3 ⋅100 = 500 ⋅ p 300 = 500 p 300 500 p = 500 500 0.6 = p 0.6% of 500 is 3. 9.18 6.8 = b 100 9.18 ⋅ 100 = b ⋅ 6.8 918 = 6.8b 918 6.8b = 6.8 6.8 135 = b 9.18 is 6.8% of 135.

ISM: Prealgebra

46.

48.

50.

a 75 = 14 100 a ⋅ 100 = 14 ⋅ 75 100 a = 1050 100 a 1050 = 100 100 a = 10.50 75% of 14 is 10.5.

66.

p 88, 542 = 110, 736 100 88, 542 ⋅ 100 = 110, 736 ⋅ p 8,854, 200 = 110, 736 p 8,854, 200 110, 736 p = 110, 736 110, 736 79.958 ≈ p 80.0% of 110,736 is 88,542.

Mid-Chapter Review 1. 0.94 = 0.94(100%) = 94% 2. 0.17 = 0.17(100%) = 17%

10.78 4.30 + 0.21 15.29

3 3 300 = ⋅ 100% = % = 37.5% 8 8 8

7 7 700% = ⋅ 100% = = 350% 2 2 2

16.37 − 2.61 13.76

5. 4.7 = 4.7(100%) = 470%

58. answers may vary 60.

36 p = 12 100 36 50 0 12 100 1 3= False 2 The percent is not 50. p 36 = 12 100 36 ⋅ 100 = 12 ⋅ p 3600 = 12 p 3600 12 p = 12 12 300 = p The percent is 300 (300%).

64. answers may vary

7 5 7 ⋅ 2 5 ⋅ 3 14 15 14 − 15 1 − = − = − = =− 12 8 12 ⋅ 2 8 ⋅ 3 24 24 24 24

56.

62.

9310 p = 3800 100 9310 ⋅ 100 = 3800 ⋅ p 931, 000 = 3800 p 931, 000 3800 p = 3800 3800 245 = p 9310 is 245% of 3800.

2 1 8 9 52. 2 + 4 = + 3 2 3 2 8⋅2 9⋅3 = + 3⋅2 2 ⋅3 16 27 = + 6 6 43 = 6 1 =7 6

54.

Chapter 7: Percent

520 65 = b 100 520 65 0 800 100 520 ⋅100 0 800 ⋅ 65 52, 000 = 52, 000 True Yes the base is 800.

6. 8 = 8(100%) = 800% 7.

9 9 900 = ⋅ 100% = % = 45% 20 20 20

53 53 5300 = ⋅ 100% = % = 106% 50 50 50

9. 6

3 27 27 2700 = = ⋅ 100% = % = 675% 4 4 4 4

257

Chapter 7: Percent

10. 3

ISM: Prealgebra

1 13 13 1300 = = ⋅ 100% = % = 325% 4 4 4 4

11. 0.02 = 0.02(100%) = 2%

26. 45% = 45(0.01) = 0.45 1 45 9 = = 45% = 45 ⋅ 100 100 20 1 27. 16 % = 16.33(0.01) ≈ 0.163 3 1 49 49 1 49 ⋅ = 16 % = % = 3 3 3 100 300

12. 0.06 = 0.06(100%) = 6% 13. 71% = 71(0.01) = 0.71 14. 31% = 31(0.01) = 0.31

2 28. 12 % = 12.66(0.01) ≈ 0.127 3 2 38 38 1 38 19 = = 12 % = % = ⋅ 3 3 3 100 300 150

15. 3% = 3(0.01) = 0.03 16. 4% = 4(0.01) = 0.04 17. 224% = 224(0.01) = 2.24 18. 700% = 700(0.01) = 7.0 or 7

29.

19. 2.9% = 2.9(0.01) = 0.029 20. 6.6% = 6.6(0.01) = 0.066 21. 7% = 7(0.01) = 0.07 1 7 = 7% = 7 ⋅ 100 100 30.

22. 5% = 5(0.01) = 0.05 1 5 1 = = 5% = 5 ⋅ 100 100 20 23. 6.8% = 6.8(0.01) = 0.068 1 6.8 6.8 ⋅ 10 68 17 6.8% = 6.8 ⋅ = = = = 100 100 100 ⋅ 10 1000 250 24. 11.25% = 11.25(0.01) = 0.1125 1 11.25% = 11.25 ⋅ 100 11.25 = 100 11.25 ⋅ 100 = 100 ⋅ 100 1125 = 10, 000 9 = 80 25. 74% = 74(0.01) = 0.74 1 74 37 = = 74% = 74 ⋅ 100 100 50

258

31.

32.

a 15 = 90 100 a ⋅ 100 = 90 ⋅ 15 100 a = 1350 100 a 1350 = 100 100 a = 13.5 15% of 90 is 13.5. 78 78 = b 100 78 ⋅ 100 = b ⋅ 78 7800 = 78b 7800 78b = 78 78 100 = b 78% of 100 is 78. 297.5 85 = b 100 297.5 ⋅ 100 = b ⋅ 85 29, 750 = 85b 29, 750 85b = 85 85 350 = b 297.5 is 85% of 350. p 78 = 65 100 78 ⋅ 100 = 65 ⋅ p 7800 = 65 p 7800 65 p = 65 65 120 = p 78 is 120% of 65.

ISM: Prealgebra

33.

23.8 p = 85 100 23.8 ⋅ 100 = 85 ⋅ p 2380 = 85 p 2380 85 p = 85 85 28 = p 23.8 is 28% of 85.

Chapter 7: Percent

38.

a 38 34. = 200 100 a ⋅ 100 = 200 ⋅ 38 100 a = 7600 100 a 7600 = 100 100 a = 76 38% of 200 is 76.

39.

a 40 35. = 85 100 a ⋅ 100 = 85 ⋅ 40 100 a = 3400 100 a 3400 = 100 100 a = 34 34 is 40% of 85.

40.

128.7 p 36. = 99 100 128.7 ⋅ 100 = 99 ⋅ p 12,870 = 99 p 12,870 99 p = 99 99 130 = p 130% of 99 is 128.7.

37.

115 p = 250 100 115 ⋅ 100 = 250 ⋅ p 11, 500 = 250 p 11, 500 250 p = 250 250 46 = p 46% of 250 is 115.

45 a = 84 100 a ⋅ 100 = 84 ⋅ 45 100 a = 3780 100 a 3780 = 100 100 a = 37.8 37.8 is 45% of 84. 63 42 = b 100 63 ⋅ 100 = b ⋅ 42 6300 = 42b 6300 42b = 42 42 150 = b 42% of 150 is 63. 58.9 95 = b 100 58.9 ⋅ 100 = b ⋅ 95 5890 = 95b 5890 95b = 95 95 62 = b 95% of 62 is 58.9.

Section 7.4 Practice Exercises 1. Method 1: What number   is 25% of 2174?

↓ ↓ ↓ ↓ ↓ x = 25% ⋅ 2174 x = 0.25 ⋅ 2174 x = 543.5 We predict 543.5 miles of the trail resides in the state of Virginia. Method 2: What number   is 25% of 2174? ↓ ↓ ↓ amount percent base a 25 = 2174 100 a ⋅100 = 2174 ⋅ 25 100a = 54, 350 100a 54, 350 = 100 100 a = 543.5 We predict 543.5 miles of the trail resides in the state of Virginia. Copyright © 2025 Pearson Education, Inc.

259

Chapter 7: Percent

ISM: Prealgebra

2. Method 1: 195 is what percent of 3326?   

4. Method 1: What number   is 1.4% of 287 million?

↓ ↓ ↓ ↓ ↓ 195 = x ⋅ 3326 195 = 3326 x 195 3326 x = 3326 3326 0.06 ≈ x 6% = x It is estimated that the number of nurses employed will increase by 6%. Method 2: 195 is what percent of 3326?    ↓ ↓ ↓ amount percent base p 195 = 3326 100 195 ⋅100 = 3326 ⋅ p 19,500 = 3326 p 19,500 3326 p = 3326 19,500 6≈ p It is estimated that the number of nurses employed will increase by 6%. 3. Method 1: 864 is 32% of what number   ?

↓ ↓ ↓ amount percent base 864 32 = b 100 864 ⋅ 100 = b ⋅ 32 86, 400 = 32b 86, 400 32b = 32 32 2700 = b There are 2700 students at Euclid University.

The increase in the number of registered vehicles on the road in 2022 was 4.018 million.

b. The total number of registered vehicles on the road in 2022 was 287 million + 4.018 million = 291.018 million

Method 2: What number   is 1.4% of 287 million? ↓ ↓ amount percent a 1.4 = 287 100 a ⋅100 = 287 ⋅1.4 100a = 401.8 100a 401.8 = 100 100 a = 4.018 a.

↓ ↓ ↓ ↓ ↓ 864 = 32% ⋅ x 864 = 0.32 ⋅ x 864 0.32 x = 0.32 0.32 2700 = x There are 2700 students at Euclid University. Method 2: 864 is 32% of what number   ?

260

↓ ↓ ↓ ↓ ↓ x = 1.4% ⋅ 287 x = 0.014 ⋅ 287 x = 4.018

↓ base

The increase in the number of registered vehicles on the road in 2022 was 4.018 million.

b. The total number of registered vehicles on the road in 2022 was 287 million + 4.018 million = 291.018 million 5. Find the amount of increase by subtracting the original number of attendants from the new number of attendants. amount of increase = 333 − 285 = 48 The amount of increase is 48 attendants. amount of increase percent of increase = original amount 48 = 285 ≈ 0.168 = 16.8% The number of attendants to the local play, Peter Pan, increased by about 16.8%.

ISM: Prealgebra

Chapter 7: Percent

6. Find the amount of decrease by subtracting 18,483 from 20,200. Amount of decrease = 20,200 − 18,483 = 1717 The amount of decrease is 1717. amount of decrease percent of decrease = original amount 1717 = 20, 200 = 0.085 = 8.5% The population decreased by 8.5%. Vocabulary, Readiness & Video Check 7.4 1. The price of the home is $375,000. 2. An improper fraction is greater than 1, so our percent increase is greater than 100%. Exercise Set 7.4 2. 28 is 35% of what number? Method 1: 28 = 35% ⋅ x 28 = 0.35 x 28 0.35 x = 0.35 0.35 80 = x 80 children attend this day care center. Method 2: 28 35 = b 100 28 ⋅ 100 = b ⋅ 35 2800 = 35b 2800 35b = 35 35 80 = b 80 children attend this day care center. 4. 60% of 220 is what number? Method 1: 60% ⋅ 220 = x 0.60 ⋅ 220 = x 132 = x The maximum weight resistance is 132 pounds.

Method 2: 60 a = 220 100 100 ⋅ a = 220 ⋅ 60 100 a = 13, 200 100 a 13, 200 = 100 100 a = 132 The maximum weight resistance is 132 pounds. 6. 154 is what percent of 3850? Method 1: 154 = x ⋅ 3850 154 3850 x = 3850 3850 0.04 = x 4% = x Pierre received 4% of his total spending at the food cooperative as a dividend. Method 2: 154 p = 3850 100 154 ⋅ 100 = p ⋅ 3850 15, 400 = 3850 p 15, 400 3850 p = 3850 3850 4= p Pierre received 4% of his total spending at the food cooperative as a dividend. 8. 62% of 43,600 is what number? Method 1: 62% ⋅ 43, 600 = x 0.62 ⋅ 43, 600 = x 27, 032 = x 27,032 screens were digital non-3-D. Method 2: a 62 = 43, 600 100 a ⋅100 = 43, 600 ⋅ 62 100a = 2, 703, 200 100a 2, 703, 200 = 100 100 a = 27, 032 27,032 screens were digital non-3-D.

261

Chapter 7: Percent

ISM: Prealgebra

10. What percent of 121,461 is 73,900? Method 1: x ⋅121, 461 = 73,900 121, 461x 73,900 = 121, 461 121, 461 x ≈ 0.61 x ≈ 61% Of the 121,461 veterinarians in the United States in 2021, 61% practiced medicine in a clinic. Method 2: 73,900 p = 121, 461 100 73,900 ⋅100 = 121, 461 ⋅ p 7,390, 000 = 121, 461 p 7,390, 000 121, 461 p = 121, 461 121, 461 61 ≈ p Of the 121,461 veterinarians in the United States in 2021, 61% practiced medicine in a clinic. 12. 5% of 7640 is what number? Method 1: 5% ⋅ 7640 = x 0.05 ⋅ 7640 = x 382 = x There are 382 fewer students this year. The current enrollment is 7640 − 382 = 7258 students. Method 2: 5 a = 7640 100 a ⋅ 100 = 7640 ⋅ 5 100 a = 38, 200 100 a 38, 200 = 100 100 a = 382 There are 382 fewer students this year. The current enrollment is 7640 − 382 = 7258 students. 14. 22% of 467 million is what number? Method 1: 22% ⋅ 467 = x 0.22 ⋅ 467 = x 102.74 = x a.

The increase in the revenue from vinyl albums was $102.74 million.

b. The 2022 revenue from vinyl albums was $467 million + $102.74 million = $569.74 million. 262

Method 2: a 22 = 467 100 a ⋅100 = 467 ⋅ 22 100a = 10, 274 100a 10, 274 = 100 100 a = 102.74 a.

The increase in the revenue from vinyl albums was $102.74 million.

b. The 2022 revenue from vinyl albums was $467 million + $102.74 million = $569.74 million. 16. 1.1% of 2,960,000 is what number? Method 1: 1.1% ⋅ 2,960, 000 = x 0.011 ⋅ 2,960, 000 = x 32,560 = x The population of Mississippi in 2023 was 2,960,000 − 32,560 = 2,927,440. Method 2: a 1.1 = 2,960, 000 100 a ⋅100 = 2,960, 000 ⋅1.1 100a = 3, 256, 000 100a 3, 256, 000 = 100 100 a = 32,560 The population of Mississippi in 2023 was 2,960,000 − 32,560 = 2,927,440. 18. 28 is what percent of 115? Method 1: 28 = x ⋅115 28 115 x = 115 115 0.24 ≈ x 24% ≈ x 24% of the runs are easy. Method 2: 28 p = 115 100 28 ⋅100 = 115 ⋅ p 2800 = 115 p 2800 115 p = 115 115 24 ≈ p 24% of the runs are easy.

ISM: Prealgebra

Chapter 7: Percent

20. 5 is what percent of 20? Method 1: 5 = x ⋅ 20 5 20 x = 20 20 0.25 = x 25% = x 25% of the total calories come from fat. Method 2: 5 p = 20 100 5 ⋅100 = 20 ⋅ p 500 = 20 p 500 20 p = 20 20 25 = p 25% of the total calories come from fat. 22. 130 is what percent of 190? Method 1: 130 = x ⋅ 190 130 190 x = 190 190 0.684 ≈ x 68.4% ≈ x 68.4% of the total calories come from fat. Method 2: p 130 = 190 100 130 ⋅ 100 = 190 ⋅ p 13, 000 = 190 p 13, 000 190 p = 190 190 68.4 ≈ p 68.4% of the total calories come from fat. 24. 35 is what percent of 130? Method 1: 35 = x ⋅ 130 35 = 130 x 35 130 x = 130 130 0.269 ≈ x 26.9% ≈ x 26.9% of the total calories come from fat.

Method 2: 35 p = 130 100 35 ⋅ 100 = 130 ⋅ p 3500 = 130 p 3500 130 p = 130 130 26.9 ≈ p 26.9% of the total calories come from fat. 26. 842.40 is 18% of what number? Method 1: 842.40 = 18% ⋅ x 842.40 = 0.18 x 842.40 0.18 x = 0.18 0.18 4680 = x The banker’s total pay is $4680. Method 2: 842.40 18 = 100 b 842.40 ⋅ 100 = b ⋅ 18 84, 240 = 18b 84, 240 18b = 18 18 4680 = b The banker’s total pay is $4680. 28. What number is 1.04% of 28,350? Method 1: x = 1.04% ⋅ 28,350 x = 0.0104 ⋅ 28,350 x = 294.84 There are expected to be 295 defective components in the batch. Method 2: a 1.04 = 28,350 100 a ⋅ 100 = 28,350 ⋅ 1.04 100 a = 29, 484 100 a 29, 484 = 100 100 a = 294.84 There are expected to be 295 defective components in the batch. 30. What number is 6.5% of 58,500? Method 1: x = 6.5% ⋅ 58,500 x = 0.065 ⋅ 58,500 x = 3802.5 The union worker can expect an increase of $3802.50 in salary. The new salary will be $58,500 + $3802.50 = $62,302.50.

263

Chapter 7: Percent

ISM: Prealgebra

Method 2: a 6.5 = 58, 500 100 a ⋅ 100 = 58, 500 ⋅ 6.5 100 a = 380, 250 100 a 380, 250 = 100 100 a = 3802.50 The union worker can expect an increase of $3802.50 in salary. The new salary will be $58,500 + $3802.50 = $62,302.50. 32. 126 is 70% of what number? Method 1: 126 = 70% ⋅ x 126 = 0.70 x 126 0.70 x = 0.70 0.70 180 = x $126 is 70% of the total price of $180. Method 2: 126 70 = b 100 126 ⋅100 = b ⋅ 70 12, 600 = 70b 12, 600 70b = 70 70 180 = b $126 is 70% of the total price of $180. 34. 43.8% of 56 million is what number? Method 1: 43.8% ⋅ 56 million = x 0.438 ⋅ 56 million = x 24.528 million = x 56 million + 24.528 million = 80.528 million The increase is 24.528 million and the projected population in 2040 is 80.528 million. Method 2: a 43.8 = 56 million 100 a ⋅100 = 56 million ⋅ 43.8 100a = 2452.8 million 100a 2452.8 million = 100 100 a = 24.528 million 56 million + 24.528 million = 80.528 million The increase is 24.528 million and the projected population in 2040 is 80.528 million. 36. What number is 3% of 1960 thousand? Method 1: x = 3% ⋅ 1960 x = 0.03 ⋅ 1960 x = 58.8 The increase is projected to be 58.8 thousand. The number of bachelor degrees awarded in 2028−2029 is projected to be 1960 thousand + 58.8 thousand = 2018.8 thousand. 264

ISM: Prealgebra

Chapter 7: Percent

Method 2: a 3 = 1960 100 a ⋅100 = 1960 ⋅ 3 100a = 5880 100a 5880 = 100 100 a = 58.8 The increase is projected to be 58.8 thousand. The number of bachelor degrees awarded in 2028−2029 is projected to be 1960 thousand + 58.8 thousand = 2018.8 thousand. Original Amount

New Amount

Amount of Increase

Percent of Increase

38.

12 − 8 = 4

4 = 0.5 = 50% 8

40.

170

170 − 68 = 102

102 = 1.5 = 150% 68

Original Amount

New Amount

Amount of Decrease

Percent of Decrease

42.

25 − 20 = 5

5 = 0.20 = 20% 25

44.

200

162

200 − 162 = 38

38 = 0.19 = 19% 200

amount of decrease original amount 530 − 477 = 530 53 = 530 = 0.10 The decrease in the number of employees was 10%.

46. percent of decrease =

amount of decrease original amount 10,845 − 10, 700 = 10,845 145 = 10,845 ≈ 0.013 The decrease in the number of cable TV systems was 1.3%.

48. percent of decrease =

265

Chapter 7: Percent

ISM: Prealgebra

amount of decrease original amount 125,536 − 105,804 = 125,536 19, 732 = 125,536 ≈ 0.157 The decrease in population is expected to be 15.7%.

50. percent of decrease =

amount of decrease original amount 419, 000 − 377, 000 = 419, 000 42, 000 = 419, 000 ≈ 0.100 The decrease in correctional officers is expected to be 10.0%.

52. percent of decrease =

amount of increase original amount 3.3 − 3.1 = 3.1 0.2 = 3.1 ≈ 0.065 The increase in registered nurses is expected to be 6.5%

54. percent of increase =

amount of decrease original amount 2045 − 1906 = 2045 139 = 2045 ≈ 0.068 The decrease in Ford vehicle sales in the United States from 2020 to 2021 was 6.8%.

56. percent of decrease =

amount of decrease original amount 767 − 658 = 767 109 = 767 ≈ 0.142 The decrease in boating accidents was 14.2%.

58. percent of decrease =

266

amount of increase original amount 6.5 − 4.7 = 4.7 1.8 = 4.7 ≈ 0.383 The increase in revenue from streaming music was 38.3%.

60. percent of increase =

62. 0.7 29.4 becomes 7

64.

42 294 −28 14 −14 0

78.00 − 19.46 58.54

3⋅5 5  3  5  3 ⋅ 5 = = 66.  −  −  =  8  12  8 ⋅ 12 8 ⋅ 3 ⋅ 4 32 4 9 4  9 68. 2 − 3 = 2 +  −3  5 10 5  10  Subtract the absolute values. 9 9 3 3 10 10 4 8 −2 −2 5 10 1 1 10 Attach the sign of the larger absolute value. 4 9 1 2 − 3 = −1 5 10 10 70. answers may vary 72. What number is 62.1% of 273,800,000? Method 1: x = 62.1% ⋅ 273,800,000 x = 0.621 ⋅ 273,800,000 x = 170,029,800 The number of Internet users in Indonesia was 170,029,800 in 2021.

ISM: Prealgebra

Chapter 7: Percent

Method 2: a 62.1 = 273,800, 000 100 a ⋅100 = 273,800, 000 ⋅ 62.1 100a = 17, 002,980, 000 100a 17, 002,980, 000 = 100 100 a = 170, 029,800 The number of Internet users in Indonesia was 170,029,800 in 2021. 74. 9,200,860 is what percent of 10,420,000? Method 1: 9, 200,860 = x ⋅10, 420, 000 9, 200,860 10, 420, 000 x = 10, 420, 000 10, 420, 000 0.883 = x 88.3% of the population of Sweden were Internet users in 2021. Method 2: 9, 200,860 p = 10, 420, 000 100 9, 200,860 ⋅100 = 10, 420, 000 ⋅ p 920, 086, 000 = 10, 420, 000 p 920, 086, 000 =p 10, 420, 000 88.3 = p 88.3% of the population of Sweden were Internet users in 2021. 76. 20,090,100 is 16.7% of what number? Method 1: 20, 090,100 = 16.7% ⋅ x 20.090,100 = 0.167 ⋅ x 20, 090,100 =x 0.167 120,300, 000 = x The population of Ethiopia was about 120,300,000 in 2021. Method 2: 20, 090,100 16.7 = b 100 20, 090,100 ⋅100 = b ⋅16.7 2, 009, 010, 000 = 16.7b 2, 009, 010, 000 16.7b = 16.7 16.7 120,300, 000 = b The population of Ethiopia was about 120,300,000 in 2021.

80. To find the percent of decrease, Payton should have divided by the original amount, which is 180. 180 − 150 30 2 = = 16 % percent of decrease = 180 180 3 Section 7.5 Practice Exercises 1. sales tax = tax rate ⋅ purchase price ↓ ↓ = 8.5% ⋅ $59.90 = 0.085 ⋅ $59.90 ≈ $5.09 The sales tax is $5.09. Total Price = purchase price + sales tax ↓ ↓ = $59.90 + $5.09 = $64.99 The sales tax on $59.90 is $5.09, and the total price is $64.99. 2. sales tax = tax rate ⋅ purchase price ↓ ↓ ↓ $1665 = r ⋅ $18,500 1665 r ⋅18,500 = 18,500 18,500 0.09 = r The sales tax rate is 9%. 3. commission = commission rate ⋅ sales ↓ ↓ = 6.6% ⋅ $47, 632 = 0.066 ⋅ $47, 632 ≈ $3143.71 The commission for the month is $3143.71. 4. commission = commission rate ⋅ sales ↓ ↓ ↓ r ⋅ = $645 $4300 645 =r 4300 0.15 = r 15% = r The commission rate is 15%.

78. answers may vary

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5. amount of discount = discount rate ⋅ original price ↓ ↓ = ⋅ 35% $700 = 0.35 ⋅ $700 = $245 The discount is $245. sale price = original price − discount ↓ ↓ = $700 − $245 = $455 The sale price is $455. Vocabulary, Readiness & Video Check 7.5 1. sales tax = tax rate ⋅ purchase price 2. total price = purchase price + sales tax 3. commission = commission rate ⋅ sales 4. amount of discount = discount rate ⋅ original price 5. sale price = original price − amount of discount 6. We rewrite the percent as an equivalent decimal. 7. We write the commission rate as a percent. 8. Replace “amount of discount” in the second equation with “discount rate ⋅ original price”: sale price = original price − (discount rate ⋅ original price). Exercise Set 7.5 2. sales tax = 6% ⋅ $188 = 0.06 ⋅ $188 = $11.28 The sales tax is $11.28. 4. sales tax = 8% ⋅ $699 = 0.08 ⋅ $699 = $55.92 Total price = $699 + 55.92 = $754.92 The total price of the stereo system is $754.92. 6.

268

$374 = r ⋅ $6800 374 =r 6800 0.055 = r The sales tax rate is 5.5%.

8. a.

$76 = 9.5% ⋅ p $76 = 0.095 p $76 =p 0.095 $800 = p The purchase price of the earrings is $800.

b. total price = $800 + $76 = $876 The total price of the earrings is $876. 10. sales tax = 8% ⋅ $1890 = 0.08 ⋅ $1890 = $151.20 total price = $1890 + $151.20 = $2041.20 The sales tax is $151.20 and the total price is $2041.20. 12. $32.85 = 9% ⋅ p $32.85 = 0.09 p $32.85 =p 0.09 $365 = p The purchase price of the LED TV with DVD player is $365. 14. $103.50 = r ⋅ $1150 103.50 =r 1150 0.09 = r The sales tax rate is 9%. 16. Purchase price = $35 + $55 + $95 = $185 Sales tax = 6.5% ⋅ $185 = 0.065 ⋅ $185 ≈ $12.03 Total price = $185 + $12.03 = $197.03 The sales tax is $12.03 and the total price of the clothing is $197.03. 18. commission = 12.8% ⋅ $1638 = 0.128 ⋅ $1638 ≈ $209.66 Her commission was $209.66. 20.

$3575 = r ⋅ $32, 500 $3575 =r $32, 500 0.11 = r The commission rate is 11%.

22. commission = 5.6% ⋅ $9638 = 0.056 ⋅ $9638 ≈ $539.73 Her commission will be $539.73.

ISM: Prealgebra

24.

Chapter 7: Percent

$1750 = 7% ⋅ sales $1750 = 0.07 ⋅ s $1750 =s 0.07 $25, 000 = s The selling price of the fertilizer was $25,000.

Original Price

Discount Rate

Amount of Discount

Sale Price

26.

$74

20%

20% ⋅ $74 = $14.80

$78 − $14.80 = $59.20

28.

$110.60

40%

40% ⋅ $110.60 = $44.24

$110.60 − $44.24 = $66.36

30.

$370

25%

25% ⋅ $370 = $92.50

$370 − $92.50 = $277.50

32.

$17,800

12%

12% ⋅ $17,800 = $2136

$17,800 − $2136 = $15,664

34. discount = 30% ⋅ $4295 = 0.30 ⋅ $4295 = $1288.50 sale price = $4295 − $1288.50 = $3006.50 The discount is $1288.50 and the sale price is $3006.50.

Purchase Price

Tax Rate

Sales Tax

Total Price

36.

$243

8% ⋅ $243 = $19.44

$243 + $19.44 = $262.44

38.

$65

8.4%

8.4% ⋅ $65 = $5.46

$65 + $5.46 = $70.46

Sale

Commission Rate

Commission

40.

$195,450

$195,450 ⋅ 5% = $9772.50

42.

$25,600

$2304 = 0.09 = 9% $25, 600

$2304

44. 500 ⋅

2 ⋅ 3 = 40 ⋅ 3 = 120 25

46. 1000 ⋅

1 ⋅ 5 = 50 ⋅ 5 = 250 20

48. 6000 ⋅ 0.06 ⋅

3 3 = 360 ⋅ = 270 4 4

50. 10% ⋅ $200 = 0.10 ⋅ $200 = $20 $200 + $20 = $220 The best estimate is c; $220.

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Bill Amount

10%

15%

20%

52.

$15.89 ≈ $16

$1.60

1 $1.60 + ($1.60) = $1.60 + $0.80 = $2.40 2

2($1.60) = $3.20

54.

$9.33 ≈ $9

$0.90

1 $0.90 + ($0.90) = $0.90 + $0.45 = $1.35 2

2($0.90) = $1.80

56. A discount of 20% plus an additional 40% is better. Answers may vary. 58. commission = 5.5% ⋅ $562, 560 = 0.055 ⋅ $562, 560 = $30, 940.80 agent's amount = 60% ⋅ $30, 940.80 = 0.60 ⋅ $30, 940.80 = $18, 564.48 The real estate agent will receive $18,564.48. Section 7.6 Practice Exercises 1. I = P ⋅ R ⋅ T I = $875 ⋅ 7% ⋅ 5 = $875 ⋅ (0.07) ⋅ 5 = $306.25 The simple interest is $306.25. 2. I = P ⋅ R ⋅ T

9 12 9 = $1500 ⋅ (0.20) ⋅ 12 = $225 She paid $225 in interest. I = $1500 ⋅ 20% ⋅

3. I = P ⋅ R ⋅ T

6 12 6 = $2100 ⋅ (0.13) ⋅ 12 = $136.50 The interest is $136.50. total amount = principal + interest = $2100 + $136.50 = $2236.50 After 6 months, the total amount paid will be $2236.50. = $2100 ⋅ 13% ⋅

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Chapter 7: Percent

4. P = $3000, r = 4% = 0.04, n = 1, t = 6 years  r A = P 1 +   n

n⋅t

1⋅6

 0.04  = $3000  1 + 1   = $3000(1.04)6 ≈ $3795.96 The total amount after 6 years is $3795.96. 1 5. P = $5500, r = 6 % = 0.0625, n = 365, t = 5 4

 r A = P 1 +   n

n⋅t

4. When interest is computed on the original principal only, it is called simple interest. 5. Total amount (paid or received) = principal + interest 6. The principal amount is the money borrowed, loaned, or invested. 7. Simple interest is calculated on the principal only. 8. semiannually; 2

365⋅5

 0.0625  = $5500  1 + 365   ≈ $7517.41 The total amount after 5 years is $7517.41.

Calculator Explorations  0.09  1. A = $600  1 + 4  

4⋅5

≈ $936.31 365⋅15

 0.04  2. A = $10, 000  1 + 365  

1⋅20

 0.11  3. A = $1200  1 + 1  

 0.07  4. A = $5800  1 + 2    0.06  5. A = $500  1 + 4  

2⋅1

4⋅4

≈ $18, 220.59

≈ $9674.77

≈ $6213.11

≈ $634.49

365⋅19

 0.05  6. A = $2500  1 + 365  

≈ $6463.85

Vocabulary, Readiness & Video Check 7.6 1. To calculate simple interest, use I = P ⋅ R ⋅ T. 2. To calculate compound interest, use r  A = P 1 +   n

3. Compound interest is computed on not only the original principal, but on interest already earned in previous compounding periods.

n⋅t

Exercise Set 7.6 2. simple interest = principal ⋅ rate ⋅ time = $800 ⋅ 9% ⋅ 3 = $800 ⋅ 0.09 ⋅ 3 = $216 4. simple interest = principal ⋅ rate ⋅ time = $950 ⋅ 12.5% ⋅ 5 = $950 ⋅ 0.125 ⋅ 5 = $593.75 6. simple interest = principal ⋅ rate ⋅ time 1 = $1500 ⋅ 14% ⋅ 2 4 = $1500 ⋅ 0.14 ⋅ 2.25 = $472.50 8. simple interest = principal ⋅ rate ⋅ time 8 = $775 ⋅ 15% ⋅ 12 2 = $775 ⋅ 0.15 ⋅ 3 = $77.50 10. simple interest = principal ⋅ rate ⋅ time 18 = $1000 ⋅ 10% ⋅ 12 = $1000 ⋅ 0.10 ⋅ 1.5 = $150 12. simple interest = principal ⋅ rate ⋅ time = $265, 000 ⋅ 8.25% ⋅ 30 = $265, 000 ⋅ 0.0825 ⋅ 30 = $655,875 $265,000 + $655,875 = $920,875 The amount of interest is $655,875. The total amount paid is $920,875.

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14. simple interest = principal ⋅ rate ⋅ time 9 = $4500 ⋅ 14% ⋅ 12 = $4500 ⋅ 0.14 ⋅ 0.75 = $472.50 Total amount = $4500 + $472.50 = $4972.50 16. Simple interest = principal ⋅ rate ⋅ time = $2000 ⋅ 8% ⋅ 5 = $2000 ⋅ 0.08 ⋅ 5 = $800 Total amount = $2000 + $800 = $2800  r 18. A = P  1 +   n

n⋅t

2⋅5

 0.08  = 2000  1 + 2   = 2000(1.04)10 ≈ 2960.49 The total amount is $2960.49.  r 28. A = P  1 +   n

n⋅t

 0.08  = 2000  1 + 365  

n⋅t

365⋅5

1825

 0.08  = 2000  1 + 365   ≈ 2983.52 The total amount is $2983.52.

⋅ 110

 0.15  = 2060  1 + 1   = 2060(1.15)10 ≈ 8333.85 The total amount is $8333.85. r  20. A = P  1 +  n  

30. Perimeter = 18 + 16 + 12 = 46 The perimeter is 46 centimeters. 32. Perimeter = 21 + 21 + 21 + 21 = 84 The perimeter is 84 miles.

n⋅t

4⋅15

 0.10  = 1450  1 + 4   = 1450(1.025)60 ≈ 6379.70 The total amount is $6379.70.  r 22. A = P  1 +   n

r  26. A = P  1 +  n  

x  x x  5  x ⋅5 5 = 34. − ÷  −  = − ⋅  −  = 4  5 4  x  4⋅ x 4 36.

3 9 1 3 22 1 3 ⋅ 2 ⋅ 11 ⋅ 1 2 ÷ ⋅ = ⋅ ⋅ = = 11 22 3 11 9 3 11 ⋅ 9 ⋅ 3 9

38. answers may vary

n⋅t

Chapter 7 Vocabulary Check

 0.08  = 3500  1 + 365  

365⋅10

1. In a mathematical statement, of usually means “multiplication.”

3650

 0.08  = 3500  1 + 365   ≈ $7788.71 The total amount is $7788.71.  r 24. A = P  1 +   n

n⋅t

2⋅1

 0.10  = 6375  1 + 2   = 6375(1.05)2 ≈ 7028.44 The total amount is $7028.44.

2. In a mathematical statement, is means “equals.” 3. Percent means “per hundred.” 4. Compound interest is computed not only on the principal, but also on interest already earned in previous compounding periods. 5. In the percent proportion

amount percent = . 100 base

6. To write a decimal or fraction as a percent, multiply by 100%. 7. The decimal equivalent of the % symbol is 0.01.

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Chapter 7: Percent

8. The fraction equivalent of the % symbol is

1 . 100

9. The percent equation is base ⋅ percent = amount. 10. percent of decrease =

11. percent of increase =

amount of decrease original amount amount of increase original amount

12. sales tax = tax rate ⋅ purchase price 13. total price = purchase price + sales tax 14. commission = commission rate ⋅ sales 15. amount of discount = discount rate ⋅ original price 16. sale price = original price − amount of discount Chapter 7 Review 1.

37 = 37% 100 37% of adults preferred pepperoni. 77 = 77% 100 77% of free throws were made.

3. 26% = 26(0.01) = 0.26 4. 75% = 75(0.01) = 0.75 5. 3.5% = 3.5(0.01) = 0.035 6. 1.5% = 1.5(0.01) = 0.015 7. 275% = 275(0.01) = 2.75 8. 400% = 400(0.01) = 4.00 or 4 9. 47.85% = 47.85(0.01) = 0.4785 10. 85.34% = 85.34(0.01) = 0.8534 11. 1.6 = 1.6(100%) = 160% 12. 5.2 = 5.2(100%) = 520%

15. 0.71 = 0.71(100%) = 71% 16. 0.65 = 0.65(100%) = 65% 17. 6 = 6(100)% = 600% 18. 9 = 9(100%) = 900%

7  1  19. 7% = 7   = 100 100   3  1  15 = 20. 15% = 15  =  100  100 20  1  25 1 = 21. 25% = 25  =  100  100 4  1  22. 8.5% = 8.5    100  8.5 = 100 8.5 ⋅ 10 = 100 ⋅ 10 85 = 1000 17 = 200  1  23. 10.2% = 10.2    100  10.2 = 100 10.2 ⋅ 10 = 100 ⋅ 10 102 = 1000 51 = 500

2 50 50  1  50 1 = = 24. 16 % = % = 3 3 3  100  300 6 1 100 100  1  100 1 %= = = 25. 33 % = 3 3 3  100  300 3 10 1  1  110 = =1 =1 26. 110% = 110   100 10  100  100

13. 0.076 = 0.076(100%) = 7.6% 14. 0.085 = 0.085(100%) = 8.5%

27.

2 2 100 200 = ⋅ %= % = 40% 5 5 1 5

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28.

7 7 100 700 = ⋅ %= % = 70% 10 10 1 10

29.

7 7 100 700 175 1 = ⋅ %= %= % = 58 % 12 12 1 12 3 3

39. x = 17% ⋅ 640 x = 0.17 ⋅ 640 x = 108.8 108.8 is 17% of 640. 40.

2 5 100 500 2 30. 1 = ⋅ %= % = 166 % 3 3 1 3 3

1 5 100 500 31. 1 = ⋅ %= % = 125% 4 4 1 4

41.

3 3 100 300 32. = ⋅ %= % = 60% 5 5 1 5

33.

1 1 100 100 = ⋅ %= % = 6.25% 16 16 1 16

34.

5 5 100 500 = ⋅ %= % = 62.5% 8 8 1 8

35.

1250 = 1.25% ⋅ x 1250 = 0.0125 x 1250 0.0125 x = 0.0125 0.0125 100, 000 = x 1250 is 1.25% of 100,000.

1 36. x = 33 % ⋅ 24, 000 3 100 1 ⋅ ⋅ 24, 000 x= 3 100 1 x = ⋅ 24, 000 3 x = 8000 1 8000 is 33 % of 24,000. 3

37. 124.2 = x ⋅ 540 124.2 540 x = 540 540 0.23 = x 23% = x 124.2 is 23% of 540. 38.

274

22.9 = 20% ⋅ x 22.9 = 0.20 ⋅ x 22.9 0.20 x = 0.20 0.20 114.5 = x 22.9 is 20% of 114.5.

42.

43.

44.

693 = x ⋅ 462 693 462 x = 462 462 1.5 = x 150% = x 693 is 150% of 462.

104.5 25 = b 100 104.5 ⋅ 100 = b ⋅ 25 10, 450 = 25b 10, 450 25b = 25 25 418 = b 104.5 is 25% of 418. 16.5 5.5 = b 100 16.5 ⋅ 100 = b ⋅ 5.5 1650 = 5.5b 1650 5.5b = 5.5 5.5 300 = b 16.5 is 5.5% of 300. 30 a = 532 100 a ⋅ 100 = 532 ⋅ 30 100 a = 15, 960 100 a 15, 960 = 100 100 a = 159.6 159.6 is 30% of 532. 63 p = 35 100 63 ⋅ 100 = 35 ⋅ p 6300 = 35 p 6300 35 p = 35 35 180 = p 63 is 180% of 35.

ISM: Prealgebra

45.

46.

Chapter 7: Percent Method 2: p 200 = 12,360 100 2000 ⋅ 100 = 12,360 ⋅ p 200, 000 = 12,360 p 200, 000 12,360 p = 12,360 12,360 16 ≈ p 16% of freshmen are enrolled in prealgebra.

93.5 p = 85 100 93.5 ⋅ 100 = 85 ⋅ p 9350 = 85 p 9350 85 p = 85 85 110 = p 93.5 is 110% of 85. 33 a = 500 100 a ⋅ 100 = 500 ⋅ 33 100 a = 16, 500 100 a 16, 500 = 100 100 a = 165 165 is 33% of 500.

amount of decrease original amount 675 − 534 = 675 141 = 675 ≈ 0.209 ≈ 20.9% Violent crime decreased 20.9%.

49. percent of decrease =

47. 1320 is what percent of 2000? Method 1: 1320 = x ⋅ 2000 1320 2000 x = 2000 2000 0.66 = x 66% = x 66% of people own microwaves. Method 2: 1320 p = 2000 100 1320 ⋅ 100 = 2000 ⋅ p 132, 000 = 2000 p 132, 000 2000 p = 2000 2000 66 = p 66% of people own microwaves. 48. 2000 is what percent of 12,360? Method 1: 2000 = x ⋅ 12,360 2000 12,360 x = 12,360 12,360 0.16 ≈ x 16% ≈ x 16% of freshmen are enrolled in prealgebra.

amount of increase original amount 33 − 16 = 16 17 = 16 = 1.0625 = 106.25% The charge will increase 106.25%.

50. percent of increase =

51. Amount of increase = percent increase ⋅ original amount = 15% ⋅ $11.50 = 0.15 ⋅ $11.50 ≈ $1.73 Total amount = $11.50 + $1.73 = $13.23 The new hourly rate is $13.23. 52. Amount of decrease = percent decrease ⋅ original amount = 4% ⋅ $215, 000 = 0.04 ⋅ $215, 000 = $8600 Total amount = $215,000 − $8600 = $206,400 $206,400 is expected to be collected next year.

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53. sales tax = tax rate ⋅ purchase price = 9.5% ⋅ $250 = 0.095 ⋅ $250 = $23.75 Total amount = $250 + $23.75 = $273.75 The total price for the coat is $273.75. 54. sales tax = tax rate ⋅ purchase price = 8.5% ⋅ $25.50 = 0.085 ⋅ $25.50 ≈ $2.17 The sales tax is $2.17. 55. commission = commission rate ⋅ sales = 5% ⋅ $100, 000 = 0.05 ⋅ $100, 000 = $5000 The sales representative’s commission is $5000. 56. commission = commission rate ⋅ sales = 7.5% ⋅ $4005 = 0.075 ⋅ $4005 ≈ $300.38 The sales clerk’s commission is $300.38. 57. Amount of discount = discount ⋅ original price = 30% ⋅ $3000 = 0.30 ⋅ $3000 = $900 Sale price = $3000 − $900 = $2100 The amount of discount is $900; the sale price is $2100. 58. Amount of discount = discount ⋅ original price = 10% ⋅ $90 = 0.10 ⋅ $90 = $9.00 Sale price = $90 − $9 = $81 The amount of discount is $9; the sale price is $81. 59. I = P ⋅ R ⋅ T

4 12 1 = $4000 ⋅ 0.12 ⋅ 3 = $160 The simple interest is $160. = $4000 ⋅ 12% ⋅

60. I = P ⋅ R ⋅ T

3 12 = $6500 ⋅ 0.20 ⋅ 0.25 = $325 The simple interest is $325. = $6500 ⋅ 20% ⋅

 r 61. A = P  1 +   n

n⋅t

⋅ 115

 0.12  = 5500  1 + 1   = 5500(1.12)15 ≈ 30,104.61 The total amount is $30,104.61. r  62. A = P  1 +   n

n⋅t

2⋅10

 0.11  = 6000  1 + 2   = 6000(1.055)20 ≈ 17, 506.54 The total amount is $17,506.54. r  63. A = P  1 +   n

n⋅t

4⋅5

 0.12  = 100  1 + 4   = 100(1.03)20 ≈ 180.61 The total amount is $180.61. r  64. A = P  1 +   n

n⋅t

4⋅20

 0.18  = 1000  1 + 4   = 1000(1.045)80 ≈ 33,830.10 The total amount is $33,830.10.

65. 3.8% = 3.8(0.01) = 0.038 66. 124.5% = 124.5(0.01) = 1.245 67. 0.54 = 0.54(100%) = 54% 68. 95.2 = 95.2(100%) = 9520%

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Chapter 7: Percent

 1  47 69. 47% = 47  =  100  100

77.

 1  70. 5.6% = 5.6    100  5.6 = 100 5.6 ⋅ 10 = 100 ⋅ 10 56 = 1000 7 = 125

78.

71.

1 1 100 100 1 = ⋅ %= % = 12 % or 12.5% 8 8 1 8 2

72.

6 6 100 600 = ⋅ %= % = 120% 5 5 1 5

73.

43 = 16% ⋅ x 43 = 0.16 x 43 0.16 x = 0.16 0.16 268.75 = x 43 is 16% of 268.75.

74.

27.5 = x ⋅ 25 27.5 25 x = 25 25 1.1 = x 110% = x 27.5 is 110% of 25.

75. x = 36% ⋅ 1968 x = 0.36 ⋅ 1968 x = 708.48 708.48 is 36% of 1968. 76.

67 = x ⋅ 50 67 50 x = 50 50 1.34 = x 134% = x 67 is 134% of 50.

79.

80.

81.

75 p = 25 100 75 ⋅ 100 = 25 ⋅ p 7500 = 25 p 7500 25 p = 25 25 300 = p 75 is 300% of 25. 16 a = 240 100 a ⋅ 100 = 240 ⋅ 16 100 a = 3840 100 a 3840 = 100 100 a = 38.4 38.4 is 16% of 240. 28 5 = b 100 28 ⋅ 100 = b ⋅ 5 2800 = 5b 2800 5b = 5 5 560 = b 28 is 5% of 560. 52 p = 16 100 52 ⋅ 100 = 16 ⋅ p 5200 = 16 p 5200 16 p = 16 16 325 = p 52 is 325% of 16. 78 = 0.26 = 26% 300 26% of the soft drinks have been sold.

82. $96,950 ⋅ 7% = $96,950 ⋅ 0.07 = $6786.50 The house has lost $6786.50 in value. 83. Sales tax = 8.75% ⋅ $568 = 0.0875 ⋅ $568 = $49.70 Total price = $568 + $49.70 = $617.70 The total price is $617.70.

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84. Amount of discount = 15% ⋅ $23.00 = 0.15 ⋅ $23.00 = $3.45 85. commission = commission rate ⋅ sales $1.60 = r ⋅ $12.80 $1.60 =r $12.80 0.125 = r 12.5% = r The rate of commission is 12.5%. 86. Simple interest = principal ⋅ rate ⋅ time 6 = $1400 ⋅ 13% ⋅ 12 = $1400 ⋅ 0.13 ⋅ 0.5 = $91 Total amount = $1400 + $91 = $1491 The total amount is $1491. 87. Simple interest = principal ⋅ rate ⋅ time = $5500 ⋅ 12.5% ⋅ 9 = $5500 ⋅ 0.125 ⋅ 9 = $6187.50 Total amount = $5500 + $6187.50 = $11,687.50 The total amount is $11,687.50. Chapter 7 Getting Ready for the Test 1. “Percent” means “per hundred” so 12% means 12 12 3⋅ 4 3 or 0.12. Also, = = . Thus, 100 100 4 ⋅ 25 25 choice D, 1.2, does not equal 12%. 2. Since 100% = 100(0.01) = 1.00 = 1, or  1  100 100% = 100  , choice A, 10, does not =  100  100 equal 100%. 3. Since 50% = 50(0.01) = 0.5, or  1  50 50 ⋅1 1 = = , choice B, 5, 50% = 50  =  100  100 50 ⋅ 2 2 does not equal 50%. 4. Since 80% = 80(0.01) = 0.80 = 0.8, or  1  80 20 ⋅ 4 4 = = , choice C, 8, 80% = 80  =  100  100 20 ⋅ 5 5 does not equal 80%. 5. Since 35 is half of 70, 50% of 70 is 35, B.

278

6. Since 8.8 is

1 of 88, 10% of 88 is 8.8, D. 10

7. Since 47 = 47, 100% of 47 is 47, A. 8. Since 7 is

1 of 28, 25% of 28 is 7, C. 4

9. Round $24.86 to $25. 10% of $25 is $2.50. 20% of $25 is 2($2.50) = $5. Of the amounts given, $5 is closest to 20% of $24.86, C. 10. amount of discount = discount rate ⋅ original price = 10% ⋅ $150 = 0.10 ⋅ $150 = $15 The amount of discount is $15, A. 11. discounted price = $150 − $15 = $135 The discounted price is $135, D. 12. 25% of a number is

1 of that number. 4

1 ⋅ $40 = $10 4 The discount amount is $10, so the sale price of the shoes before tax is $40 − $10 = $30, C. 25% ⋅ $40 =

Chapter 7 Test 1. 85% = 85(0.01) = 0.85 2. 500% = 500(0.01) = 5 3. 0.6% = 0.6(0.01) = 0.006 4. 0.056 = 0.056(100%) = 5.6% 5. 6.1 = 6.1(100%) = 610% 6. 0.35 = 0.35(100%) = 35%

1  1  120 =1 7. 120% = 120  = 5  100  100 77  1  38.5 385 = = = 8. 38.5% = 38.5    100  100 1000 200 2 1  1  0.2 = = 9. 0.2% = 0.2  =  100  100 1000 500

ISM: Prealgebra

Chapter 7: Percent

10.

11 11 100 1100 = ⋅ %= % = 55% 20 20 1 20

11.

3 3 100 300 = ⋅ %= % = 37.5% 8 8 1 8

3 7 100 700 12. 1 = ⋅ %= % = 175% 4 4 1 4

13.

3 3 100 300 = ⋅ %= % = 75% 4 4 1 4

 1  40 2 = 14. 40% = 40  =  100  100 5 15. Method 1: x = 42% ⋅ 80 x = 0.42 ⋅ 80 x = 33.6 33.6 is 42% of 80. Method 2: a 42 = 80 100 a ⋅ 100 = 80 ⋅ 42 100 a = 3360 100 a 3360 = 100 100 a = 33.6 33.6 is 42% of 80. 16. Method 1: 0.6% ⋅ x = 7.5 0.006 x = 7.5 0.006 x 7.5 = 0.006 0.006 x = 1250 0.6% of 1250 is 7.5. Method 2: 7.5 0.6 = b 100 7.5 ⋅ 100 = b ⋅ 0.6 750 = 0.6b 750 0.6b = 0.6 0.6 1250 = b 0.6% of 1250 is 7.5.

17. Method 1: 567 = x ⋅ 756 567 x ⋅ 756 = 756 756 0.75 = x 75% = x 567 is 75% of 756. Method 2: 567 p = 756 100 567 ⋅ 100 = 756 ⋅ p 56, 700 = 756 p 56, 700 756 p = 756 756 75 = p 567 is 75% of 756. 18. 12% of 320 is what number? 12% ⋅ 320 = x 0.12 ⋅ 320 = x 38.4 = x There are 38.4 pounds of copper. 19. 20% of what number is $11,350? 20% ⋅ x = $11,350 0.20 x = $11,350 0.2 x $11,350 = 0.2 0.2 x = $56, 750 The value of the potential crop is $56,750. 20. tax = 8.25% ⋅ $354 = 0.0825 ⋅ $354 ≈ $29.21 total amount = $354 + $29.21 = $383.21 The total amount of the stereo is $383.21.

amount of increase original amount 26, 460 − 25, 200 = 25, 200 1260 = 25, 200 = 0.05 The increase in population was 5%.

21. percent of increase =

22. Amount of discount = 15% ⋅ $120 = 0.15 ⋅ $120 = $18 Sale price = $120 − $18 = $102 The amount of the discount is $18; the sale price is $102.

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ISM: Prealgebra 2.

23. commission = 4% ⋅ $9875 = 0.04 ⋅ $9875 = $395 The commission was $395. 24.

$13.77 = rate ⋅ $152.99 $13.77 =r $152.99 0.09 ≈ r 9% ≈ r The sales tax rate is 9%.

409 × 76 2 454 28 630 31, 084

3. −3 − 7 = −3 + (−7) = −10 4. 8 − (−2) = 8 + 2 = 10 5.

25. simple interest = principal ⋅ rate ⋅ time 1 = $2000 ⋅ 9.25% ⋅ 3 2 = $2000 ⋅ 0.0925 ⋅ 3.5 = $647.5

x − 2 = −1 x − 2 + 2 = −1 + 2 x =1

x+4 =3 x +4−4 = 3−4 x = −1

n⋅t

3(2 x − 6) + 6 = 0 3⋅ 2x − 3⋅ 6 + 6 = 0 6 x − 18 + 6 = 0 6 x − 12 = 0 6 x − 12 + 12 = 0 + 12 6 x = 12 6 x 12 = 6 6 x=2

5( x − 2) = 3 x 5 ⋅ x − 5 ⋅ 2 = 3x 5 x − 10 = 3 x 5 x − 5 x − 10 = 3 x − 5 x −10 = −2 x −10 −2 x = −2 −2 5= x

r  26. A = P  1 +   n

1⋅5

 0.08  = 1365  1 + 1   = 1365(1.08)5 ≈ 2005.63 The total amount of $2005.63.

27. Simple interest = principal ⋅ rate ⋅ time 6 = $400 ⋅ 13.5% ⋅ 12 = $400 ⋅ 0.135 ⋅ 0.5 = $27 Total amount = $400 + $27 = $427

amount of decrease original amount 43,150 − 40,870 = 43,150 2280 = 43,150 ≈ 0.053 The number of crimes decreased 5.3%.

28. percent of decrease =

Cumulative Review Chapters 1−7 1.

280

236 × 86 1 416 18 880 20, 296

3 7 3 ⋅ 7 21 9. 3 = ⋅ = = 1 7 1⋅ 7 7 8 5 8 ⋅ 5 40 10. 8 = ⋅ = = 1 5 1⋅ 5 5

11. −

10 2⋅5 =− 27 3⋅3⋅3

Since 10 and 27 have no common factors, − is already in simplest form. 12.

10 y 5 ⋅ 2 ⋅ y 5 y = = 32 16 ⋅ 2 16

10 27

ISM: Prealgebra

13. −

Chapter 7: Percent

7 5 7 6 7⋅6 7 ÷− = − ⋅− = = 12 6 12 5 6 ⋅ 2 ⋅ 5 10

20.

2 7 2 10 2⋅2⋅5 4 14. − ÷ = − ⋅ = − =− 5 10 5 7 5⋅7 7

15. y − x = −

8  3 8 3 5 1 − − =− + =− =− 10  10  10 10 10 2

2+1 2⋅2 + 1 3 6 = 3 2 6 3−3 3⋅5−3⋅4 4 5 4 5 5 4 4+1 = 6 6 15 − 12 20 20 5 6 = 3 20

5 3 ÷ 6 20 5 20 = ⋅ 6 3 5 ⋅ 2 ⋅ 10 = 2 ⋅3⋅3 50 = 9 =

2  1 4  3 1 16. 2 x + 3 y = 2   + 3  −  = +  −  =  5  5 5  5 5 3 1 6 3 7 1 2 6 4 17. − − + = − ⋅ − ⋅ + ⋅ 4 14 7 4 7 14 2 7 4 21 2 24 =− − + 28 28 28 1 = 28 18.

19.

21.

2 7 1 2 5 7 3 1 15 + − = ⋅ + ⋅ − ⋅ 9 15 3 9 5 15 3 3 15 10 21 15 = + − 45 45 45 16 = 45 1+3 1⋅4+3 2 8 = 2 4 8 3−1 3⋅3−1⋅2 4 6 4 3 6 2 4+3 = 8 8 9 − 2 12 12 7 = 8 7 12

7 7 ÷ 8 12 7 12 = ⋅ 8 7 7⋅ 4 ⋅3 = 4⋅2⋅7 3 = 2 =

22.

x x 1 = + 2 3 2 x x 1 6  = 6 +  2   3 2 1 x 3x = 6 ⋅ + 6 ⋅ 3 2 3x = 2 x + 3 3x − 2 x = 2 x + 3 − 2 x x =3 x 1 x + = 3− 2 5 5 x  x 1  10  +  = 10  3 −  2 5 5     x x 1 10 ⋅ + 10 ⋅ = 10 ⋅ 3 − 10 ⋅ 2 5 5 5 x + 2 = 30 − 2 x 5 x + 2 + 2 x = 30 − 2 x + 2 x 7 x + 2 = 30 7 x + 2 − 2 = 30 − 2 7 x = 28 7 x 28 = 7 7 x=4

23. a.

2 9 ⋅ 4 + 2 36 + 2 38 = = = 9 9 9 9

8 11 ⋅ 1 + 8 11 + 8 19 = = = 11 11 11 11

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24. a.

2 5 ⋅ 3 + 2 15 + 2 17 = = = 5 5 5 5

38. mean =

2 7 ⋅ 6 + 2 42 + 2 44 = = = 7 7 7 7

39.

2.5 2.5 ⋅ 100 250 50 = = = 3.15 3.15 ⋅ 100 315 63

40.

5.8 5.8 ⋅ 10 58 29 = = = 7.6 7.6 ⋅ 10 76 38

25. 0.125 = 26. 0.85 =

125 1 = 1000 8

0.21 41. 16 3.36 −32 16 −16 0 The unit price is $0.21/ounce.

85 17 = 100 20

27. −105.083 = −105 28. 17.015 = 17

83 1000

15 3 = 17 1000 200

29.

85.00 − 17.31 67.69

Check:

67.69 + 17.31 85.00

30.

38.00 − 10.06 27.94

Check:

27.94 + 10.06 38.00

2.25 90.00 −80 10 0 −8 0 2 00 −2 00 0 The price is $2.25 per tile, or $2.25 per square foot.

42. 40

31. 7.68 × 10 = 76.8 32. 12.483 × 100 = 1248.3

43.

33. (−76.3)(1000) = −76,300 34. −853.75 × 10 = −8537.5 35. x ÷ y = 2.5 ÷ 0.05

50 0.05 2.5 becomes 5 250 −25 00

36.

x = 4.75 100 470 0 4.75 100 4.7 = 4.75 False No, 470 is not a solution.

37. 55, 67, 75, 86, 91, 91 75 + 86 161 median = = = 80.5 2 2

282

36 + 40 + 86 + 30 192 = = 48 4 4

44.

45.

4.1 2.9 0 7 5 4.1(5) 0 7(2.9) 20.5 ≠ 20.3 No, it is not a true proportion.

6.3 3.5 0 9 5 6.3(5) 0 9(3.5) 31.5 = 31.5 Yes, it is a true proportion. 5 miles x miles = 2 inches 7 inches 5⋅ 7 = 2 ⋅ x 35 = 2 x 35 2 x = 2 2 17.5 = x 17.5 miles corresponds to 7 inches.

ISM: Prealgebra

46.

Chapter 7: Percent

7 problems x problems = 6 minutes 30 minutes 7 ⋅ 30 = 6 ⋅ x 210 = 6 x 210 6 x = 6 6 35 = x The student can complete 35 problems in 30 minutes.

23  1  2.3 48. 2.3% = 2.3   = 100 = 1000 100   1 100  1  100 1 = = 49. 33 % = 3 3  100  300 3 2  1  108 = =1 50. 108% = 108   25  100  100

19  1  1.9 47. 1.9% = 1.9   = 100 = 1000 100  

283

Chapter 8 Section 8.1 Practice Exercises 1. a.

Class Interval (credit card balances)

Tally

Class Frequency (number of months)

$0−$49

|||

$50−$99

||||

$100−$149

$150−$199

$200−$249

$250−$299

Hindi has 7 symbols and each symbol represents 50 million speakers, so Hindi is spoken by 7(50) = 350 million people.

b. Spanish has 9.5 symbols, or 9.5(50) = 475 million speakers. So, 475 million − 350 million = 125 million more people speak Spanish than Hindi. 2. a.

The height of the bar for crustaceans is 30, so approximately 30 endangered species are crustaceans.

b. The shortest bar corresponds to arachnids, so arachnids have the fewest endangered species. 3.

8. a. 4. The height of the bar for 80−89 is 12, so 12 students scored 80−89 on the test. 5. The height of the bar for 40−49 is 1, for 50−59 is 3, for 60−69 is 2, for 70−79 is 10. So, 1 + 3 + 2 + 10 = 16 students scored less than 80 on the test.

The lowest point on the graph corresponds to January, so the average daily temperature is the lowest during January.

b. The point on the graph that corresponds to 25 is December, so the average daily temperature is 25°F in December. c.

The points on the graph that are greater than 70 are June, July, and August. So, the average daily temperature is greater than 70°F in June, July, and August.

9. range = largest data value − smallest data value = $102 − $32 = $70 The range of the salaries is $70 thousand. 10. a.

284

range = largest data value − smallest data value = 67 − 50 = 17 The range of this data set is 17.

ISM: Prealgebra b.

Chapter 8: Graphing and Introduction to Statistics and Probability

mean sum of items = number of items 50 ⋅1 + 59 ⋅ 3 + 60 ⋅ 3 + 62 ⋅ 5 + 63 ⋅ 2 + 67 ⋅1 = 1+ 3 + 3 + 5 + 2 +1 910 = 15 ≈ 60.7 The mean of the data set is approximately 60.7.

Since there are 15 data items and the items are arranged in numerical order, the middle 15 + 1 16 item can be located using = = 8. 2 2 The median is the eighth item, which is 62.

d. The mode has the greatest frequency, so the mode is 62. Vocabulary, Readiness & Video Check 8.1

Exercise Set 8.1 2. South Dakota has the least number of wheat symbols, so the least amount of wheat acreage was planted in South Dakota. 4. Kansas is represented by 7.5 wheat symbols, and each symbol represents 1 million acres, so there were approximately 7.5(1 million) = 7.5 million or 7,500,000 acres of wheat planted in Kansas. 6. Each wheat symbol represents 1 million acres, so look for any states that have more than 6 million = 6 wheat symbols. Texas and Kansas 1 million have more than 6 wheat symbols, so these states plant more than 6,000,000 acres of wheat. 8. North Dakota and Oklahoma each have 6 wheat symbols. The same acreage of wheat is planted in North Dakota and Oklahoma.

1. A bar graph presents data using vertical or horizontal bars.

10. 2022 has 5.5 flames and each flame represents 12,000 wildfires, so there were approximately 5.5(12,000) = 66,000 wildfires in 2022.

2. A pictograph is a graph in which pictures or symbols are used to visually present data.

12. The year with the fewest flames is 2019, so the fewest wildfires occurred in 2019.

3. A line graph displays information with a line that connects data points.

14. 2021 has 5 flames and 2022 has 5

4. A histogram is a special bar graph in which the width of each bar represents a class interval and the height of each bar represents the class frequency. 5. range = greatest data value − smallest data value n +1 can be used to find the 2 position of the median.

6. The formula

7. Count the number of symbols and multiply this number by how much each symbol stands for (from the key). 8. A bar graph lets you see and visually compare the data. 9. A histogram is a special kind of bar graph. 10. 2021; 9.4 goals per match 11. answers may vary; for example: 6, 6, 6, 6

1 flames, 2

1 flame more. The increase in 2 wildfires from 2021 to 2022 was 1 (12, 000) = 6000 wildfires. 2

which is

16. Since each flame represents 12,000 wildfires, look for the years that have fewer than 60, 000 = 5 flames. There were fewer than 12, 000 60,000 wildfires in 2018 and 2019. 18. The shortest bar corresponds to November, so the month in which the fewest hurricanes made landfall is November. 20. The length of the bar for October is 61, so approximately 61 hurricanes made landfall in October. 22. One of 111 hurricanes made landfall in 1 ≈ 0.009 or September 2007. The percent is 111 about 0.9%.

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ISM: Prealgebra

24. The shortest bar corresponds to Sao Paulo, Brazil and the length of the bar is about 21.7. The metropolitan area with the smallest population is Sao Paulo, Brazil and its population is about 21.7 million or 21,700,000.

40. 29 of the adults drive 0−49 miles per week and 17 of the adults drive 50−99 miles per week, so 29 − 17 = 12 more adults drive 0−49 miles per week than 50−99 miles per week.

26. The bar between 24 and 25 corresponds to Seoul, South Korea.

42. 17 of the 100 adults surveyed drive 50−99 miles 17 per week, so the ratio is . 100

28. The bar corresponding to Jakarta, Indonesia has length about 30 and the bar corresponding to Seoul, South Korea has length about 24. Thus, Jakarta, Indonesia is about 30 − 24 = 6 million larger than Seoul, South Korea.

44. The shortest bar corresponds to under 25, so the least number of U.S. householders were in the under 25 age range. 46. According to the graph, approximately 23 million U.S. householders were 35−44 years old.

30.

48. The sum of the last three bars is 24 + 21 + 15 = 60, so approximately 60 million U.S. householders were 55 years old or older. 50. The height of the bar for 45−54 is 22 and the height of the bar for 75 and over is 15. So, approximately 22 − 15 = 7 million more U.S. householders were 45−54 years old than 75 and over.

32.

Class Interval (scores)

Tally

Class Frequency (number of games)

52.

80−89

||||

54.

100−109

Class Interval (account balances) 34. The height of the bar for 200−249 miles per week is 9, so 9 of the adults drive 200−249 miles per week. 36. 9 of the adults drive 200−249 miles per week and 21 of the adults drive 250−299 miles per week, so 9 + 21 = 30 of the adults drive 200 miles or more per week.

Class Frequency (number of people)

56.

$100−$199

|||| |

58.

$300−$399

||||

60.

$500−$599

38. 9 of the adults drive 150−199 miles per week and 9 of the adults drive 200−249 miles per week, so 9 + 9 = 18 of the adults drive 150−249 miles per week.

286

Tally

ISM: Prealgebra

Chapter 8: Graphing and Introduction to Statistics and Probability 84. There are 2 + 1 + 4 + 7 + 2 + 1 = 17 data items.

62.

64. The point on the graph corresponding to 2017 is 8.3, so the average number of goals per match in 2017 was 8.3. 66. The lowest point on the graph corresponds to 2013, so the average number of goals per match was the lowest in 2013.

3 ⋅ 2 + 4 ⋅1 + 5 ⋅ 4 + 6 ⋅ 7 + 7 ⋅ 2 + 8 ⋅1 17 94 = 17 ≈ 5.5

mean =

n + 1 17 + 1 = =9 2 2 The data item in the ninth position is 6, so the median is 6.

The data item with the greatest frequency is the mode, so the mode is 6.

86. There are 3 + 8 + 5 + 8 + 2 = 26 data items. a.

4 ⋅3 + 5⋅8 + 6 ⋅5 + 7 ⋅8 + 8⋅ 2 26 154 = 26 ≈ 5.9

mean =

68. The graph decreases between 2011 and 2013, so the average number of goals per match decreased from 2011 to 2013. 70. The dots for 2009, 2011, 2017, 2019, and 2021 are above the 8-level, so the average number of goals per match was greater than 8 in 2009, 2011, 2017, 2019, and 2021.

72. The point on the graph corresponding to 2029 is at about 450, so the amount spent on organic food in 2029 is predicted to be $450 billion. 74. The dots for 2029 and 2030 are above the 400-level, so the amount spent on organic food is predicted to be more than $400 billion in 2029 and 2030. 76. range = largest data value − smallest data value = 40 − 11 = 29

The data items with the greatest frequency are the mode, so the mode is 5 and 7.

88. There are 1 + 4 + 5 + 3 + 2 = 15 data items. a.

78. range = largest data value − smallest data value = 276 − 129 = 147

80. range = largest data value − smallest data value = 10 − 7 =3

82. range = largest data value − smallest data value = 90 − 40 = 50

n + 1 26 + 1 = = 13.5 2 2 The mean of the data items in the 13th and 6+6 = 6, so the median is 14th positions is 2 6.

10 ⋅1 + 20 ⋅ 4 + 30 ⋅ 5 + 40 ⋅ 3 + 50 ⋅ 2 15 460 = 15 ≈ 30.7

mean =

n + 1 15 + 1 = =8 2 2 The data item in the eighth position is 30, so the median is 30.

The data item with the greatest frequency is the mode, so the mode is 30.

287

Chapter 8: Graphing and Introduction to Statistics and Probability 90. There are 6 + 6 + 4 + 3 + 3 = 22 data items. a.

5 ⋅ 6 + 10 ⋅ 6 + 15 ⋅ 4 + 20 ⋅ 3 + 25 ⋅ 3 22 285 = 22 ≈ 13.0

mean =

n + 1 22 + 1 = = 11.5 2 2 The mean of the data items in the eleventh 10 + 10 = 10, so the and twelfth positions is 2 median is 10.

The data items with the greatest frequency are the mode, so the mode is 5 and 10.

92. 45% of 120 is 0.45 ⋅ 120 = 54.

ISM: Prealgebra

Section 8.2 Practice Exercises 1. Eight of the 100 adults prefer golf. The ratio is adults preferring golf 8 2 = = total adults 100 25 2. Add the percents corresponding to South America & Caribbean and India. 13% + 2% = 15% 15% of visitors to the U.S. come from South America & Caribbean and India. 3. amount = percent ⋅ base = 0.47 ⋅ 91, 000, 000 = 42, 770, 000 Thus, 42,770,000 tourists might come from Mexico in 2027. 4.

Degrees in Sector

Year

Percent

Freshmen

30%

30% of 360° = 0.30(360°) = 108°

Sophomores

27%

27% of 360° = 0.27(360°) = 97.2°

100. The point on the low temperature graph corresponding to Thursday is 74, so the low temperature reading on Thursday was 74°F.

Juniors

25%

25% of 360° = 0.25(360°) = 90°

102. The highest point on the graph of high temperature corresponds to Tuesday. The high temperature on Tuesday was 86°F.

Seniors

18%

18% of 360° = 0.18(360°) = 64.8°

94. 95% of 50 is 0.95 ⋅ 50 = 47.5. 96.

2 2 2 ⋅ 5 ⋅ 20 = ⋅100% = % = 40% 5 5 5

9 9 9 ⋅10 ⋅10 98. = ⋅100% = % = 90% 10 10 10

104. The difference between the graphs is the least for Friday. The high temperature was 76°F and the low temperature was 70°F, so the difference is 76 − 70 = 6°F. 106. Kansas is represented by 7.5 wheat symbols, and each symbol represents 1 million acres, so there were approximately 7.5(1 million) = 7.5 million or 7,500,000 acres of wheat planted. This represents 20% of the wheat acreage in the U.S., 7.5 million so = 37.5 ≈ 38 million acres of 0.20 wheat were planted in the U.S.

Vocabulary, Readiness & Video Check 8.2 1. In a circle graph, each section (shaped like a piece of pie) shows a category and the relative size of the category. 2. A circle graph contains pie-shaped sections, each called a sector. 3. The number of degrees in a whole circle is 360.

288

ISM: Prealgebra

Chapter 8: Graphing and Introduction to Statistics and Probability

4. If a circle graph has percent labels, the percents should add up to 100. 5. 100% 6. 360° Exercise Set 8.2 2. The smallest sector corresponds to the category “own off-campus housing,” thus the least of these college students own off-campus housing. 4. 124 of the 700 total students live in off-campus rentals. 124 31 = 700 175 31 The ratio is . 175 6. 124 of the students live in off-campus rentals while 320 live in a parent or guardian’s home. 124 31 = 320 80 31 The ratio is . 80 8. The smallest sector corresponds to Australia. Thus, the smallest continent is Australia. 10. 16% + 12% = 28% 28% of the continental land area on Earth is accounted for by North and South America. 12. South America accounts for 12% of the continental land area on Earth. 12% of 57,000,000 = 0.12 ⋅ 57, 000, 000 = 6,840, 000 South America is 6,840,000 square miles. 14. Europe accounts for 7% of the continental land area on Earth. 7% of 57,000,000 = 0.07 ⋅ 57, 000, 000 = 3,990, 000 Europe is 3,990,000 square miles. 16. The second-largest sector corresponds to dairy and eggs, so the second-greatest amount of money spent on organic food is spent on dairy and eggs. 18. Breads and grains account for 9.4% of the money spent on organic food. 0.094 ⋅ 400 = 37.6 It is predicted that $37.6 billion will be spent on organic breads and grains in 2028. 20. Add the percent for nonfiction (25%) to the percent for reference (17%). 25% + 17% = 42% Thus, 42% of books are nonfiction or reference. 22. The third-largest sector corresponds to children’s fiction, so the third-largest category of books is children’s fiction. 24. Reference accounts for 17% of the books. 17% of 125,600 = 0.17 ⋅125, 600 = 21,352 The library has 21,352 reference books.

289

Chapter 8: Graphing and Introduction to Statistics and Probability 26. Adult’s fiction accounts for 33% of the books. 33% of 125,600 = 0.33 ⋅125, 600 = 41, 448 The library has 41,448 adult fiction books. 28. Nonfiction or other accounts for 25% + 3% = 28% of the books. 28% of 125,600 = 0.28 ⋅ 125,600 = 35,168 The library has 35,168 nonfiction or other books. 30.

290

Color Distribution of M&M’s Milk Chocolate Color

Percent

Degrees in Sector

Blue

24%

24% of 360° = 0.24(360°) ≈ 86°

Orange

20%

20% of 360° = 0.20(360°) = 72°

Green

16%

16% of 360° = 0.16(360°) ≈ 58°

Yellow

14%

14% of 360° = 0.14(360) ≈ 50°

Red

13%

13% of 360° = 0.13(360°) ≈ 47°

Brown

13%

13% of 360° = 0.13(360°) ≈ 47°

ISM: Prealgebra

32.

Chapter 8: Graphing and Introduction to Statistics and Probability

Distribution of Department of the Interior Public Lands by Management Bureau of Management

Percent

Degrees in Sector

Bureau of Indian Affairs

9% of 360° = 0.09(360°) ≈ 32°

National Park Service

11%

11% of 360° = 0.11(360°) ≈ 40°

Fish and Wildlife

20%

20% of 360° = 0.20(360°) = 72°

U.S. Forest Service

31%

31% of 360° = 0.31(360°) ≈ 112°

Bureau of Land Management

29%

29% of 360° = 0.29(360°) ≈ 104°

34. 25 = 5 ⋅ 5 = 52 36. 16 = 4 ⋅ 4 = 2 ⋅ 2 ⋅ 2 ⋅ 2 = 24 38. 105 = 3 ⋅ 35 = 3 ⋅ 5 ⋅ 7 40. answers may vary 42. Atlantic Ocean: 26% ⋅ 264, 489,800 = 0.26 ⋅ 264, 489,800 = 68, 767, 348 square kilometers 44. Arctic Ocean: 4% ⋅ 264, 489,800 = 0.04 ⋅ 264, 489,800 = 10, 579,592 square kilometers 46. 9% of 2800 = 0.09 ⋅ 2800 = 252 252 members of the community would be expected to buy online daily. 48. 56% of 2800 = 0.56 ⋅ 2800 = 1568 1568 members of the community would be expected to buy online monthly or less than monthly.

291

Chapter 8: Graphing and Introduction to Statistics and Probability

50.

percent who never buy online percent who buy monthly or less than monthly 2% = 56% 2 = 56 1 = 28

y = −5 x 0 = −5 x 0 −5 x = −5 −5 0=x The ordered pair solution is (0, 0).

y = −5x y = −5(−3) y = 15 The ordered pair solution is (−3, 15).

6. a.

y = 5x + 2 y = 5(0) + 2 y=0+2 y=2 The ordered pair solution is (0, 2).

52. true; answers may vary Section 8.3 Practice Exercises 1.

2. Point A has coordinates (5, 0). Point B has coordinates (−5, 4) Point C has coordinates (−3, −4). Point D has coordinates (0, −1). Point E has coordinates (2, 1). 3.

ISM: Prealgebra

y = 5x + 2 −3 = 5 x + 2 −3 − 2 = 5 x + 2 − 2 −5 = 5 x −5 5 x = 5 5 −1 = x The ordered pair solution is (−1, −3).

Vocabulary, Readiness & Video Check 8.3 1. In the ordered pair (−1, 2), the x-value is −1 and the y-value is 2.

x + 3 y = −12 0 + 3(−4) 0 − 12 −12 = −12 True

2. In the rectangular coordinate system, the point of intersection of the x-axis and the y-axis is called the origin.

Yes, (0, −4) is a solution of x + 3y = −12.

3. The axes divide the plane into four regions, called quadrants.

4. Every ordered pair of numbers, such as 1   , − 3  , corresponds to how many points in 2  the plane? one 5. The process of locating a point on the rectangular coordinate system is called plotting the point. 6. The origin corresponds to the ordered pair (0, 0). 5. a.

292

y = −5x y = −5(5) y = −25 The ordered pair solution is (5, −25).

7. A plane is a flat surface that extends indefinitely in all directions. 8. Every point in the rectangular coordinate system corresponds to an ordered pair of numbers.

ISM: Prealgebra

Chapter 8: Graphing and Introduction to Statistics and Probability

9. In a rectangular coordinate system, the x-axis and the y-axis number lines intersect at a right angle and at the zero coordinate of each line.

12. y = − x 1 = −1 False No, (1, 1) is not a solution of y = −x.

10. When replacing values for x and y in an equation, we need to be sure that we use the first number in the ordered pair for x and the second number for y.

14.

11. −7

x = −9 y −9 0 − 9 ⋅ 1 −9 = −9 True

Yes, (−9, 1) is a solution of x = −9y. 16.

−5 y + 4 x = −1 −5 ⋅1 + 4 ⋅1 0 − 1 −5 + 4 0 − 1 −1 = −1 True Yes, (1, 1) is a solution of −5y + 4x = −1.

18.

x − 7 y = −15 0 − 7(2) 0 − 15 −14 = −15 False

Exercise Set 8.3 2.

No, (0, 2) is not a solution of x − 7y = −15. 20. 4.

22. 6. Point A has coordinates (2, 0). Point B has coordinates (3, −2). Point C has coordinates (−5, −4).  1  Point D has coordinates  −2 , 0  .  2  Point E has coordinates (0, −3). Point F has coordinates (−2, 3). Point G has coordinates (4, 1). 8. x = 14 y 1 0 14(14) 1 = 196 False No, (1, 14) is not a solution of x = 14y. 10.

24.

x+ y =8 −1 + 9 0 8 8 = 8 True

293

Chapter 8: Graphing and Introduction to Statistics and Probability 26. x = −7 y x = −7(2) x = −14 The solution is (−14, 2). x = −7 y 14 = −7 y 14 −7 y = −7 −7 −2 = y The solution is (14, −2). x = −7y x = −7(−1) x=7 The solution is (7, −1). 28. x − y = 8 0− y =8 −y = 8 y = −8 The solution is (0, −8). x− y =8 x−0 =8 x=8 The solution is (8, 0). x− y =8 5− y = 8 5−5− y = 8−5 −y = 3 y = −3

The solution is (5, −3). 30. x + y = −3 x + 0 = −3 x = −3 The solution is (−3, 0). x + y = −3 0 + y = −3 y = −3 The solution is (0, −3). x + y = −3 4 + y = −3 4 − 4 + y = −3 − 4 y = −7 The solution is (4, −7). 32. y = x + 5 y=1+5 y=6 The solution is (1, 6).

y = x+5 7 = x+5 7−5 = x+5−5 2=x The solution is (2, 7). y=x+5 y=3+5 y=8 The solution is (3, 8).

34. x = −12y + 5 x = −12(−1) + 5 x = 12 + 5 x = 17 The solution is (17, −1). x = −12y + 5 x = −12(1) + 5 x = −12 + 5 x = −7 The solution is (−7, 1). x = −12 y + 5 41 = −12 y + 5 41 − 5 = −12 y + 5 − 5 36 = −12 y 36 −12 y = −12 −12 −3 = y

The solution is (41, −3). 36. y = − x 0 = −x 0=x The solution is (0, 0). y = −x y = −2 The solution is (2, −2). y = −x 2 = −x −2 = x The solution is (−2, 2). 38.

−5 x + y = 1 −5(0) + y = 1 0+ y =1 y =1 The solution is (0, 1). −5 x + y = 1 −5(−1) + y = 1 5+ y =1 −5 + 5 + y = −5 + 1 y = −4

The solution is (−1, −4). 294

ISM: Prealgebra

Chapter 8: Graphing and Introduction to Statistics and Probability

−5 x + y = 1 −5(2) + y = 1 −10 + y = 1 10 − 10 + y = 10 + 1 y = 11 The solution is (2, 11).

40.

Mid-Chapter Review 1. Software developers has 5 figures and each figure represents 75,000 workers. Thus, the increase in the number of software developers is approximately 5 ⋅ 75,000 = 375,000. 2. Registered nurses has 2.5 figures and each figure represents 75,000 workers. Thus, the increase in the number of registered nurses is approximately 2.5 ⋅ 75,000 = 187,500.

5.00 2.54 + 8.70 16.24

42. 0.8 0.56 becomes 8

3. Home health and personal care aide has the greatest number of figures, so the greatest increase is expected for home health and personal care aides.

0.7 5.6 −56 0

4. Food service has the second greatest number of figures, so the second greatest increase is expected for food service.

0.56 ÷ 0.8 = 0.7 44. 44.72 ÷ 100 = 0.4472 46. To plot (−a, −b), start at the origin and move a units to the left and b units down. The point will be in quadrant III. Thus, (−a, −b) lies in quadrant IV is a false statement. 48. To plot (a, 0), start at the origin and move a units to the right and 0 units up. The point will lie on the x-axis. Thus, (a, 0) lies on the x-axis is a true statement. 50. To plot (−a, 0), start at the origin and move a units to the left and 0 units up. The point will lie on the x-axis. Thus, (−a, 0) lies on the y-axis is a false statement. 52. To plot (a, −b), start at the origin and move a units to the right and b units down. The point will lie in quadrant IV. Thus, (a, −b) lies in quadrant IV is a true statement. 54. below; answer may vary 56. length = 4 − (−2) = 4 + 2 = 6 units width = 3 − (−1) = 3 + 1 = 4 units 58. Area = l ⋅ w Let l = 6 and w = 4. Area = 6 ⋅ 4 = 24 The area is 24 square units.

5. The tallest bar corresponds to Oroville, CA. Thus, the U.S. dam with the greatest height is the Oroville Dam, which is approximately 770 feet. 6. The bar whose height is between 625 and 650 feet corresponds to New Bullards Bar, CA. Thus, the U.S. dam with a height between 625 and 650 feet is the New Bullards Bar Dam, which is approximately 645 feet. 7. From the graph, the Hoover Dam is approximately 725 feet and the Glen Canyon Dam is approximately 710 feet. The Hoover Dam is 725 − 710 = 15 feet taller than the Glen Canyon Dam. 8. There are 4 bars that are taller than the 700-feet level, so there are 4 dams with heights over 700 feet. 9. The highest points on the graph correspond to Thursday and Saturday. Thus, the highest temperature occurs on Thursday and Saturday and is 100°F. 10. The lowest point on the graph corresponds to Monday. Thus, the lowest temperature occurs on Monday, and is 82°F. 11. The points on the graph that are lower than the 90 level correspond to Sunday, Monday, and Tuesday. Thus, the temperature was less than 90°F on Sunday, Monday, and Tuesday.

295

Chapter 8: Graphing and Introduction to Statistics and Probability 12. The points on the graph that are higher than the 90 level correspond to Wednesday, Thursday, Friday, and Saturday. Thus, the temperature was greater than 90°F on Wednesday, Thursday, Friday, and Saturday.

ISM: Prealgebra

26.

13. The highest points on the graph correspond to 2026 and 2027, so those years have the highest predicted percent of food purchased being organic. 14. The lowest point on the graph corresponds to 2020, so 2020 has the lowest percent of food purchased being organic.

27.

15. The point for 2022 is below the point for 2021, so there was a decrease in the percent of organic food purchased from 2021 to 2022. 16. The steepest upward segment on the graph is between 2020 and 2021, so the greatest increase in the percent of organic food purchased is from 2020 to 2021. 17. The sector corresponding to whole milk is 35%. 35% of 200 = 0.35 ⋅ 200 = 70 Thus, 70 quart containers of whole milk are sold. 18. The sector corresponding to skim milk is 26%. 26% of 200 = 0.26 ⋅ 200 = 52 Thus, 52 quart containers of skim milk are sold. 19. The sector corresponding to buttermilk is 1%. 1% of 200 = 0.01 ⋅ 200 = 2 Thus, 2 quart containers of buttermilk are sold.

28. x = 3 y 1 0 3⋅3 1 = 9 False No, (1, 3) is not a solution of x = 3y. 29.

x + y = −6 −2 + (−4) 0 − 6 −6 = −6 True

Yes, (−2, −4) is a solution of x + y = −6.

Class Interval (scores)

Tally

Class Frequency (number of quizzes)

21.

50−59

22.

60−69

30. x − y = 6 0− y = 6 −y = 6 y = −6 (0, −6) is a solution. x− y =6 x−0 = 6 x=6 (6, 0) is a solution. x− y =6 2− y = 6 2−2− y = 6−2 −y = 4 y = −4

23.

70−79

|||

(2, −4) is a solution.

24.

80−89

|||| |

25.

90−99

||||

20. The sector corresponding to Flavored reduced fat and skim milk is 3%. 3% of 200 = 0.03 ⋅ 200 = 6 Thus, 6 quart containers of flavored reduced fat and skim milk are sold.

296

ISM: Prealgebra

Chapter 8: Graphing and Introduction to Statistics and Probability

Section 8.4 Practice Exercises 1.

2. y − x = 4 Find any 3 ordered pair solutions. Let x = 0. y−x=4 y−0 = 4 y=4 (0, 4) Let x = 2. y−x=4 y−2 = 4 y−2+2 = 4+2 y=6 (2, 6) Let y = 0. y−x=4 0− x = 4 x = −4 (−4, 0) Plot (0, 4), (2, 6), and (−4, 0). Then draw a line through them.

(0, 2) Let x = 1. y = −3x + 2 y = −3(1) + 2 y = −3 + 2 y = −1 (1, −1) Let x = −1. y = −3x + 2 y = −3(−1) + 2 y=3+2 y=5 (−1, 5) Plot (0, 2), (1, −1), and (−1, 5). Then draw a line through them.

4. y = −4 No matter what x-value we choose, y is always −4.

−2

−4

3. y = −3x + 2 Find any 3 ordered pair solutions. Let x = 0. y = −3x + 2 y = −3 ⋅ 0 + 2 y=2

297

Chapter 8: Graphing and Introduction to Statistics and Probability 5. x = 5 No matter what y-value we choose, x is always 5.

−2

ISM: Prealgebra

x+ y = 2 x+0 = 2 x=2 (2, 0) Let x = −1. x+ y = 2 −1 + y = 2 1 −1+ y = 1+ 2 y=3 (−1, 3) Plot (0, 2), (2, 0), and (−1, 3). Then draw a line through them.

Vocabulary, Readiness & Video Check 8.4 1. A linear equation in two variables can be written in the form ax + by = c. 2. The graph of the equation x = 5 is a vertical line. 3. The graph of the equation y = −2 is a horizontal line. 4. The graph of 3x − 2y = 6 is a line that is not vertical or horizontal. 5. There is an infinite number of ordered pair solutions to a linear equation in two variables. 6. It’s a check point to make sure all three points lie on the same line. 7. Because there’s no solving to do, only evaluating, since the equation is solved for y. Exercise Set 8.4 2. x + y = 2 Find any 3 ordered pair solutions. Let x = 0. x+ y = 2 0+ y = 2 y=2 (0, 2) Let y = 0.

298

4. y − x = 6 Find any 3 ordered pair solutions. Let x = 0. y−x=6 y−0 = 6 y=6 (0, 6) Let x = −2. y−x=6 y − (−2) = 6 y+2=6 y+2−2 = 6−2 y=4

(−2, 4) Let y = 0. y−x=6 0− x = 6 x = −6 (−6, 0) Plot (0, 6), (−2, 4), and (−6, 0). Then draw a line through them.

ISM: Prealgebra

Chapter 8: Graphing and Introduction to Statistics and Probability

y = 2x + 5 y = 2(−2) + 5 y = −4 + 5 y=1 (−2, 1) Plot (0, 5), (−1, 3), and (−2, 1). Then draw a line through them.

6. x = 2y Find any 3 ordered pair solutions. Let y = 0. x = 2y x=2⋅0 x=0 (0, 0) Let y = 1. x = 2y x = 2(1) x=2 (2, 1) Let y = 2. x = 2y x=2⋅2 x=4 (4, 2) Plot (0, 0), (2, 1), and (4, 2). Then draw a line through them.

8. y = 2x + 5 Find any 3 ordered pair solutions. Let x = 0. y = 2x + 5 y=2⋅0+5 y=5 (0, 5) Let x = −1. y = 2x + 5 y = 2 ⋅ −1 + 5 y = −2 + 5 y=3 (−1, 3) Let x = −2.

10. y = 1 No matter what x-value we choose, y is always 1.

−1

12. x = −7 No matter what y-value we choose, x is always −7.

−7

−2

−7

299

Chapter 8: Graphing and Introduction to Statistics and Probability

14. y = 0 No matter what x-value we choose, y is always 0.

18. x = 1 No matter what y-value we choose, x is always 1.

−3

−4

16. x = y Find any 3 ordered pair solutions. Let x = 0. x= y 0= y (0, 0) Let x = −2. x= y −2 = y

(−2, −2) Let x = 2. x= y 2= y (2, 2) Plot (0, 0), (−2, −2), and (2, 2). Then draw a line through them.

300

ISM: Prealgebra

20. 3x − y = 3 Find any 3 ordered pair solutions. Let x = 0. 3x − y = 3 3⋅ 0 − y = 3 −y = 3 y = −3 (0, −3) Let x = 3. 3x − y = 3 3⋅3 − y = 3 9− y = 3 9−9− y = 3−9 − y = −6 y=6 (3, 6) Let y = 0.

ISM: Prealgebra

Chapter 8: Graphing and Introduction to Statistics and Probability

3x − y = 3 3x − 0 = 3 3x = 3 3x 3 = 3 3 x =1 (1, 0) Plot (0, −3), (3, 6), and (1, 0). Then draw a line through them.

22. y = 7 No matter what x-value we choose, y is always 7.

−5

y = x −1 0 = x −1 0 +1 = x −1+1 1= x (1, 0) Plot (0, −1), (4, 3), and (1, 0). Then draw a line through them.

26. x = 5 − y Find any 3 ordered pair solutions. Let y = 0. x=5−y x=5−0 x=5 (5, 0) Let y = 2. x=5−y x=5−2 x=3 (3, 2) Let y = 4. x=5−y x=5−4 x=1 (1, 4) Plot (5, 0), (3, 2), and (1, 4). Then draw a line through them.

24. y = x − 1 Find any 3 ordered pair solutions. Let x = 0. y=x−1 y=0−1 y = −1 (0, −1) Let x = 4. y=x−1 y=4−1 y=3 (4, 3) Let y = 0.

301

Chapter 8: Graphing and Introduction to Statistics and Probability

ISM: Prealgebra

28. y + 4 = 0 or y = −4 No matter what x-value we choose, y is always −4.

−2

−4

1 x 4 Find any 3 ordered pair solutions. Let x = −4. 1 y= x 4 1 y = ⋅ ( −4) 4 y = −1 (−4, −1) Let x = 0. 1 y= x 4 1 y = ⋅0 4 y=0 (0, 0) Let x = 4. 1 y= x 4 1 y = ⋅4 4 y =1 (4, 1) Plot (−4, −1), (0, 0), and (4, 1). Then draw a line through them.

30. y =

302

1 32. y = − x 3 Find any 3 ordered pair solutions. Let x = −3 1 y=− x 3 1 y = − ( −3) 3 y =1 (−3, 1) Let x = 0. 1 y=− x 3 1 y = − ⋅0 3 y=0 (0, 0) Let x = 3. 1 y=− x 3 1 y = − (3) 3 y = −1 (3, −1) Plot (−3, 1), (0, 0), and (3, −1). Then draw a line through them.

ISM: Prealgebra

Chapter 8: Graphing and Introduction to Statistics and Probability

34. y = −2x + 3 Find any 3 ordered pair solutions. Let x = 0. y = −2x + 3 y = −2 ⋅ 0 + 3 y=0+3 y=3 (0, 3) Let x = −1. y = −2x + 3 y = −2(−1) + 3 y=2+3 y=5 (−1, 5) Let x = 2. y = −2x + 3 y = −2 ⋅ 2 + 3 y = −4 + 3 y = −1 (2, −1) Plot (0, 3), (−1, 5), and (2, −1). Then draw a line through them.

Let x = 4. 5 x − 2 y = 10 5 ⋅ 4 − 2 y = 10 20 − 2 y = 10 20 − 20 − 2 y = 10 − 20 −2 y = −10 −2 y −10 = −2 −2 y=5 (4, 5) Plot (0, −5), (2, 0), and (4, 5). Then draw a line through them.

38. y = 5.5 No matter what x-value we choose, y is always 5.5.

−3

5.5

36. 5x − 2y = 10 Find any 3 ordered pair solutions. Let x = 0. 5 x − 2 y = 10 5 ⋅ 0 − 2 y = 10 −2 y = 10 −2 y 10 = −2 −2 y = −5 (0, −5) Let y = 0. 5 x − 2 y = 10 5 x − 2 ⋅ 0 = 10 5 x = 10 5 x 10 = 5 5 x=2 (2, 0)

303

Chapter 8: Graphing and Introduction to Statistics and Probability

(0, 8) Let y = 0. 4 x + 2 y = 16 4 x + 2 ⋅ 0 = 16 4 x = 16 4 x 16 = 4 4 x=4 (4, 0) Let x = 2. 4 x + 2 y = 16 4 ⋅ 2 + 2 y = 16 8 + 2 y = 16 8 − 8 + 2 y = 16 − 8 2y = 8 2y 8 = 2 2 y=4 (2, 4) Plot (0, 8), (4, 0), and (2, 4). Then draw a line through them.

1 2 No matter what y-value we choose, x is always 1 − . 2

42. x = −

304

− 12

−2

− 12

ISM: Prealgebra

1 y−2 3 Find any 3 ordered pair solutions. Let y = 3. 1 x = y−2 3 1 x = (3) − 2 3 x = 1− 2 x = −1 (−1, 3) Let y = 0. 1 x = y−2 3 1 x = (0) − 2 3 x = −2 (−2, 0) Let y = 6. 1 x = y−2 3 1 x = (6) − 2 3 x = 2−2 x=0 (0, 6) Plot (−1, 3), (−2, 0), and (0, 6). Then draw a line through them.

44. x =

ISM: Prealgebra

46.

3 19 3 5 19 − = ⋅ − 4 20 4 5 20 15 19 = − 20 20 15 − 19 = 20 −4 = 20 1⋅ 4 =− 5⋅4 1 =− 5

3 3 3 48. 0.75 + = + 10 4 10 3 5 3 2 = ⋅ + ⋅ 4 5 10 2 15 6 = + 20 20 15 + 6 = 20 21 = 20

50.

Chapter 8: Graphing and Introduction to Statistics and Probability

Let x = 6. 15 x − 18 y = 270 15 ⋅ 6 − 18 y = 270 90 − 18 y = 270 −18 y = 180 −18 y 180 = −18 −18 y = −10 x

−15

−5

−10

2 x 2⋅ x 2x =− −  = − 11  11  11 ⋅11 121

52. 15x − 18y = 270 Let x = 0. 15 x − 18 y = 270 15 ⋅ 0 − 18 y = 270 −18 y = 270 −18 y 270 = −18 −18 y = −15 Let y = 0. 15 x − 18 y = 270 15 x − 18 ⋅ 0 = 270 15 x = 270 15 x 270 = 15 15 x = 18 Let x = 12. 15 x − 18 y = 270 15 ⋅12 − 18 y = 270 180 − 18 y = 270 −18 y = 90 −18 y 90 = −18 −18 y = −5

54. answers may vary 56. Since the line corresponding to BEVs is increasing from left to right, the number of BEVs sold is increasing overall. 58. The BEVs line is at about 7700 above 2022. So, approximately 7700 thousand or 7,700,000 BEVs were sold in 2022. 60. answers may vary 62. The peak for the line corresponding to light trucks is at the year 2019. Therefore, light truck sales were at a maximum in 2019. 64. The car line is at about 3.5 above the year 2019. So, approximately 3.5 million or 3,500,000 passenger cars were sold in 2019. 66. The line for passenger cars is above the line for light trucks for the year 2009. So, the number of passenger cars sold was greater than the number of light trucks sold for the year 2009.

305

Chapter 8: Graphing and Introduction to Statistics and Probability

6. A probability of 1 means that an event is certain to occur.

Section 8.5 Practice Exercises H T H T H T H T

H H T

H T T

ISM: Prealgebra

7. The number of outcomes equals the ending number of branches drawn. 8. 0; Having the die land on a 7 is impossible, and the probability of something impossible is 0. Exercise Set 8.5

8 outcomes

1 2 3 12 outcomes 4 5 6

1 2 3 4 5 6

2. 2

3. The possibilities are: H, H, H, H, H, T, H, T, H, H, T, T, T, H, H, T, H, T, T, T, H, T, T, T T, H, T is one of the 8 possible outcomes, so the 1 probability is . 8

1 2 3 4

4 outcomes

Red Blue Yellow Red Blue Yellow Red Blue Yellow

Red

Blue Yellow

4. A 2 or a 5 are two of the six possible outcomes. 2 1 The probability is = . 6 3 5. A blue is 2 out of the 4 possible marbles. The 2 1 probability is = . 4 2

a e i o u 10 outcomes a e i o u

Red Blue Yellow Red Blue Yellow 12 outcomes Red Blue Yellow Red Blue Yellow

1 2

Vocabulary, Readiness & Video Check 8.5 4

1. A possible result of an experiment is called an outcome. 2. A tree diagram shows each outcome of an experiment as a separate branch. 3. The probability of an event is a measure of the likelihood of it occurring. 4. Probability is calculated by number of ways that an event can occur divided by number of possible outcomes. 5. A probability of 0 means that an event won’t occur.

306

Red

10.

Blue Yellow

9 outcomes

H T H T

6 outcomes

H T

12. A 9 is zero of the six possible outcomes. The 0 probability is = 0. 6

ISM: Prealgebra

Chapter 8: Graphing and Introduction to Statistics and Probability

14. A 2 or a 3 are two of the six possible outcomes. 2 1 The probability is = . 6 3

42. One of the 52 cards is the 10 of spades. The 1 probability is . 52

16. Three of the six possible outcomes are odd. The 3 1 probability is = . 6 2

44. Four of the 52 cards are 10s. The probability is 4 1 = . 52 13

18. Five of the six possible outcomes are less than 6. 5 The probability is . 6

46. Thirteen of the 52 cards are clubs. The 13 1 probability is = . 52 4

20. A 3 is one of three possible outcomes. The 1 probability is . 3

48. Eight of the cards are queens or aces. The 8 2 probability is = . 52 13

22. Not 3 is a 1 or a 2, which are two of three 2 possible outcomes. The probability is . 3 24. An even number is a 2, which is one of three 1 possible outcomes. The probability is . 3 26. One of the seven marbles is blue. The probability 1 is . 7 28. Three of the seven marbles are green. The 3 probability is . 7 30. Three of the seven marbles are either blue or 3 yellow. The probability is . 7 32. The blood pressure was lower for 152 of the 152 19 200 people. The probability is = . 200 25 34.

38 152 10 200 + + = = 1; answers may vary 200 200 200 200

36.

7 2 7 2 2 7 4 7−4 3 − = − ⋅ = − = = 10 5 10 5 2 10 10 10 10

38.

7 2 7 5 7 ⋅5 7⋅5 7 3 ÷ = ⋅ = = = or 1 10 5 10 2 10 ⋅ 2 5 ⋅ 2 ⋅ 2 4 4

40.

3 3 10 3 ⋅10 3 ⋅ 5 ⋅ 2 6 ⋅10 = ⋅ = = = =6 5 5 1 5 ⋅1 5 ⋅1 1

307

Chapter 8: Graphing and Introduction to Statistics and Probability

Tree diagram for 50.−52. Die 1 1

54. answers may vary Die 2 Sum 1 2 2

Chapter 8 Vocabulary Check 1. A bar graph presents data using vertical or horizontal bars. 2. The possible results of an experiment are the outcomes. 3. A pictograph is a graph in which pictures or symbols are used to visually present data. 4. A line graph displays information with a line that connects data points. 5. In the ordered pair (a, b), the x-value is a and the y-value is b. 6. A tree diagram is one way to picture and count outcomes. 7. An experiment is an activity being considered, such as tossing a coin or rolling a die. 8. In a circle graph, each section (shaped like a piece of pie) shows a category and the relative size of the category. 9. The probability of an event is number of ways that event can occur . number of possible outcomes 10. A histogram is a special bar graph in which the width of each bar represents a class interval and the height of each bar represents the class frequency. 11. The point of intersection of the x-axis and the y-axis in the rectangular coordinate system is called the origin. 12. The axes divide the plane into 4 regions, called quadrants. 13. The process of locating a point on the rectangular coordinate system is called plotting the point.

36 outcomes

50. Three of the 36 sums are 10. The probability is 3 1 = . 36 12 52. One of the 36 sums is 2. The probability is

308

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14. A linear equation in two variables can be written in the form ax + by = c.

1 . 36

ISM: Prealgebra

Chapter 8: Graphing and Introduction to Statistics and Probability

Chapter 8 Review

13. The point on the graph corresponding to 2016 is about 970. There were approximately 970 medals awarded at the Summer Olympics in 2016.

1. Midwest has 2.5 houses, and each house represents 60,000 homes, so there were 2.5(60,000) = 150,000 new homes constructed in the Midwest. 2. South has 10 houses, and each house represents 60,000 homes, so there were 10(60,000) = 600,000 new homes constructed in the South. 3. South has the greatest number of houses, so the most new homes constructed were in the South. 4. Northeast has the least number of houses, so the fewest new homes constructed were in the Northeast. 5. Each house represents 60,000 homes, so look for 300, 000 = 5 or more houses. the regions with 60, 000 The South had 300,000 or more new homes constructed. 6. Each house represents 60,000 homes, so look for 300, 000 = 5 houses. the regions with fewer than 60, 000 The Northeast, Midwest, and West had fewer than 300,000 new homes constructed. 7. The height of the bar representing 2010 is 30. Thus, approximately 30% of persons completed four or more years of college in 2010. 8. The tallest bar corresponds to 2020. Thus, the greatest percent of persons completing four or more years of college was in 2020.

14. The point on the graph corresponding to 2020 is about 1080. There were approximately 1080 medals awarded at the Summer Olympics in 2020. 15. The points on the graph corresponding to 2008 and 2004 are 960 and 930, respectively. There were 960 − 930 = 30 more medals awarded in 2008 than in 2004. 16. The points on the graph corresponding to 2020 and 2016 are 1080 and 970, respectively. There were 1080 − 970 = 110 more medals awarded in 2020 than in 2016. 17. The height of the bar corresponding to 41−45 is 1. Thus, 1 employee works 41−45 hours per week. 18. The height of the bar corresponding to 21−25 is 4. Thus, 4 employees work 21−25 hours per week. 19. Add the heights of the bars corresponding to 16−20, 21−25, and 26−30. Thus, 6 + 4 + 8 = 18 employees work 30 hours or less per week. 20. Add the heights of the bars corresponding to 36−40 and 41−45. Thus, 8 + 1 = 9 employees work 36 or more hours per week. Class Interval (Temperatures)

Tally

Class Frequency (Number of Months)

9. The bars whose height is at a level of 20 or more are 1990, 2000, 2010, and 2020. Thus, 20% or more persons completed four or more years of college in 1990, 2000, 2010, and 2020.

21.

80°−89°

||||

10. answers may vary

22.

90°−99°

|||

11. The point on the graph corresponding to 2012 is about 960. There were approximately 960 medals awarded at the Summer Olympics in 2012.

23.

100°−109°

||||

12. The point on the graph corresponding to 2000 is about 930. There were approximately 930 medals awarded at the Summer Olympics in 2000.

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Chapter 8: Graphing and Introduction to Statistics and Probability

32. There are 5 + 7 + 8 + 10 + 15 = 45 data items.

24.

b. 25. range = largest data value − smallest data value = 6−3 =3 26. range = largest data value − smallest data value = 8 −1 =7

28. range = largest data value − smallest data value = 99 − 80 = 19

29. range = largest data value − smallest data value = 9−4 =5 30. range = largest data value − smallest data value = 2.86 − 0.02 = 2.84 31. There are 2 + 10 + 5 + 11 + 3 = 31 data items. a.

60 ⋅ 2 + 61 ⋅10 + 62 ⋅ 5 + 63 ⋅11 + 64 ⋅ 3 31 1925 = 31 ≈ 62.1

n + 1 45 + 1 = = 23 2 2 The data item in the 23rd place is 63, so the median is 63.

The data item with the greatest frequency is the mode, so the mode is 64.

mean 11 ⋅ 5 + 12 ⋅ 5 + 13 ⋅ 3 + 14 ⋅1 + 16 ⋅ 3 + 17 ⋅1 = 18 233 = 18 ≈ 12.9 n + 1 18 + 1 = = 9.5 2 2 The mean of the data items in the ninth and 12 + 12 tenth positions is = 12, so the 2 median is 12.

The data items with the greatest frequency are the mode, so the mode is 11 and 12.

34. There are 3 + 2 + 1 + 6 + 4 + 6 = 22 data items. a.

mean =

n + 1 31 + 1 = = 16 2 2 The data item in the 16th position is 62, so the median is 62.

60 ⋅ 5 + 61 ⋅ 7 + 62 ⋅8 + 63⋅10 + 64 ⋅15 45 2813 = 45 ≈ 62.5

mean =

33. There are 5 + 5 + 3 + 1 + 3 + 1 = 18 data items.

27. range = largest data value − smallest data value = 95 − 60 = 35

The data item with the greatest frequency is the mode, so the mode is 63. c.

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mean 15 ⋅ 3 + 16 ⋅ 2 + 17 ⋅1 + 18 ⋅ 6 +19 ⋅ 4 + 20 ⋅ 6 = 22 398 = 22 ≈ 18.1 n + 1 22 + 1 = = 11.5 2 2 The mean of the data items in the eleventh 18 + 18 and twelfth positions is = 18, so the 2 median is 18.

The data items with the greatest frequency are the mode, so the mode is 18 and 20.

ISM: Prealgebra

Chapter 8: Graphing and Introduction to Statistics and Probability

35. There are 5 + 3 + 8 + 6 + 3 = 25 data items. a.

50 ⋅ 5 + 55 ⋅ 3 + 60 ⋅ 8 + 65 ⋅ 6 + 70 ⋅ 3 25 1495 = 25 = 59.8

mean =

n + 1 25 + 1 = = 13 2 2 The data item in the 13th position is 60, so the median is 60.

The data item with the greatest frequency is the mode, so the mode is 60.

36. There are 7 + 3 + 5 + 2 + 7 = 24 data items. a.

42.

10 ⋅ 7 + 12 ⋅ 3 + 14 ⋅ 5 + 16 ⋅ 2 + 18 ⋅ 7 24 334 = 24 ≈ 13.9

mean =

n + 1 24 + 1 = = 12.5 2 2 The mean of the data items in the 12th and 14 + 14 13th positions is = 14, so the 2 median is 14.

The data items with the greatest frequency are the mode, so the mode is 10 and 18.

37. The largest sector corresponds to the category “Mortgage payment,” thus the largest budget item is mortgage payment. 38. The smallest sector corresponds to the category “Utilities,” thus the smallest budget item is utilities. 39. Add the amounts for mortgage payment and utilities. Thus, $975 + $250 = $1225 is budgeted for the mortgage payment and utilities.

food $700 7 ⋅100 7 = = = total $4000 40 ⋅ 100 40 7 The ratio is . 40

43. The sector corresponding to China is 55%. 55% of 102 = 0.55 ⋅ 102 ≈ 56 56 of the completed tall buildings are in China. 44. The sector corresponding to United Arab Emirates is 12%. 12% of 102 = 0.12 ⋅ 102 ≈ 12 12 of the completed tall buildings are in United Arab Emirates. 45. The sector corresponding to South Korea is 4%. 4% of 102 = 0.04 ⋅ 102 ≈ 4 4 of the completed tall buildings are in South Korea. 46. The sector corresponding to the United States is 10%. 10% of 102 = 0.10 ⋅ 102 ≈ 10 10 of the completed tall buildings are in the United States. 47.

x = −6 y 0 = −6 y 0 −6 y = −6 −6 0= y The solution is (0, 0). x = −6y x = −6(−1) x=6 The solution is (6, −1). x = −6 y −6 = −6 y −6 −6 y = −6 −6 1= y The solution is (−6, 1).

40. Add the amounts for savings and contributions. Thus, $400 + $300 = $700 is budgeted for savings and contributions. 41.

mortgage payment $975 39 ⋅ 25 39 = = = total $4000 160 ⋅ 25 160 39 The ratio is . 160

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Chapter 8: Graphing and Introduction to Statistics and Probability

48. y = 3x − 2 y=3⋅0−2 y=0−2 y = −2 The solution is (0, −2). y = 3x − 2 y = 3(−1) − 2 y = −3 − 2 y = −5 The solution is (−1, −5). y = 3x − 2 y=3⋅2−2 y=6−2 y=4 The solution is (2, 4).

49.

312

x + y = −4 −1 + y = −4 1 −1 + y = 1 − 4 y = −3 The solution is (−1, −3). x + y = −4 x + 0 = −4 x = −4 The solution is (−4, 0). x + y = −4 −5 + y = −4 5−5+ y = 5−4 y =1 The solution is (−5, 1).

50.

x− y =3 4− y = 3 −4 + 4 − y = −4 + 3 − y = −1 y =1 The solution is (4, 1). x− y =3 0− y = 3 −y = 3 y = −3 The solution is (0, −3). x− y =3 x −3 = 3 x −3+3 = 3+3 x=6 The solution is (6, 3).

51. y = 3x y=3⋅1 y=3 The solution is (1, 3). y = 3x y = 3(−2) y = −6 The solution is (−2, −6). y = 3x 0 = 3x 0 3x = 3 3 0=x The solution is (0, 0).

ISM: Prealgebra

ISM: Prealgebra 52.

Chapter 8: Graphing and Introduction to Statistics and Probability

x = y+6 1= y+6 1− 6 = y + 6 − 6 −5 = y The solution is (1, −5). x = y+6 6 = y+6 6−6 = y+6−6 0= y The solution is (6, 0). x=y+6 x = −4 + 6 x=2 The solution is (2, −4).

53. x = −5 No matter what y-value we choose, x is always −5.

−5

−2

−5

−2

3 2

55. x + y = 11 Find any 3 ordered pair solutions. Let x = 0. x + y = 11 0 + y = 11 y = 11 (0, 11) Let y = 0. x + y = 11 x + 0 = 11 x = 11 (11, 0) Let x = 5. x + y = 11 5 + y = 11 5 + y = 11 5 − 5 + y = 11 − 5 y=6 (5, 6) Plot (0, 11), (11, 0), and (5, 6). Then draw a line through them.

3 2 No matter what x-value we choose, y is always 3 . 2

54. y =

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56. x − y = 11 Find any 3 ordered pair solutions. Let x = 0. x − y = 11 0 − y = 11 − y = 11 y = −11 (0, −11) Let y = 0. x − y = 11 x − 0 = 11 x = 11 (11, 0) Let x = 5. x − y = 11 5 − y = 11 5 − 5 − y = 11 − 5 −y = 6 y = −6 (5, −6) Plot (0, −11), (11, 0), and (5, −6). Then draw a line through them.

57. y = 4x − 2 Find any 3 ordered pair solutions. Let x = 0. y = 4x − 2 y=4⋅0−2 y=0−2 y = −2 (0, −2) Let y = 0. y = 4x − 2 0 = 4x − 2 0 + 2 = 4x − 2 + 2 2 = 4x 2 4x = 4 4 1 =x 2

314

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1   , 0 2  Let x = 1. y = 4x − 2 y=4⋅1−2 y=4−2 y=2 (1, 2) 1  Plot (0, −2),  , 0  , and (1, 2). Then draw a 2  line through them.

58. y = 5x Find any 3 ordered pair solutions. Let x = 0. y = 5x y=5⋅0 y=0 (0, 0) Let x = 1. y = 5x y = 5(1) y=5 (1, 5) Let x = −1. y = 5x y = 5(−1) y = −5 (−1, −5) Plot (0, 0), (1, 5), and (−1, −5). Then draw a line through them.

ISM: Prealgebra

Chapter 8: Graphing and Introduction to Statistics and Probability

59. x = −2y Find any 3 ordered pair solutions. Let y = 0. x = −2y x = −2 ⋅ 0 x=0 (0, 0) Let y = −1. x = −2y x = −2(−1) x=2 (2, −1) Let y = 1. x = −2y x = −2(1) x = −2 (−2, 1) Plot (0, 0), (2, −1), and (−2, 1). Then draw a line through them.

60. x + y = −1 Find any 3 ordered pair solutions. Let x = 0. x + y = −1 0 + y = −1 y = −1 (0, −1) Let y = 0. x + y = −1 x + 0 = −1 x = −1 (−1, 0) Let x = −2. x + y = −1 −2 + y = −1 −2 + 2 + y = −1 + 2 y =1 (−2, 1) Plot (0, −1), (−1, 0), and (−2, 1). Then draw a line through them.

61. 2x − 3y = 12 Find any 3 ordered pair solutions. Let x = 0. 2 x − 3 y = 12 2 ⋅ 0 − 3 y = 12 0 − 3 y = 12 −3 y = 12 −3 y 12 = −3 −3 y = −4 (0, −4) Let y = 0. 2 x − 3 y = 12 2 x − 3 ⋅ 0 = 12 2 x − 0 = 12 2 x = 12 2 x 12 = 2 2 x=6 (6, 0) Let x = 3. 2 x − 3 y = 12 2 ⋅ 3 − 3 y = 12 6 − 3 y = 12 6 − 6 − 3 y = 12 − 6 −3 y = 6 −3 y 6 = −3 −3 y = −2 (3, −2) Plot (0, −4), (6, 0), and (3, −2). Then draw a line through them.

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Chapter 8: Graphing and Introduction to Statistics and Probability

ISM: Prealgebra 1 2 3 4 5 10 outcomes 1 2 3 4 5

63. T

Red

64. 1 62. x = y 2 Find any 3 ordered pair solutions. Let y = 0. 1 x= y 2 1 x = ⋅0 2 x=0 (0, 0) Let y = 2. 1 x= y 2 1 x = ⋅2 2 x =1 (1, 2) Let y = 4. 1 x= y 2 1 x = ⋅4 2 x=2 (2, 4) Plot (0, 0), (1, 2), and (2, 4). Then draw a line through them.

Blue

T H

4 outcomes

1 2 3 4 5 1 2 3 4 5 1 2 3 25 outcomes 4 5 1 2 3 4 5 1 2 3 4 5

65.

Red

66. Blue

1 2

67.

3 4 5

316

Red Blue Red

4 outcomes

Blue

Red Blue Red Blue Red Blue Red Blue Red Blue

10 outcomes

ISM: Prealgebra

Chapter 8: Graphing and Introduction to Statistics and Probability

68. One of the six possible outcomes is 4. The 1 probability is . 6 69. One of the six possible outcomes is 3. The 1 probability is . 6 70. One of the five possible outcomes is 4. The 1 probability is . 5 71. One of the five possible outcomes is 3. The 1 probability is . 5 72. Three of the five possible outcomes are a 1, 3, or 3 5. The probability is . 5

79. y = 3 No matter what x-value we choose, y is always 3.

−2

73. Two of the five possible outcomes are a 2 or a 4. 2 The probability is . 5 74. Two of the eight marbles are blue. The 2 1 probability is = . 8 4 75. Three of the eight marbles are yellow. The 3 probability is . 8 76. Two of the eight marbles are red. The probability 2 1 is = . 8 4 77. One of the eight marbles is green. The 1 probability is . 8 78. x = −4 No matter what y-value we choose, x is always −4.

−4

−2

−4

80. x − 2y = 8 Find any 3 ordered pair solutions. Let x = 0. x − 2y = 8 0 − 2y = 8 −2 y = 8 −2 y 8 = −2 −2 y = −4

(0, −4) Let x = 4. x − 2y = 8 4 − 2y = 8 −4 + 4 − 2 y = −4 + 8 −2 y = 4 −2 y 4 = −2 −2 y = −2 (4, −2) Let y = 0.

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ISM: Prealgebra

x − 2y = 8 x − 2⋅0 = 8 x−0 =8 x=8 (8, 0) Plot (0, −4), (4, −2), and (8, 0). Then draw a line through them.

81. 2x + y = 6 Find any 3 ordered pair solutions. Let x = 0. 2x + y = 6 2⋅0 + y = 6 0+ y = 6 y=6 (0, 6) Let y = 0. 2x + y = 6 2x + 0 = 6 2x = 6 2x 6 = 2 2 x=3 (3, 0) Let x = −1. 2x + y = 6 2(−1) + y = 6 −2 + y = 6 2−2+ y = 2+6 y=8 (−1, 8) Plot (0, 6), (3, 0), and (−1, 8). Then draw a line through them.

318

82. x = y + 3 Find any 3 ordered pair solutions. Let x = 0. x = y+3 0 = y+3 0−3 = y +3−3 −3 = y (0, −3) Let y = 0. x=y+3 x=0+3 x=3 (3, 0) Let y = −1. x=y+3 x = −1 + 3 x=2 (2, −1) Plot (0, −3), (3, 0), and (2, −1). Then draw a line through them.

83. x + y = −4 Find any 3 ordered pair solutions. Let x = 0. x + y = −4 0 + y = −4 y = −4 (0, −4) Let y = 0. x + y = −4 x + 0 = −4 x = −4

ISM: Prealgebra

Chapter 8: Graphing and Introduction to Statistics and Probability

(−4, 0) Let x = −2. x + y = −4 −2 + y = −4 2−2+ y = 2−4 y = −2 (−2, −2) Plot (0, −4), (−4, 0), and (−2, −2). Then draw a line through them.

3 x 4 Find any 3 ordered pair solutions. Let x = 0. 3 y= x 4 3 y = ⋅0 4 y=0 (0, 0) Let x = 4. 3 y= x 4 3 y = ⋅4 4 y=3 (4, 3) Let x = −4. 3 y= x 4 3 y = (−4) 4 y = −3

84. y =

3 85. y = − x 4 Find any 3 ordered pair solutions. Let x = 0. 3 y=− x 4 3 y = − ⋅0 4 y=0 (0, 0) Let x = 4. 3 y=− x 4 3 y = − ⋅4 4 y = −3 (4, −3) Let x = −4. 3 y=− x 4 3 y = − ⋅ (−4) 4 y=3 (−4, 3) Plot (0, 0), (4, −3), and (−4, 3). Then draw a line through them.

(−4, −3) Plot (0, 0), (4, 3), and (−4, −3). Then draw a line through them.

319

Chapter 8: Graphing and Introduction to Statistics and Probability 86. There are 2 + 5 + 4 + 1 + 6 = 18 data items. a.

10 ⋅ 2 + 12 ⋅ 5 + 14 ⋅ 4 + 16 ⋅1 + 18 ⋅ 6 18 260 = 18 ≈ 14.44

mean =

n + 1 18 + 1 = = 9.5 2 2 The mean of the data items in the ninth and 14 + 14 tenth places is = 14, so the median 2 is 14.

The data item with the greatest frequency is the mode, so the mode is 18.

range = greatest data value − smallest data value = 18 − 10 =8

87. There are 5 + 2 + 3 + 4 + 1 = 15 data items. a.

5 ⋅ 5 + 10 ⋅ 2 + 15 ⋅ 3 + 20 ⋅ 4 + 25 ⋅1 15 195 = 15 = 13

mean =

n + 1 15 + 1 = =8 2 2 The data item in the eighth position is 15, so the median is 15.

3. Orchard 4 has 2.5 symbols and each symbol corresponds to 28 thousand bushels, so orchard 4 produced 2.5(4) = 10 thousand bushels; D. 4. Orchard 2 has 8.5 symbols and orchard 4 has 2.5 symbols, which is 6 fewer symbols. Each symbol represents 4 thousand bushels, so orchard 2 produced 6(4) = 24 thousand bushels more than orchard 4. 24 thousand = 24,000; C. 5. The point (−2, −2) is 2 units to the left of the y-axis and 2 units below the x-axis. This is the location of point F. 6. The point (0, 3) is on the y-axis, 3 units above the x-axis. This is the location of point C. 7. The point (3, 0) is 3 units to the right of the y-axis and on the x-axis. This is the location of point B. 8. The point (−5, 0) is 5 units to the left of the y-axis and on the x-axis. This is the location of point E. 9. the point (0, −5) is on the y-axis and 5 units below the x-axis. This is the location of point G. 10. The point (0, 0) is the origin⎯on both the x- and y-axes. This is the location of point A. 11. A vertical line has an equation of the form x = c, where c is any number. B is the equation of a vertical line. 12. A.

x − y = 10 0 − 10 0 10 −10 0 10 False

The data item with the greatest frequency is the mode, so the mode is 5.

range = greatest data value − smallest data value = 25 − 5 = 20

x − y = 10 12 − 2 0 10 10 = 1 True

x − y = 10 10 − 0 0 10 10 = 10 True

x − y = 10 5 − (−5) 0 10 5 + 5 0 10 10 = 10 True

Chapter 8 Getting Ready for the Test 1. Orchard 1 has 7 symbols, and each symbol represents 4 thousand bushels, so orchard 1 produced 7 ⋅ 4 = 28 thousand bushels. 28 thousand = 28,000; C. 2. Orchard 5 has the most symbols, so orchard 5 produced the most bushels; C. 320

ISM: Prealgebra

Choice A is not a solution of x − y = 10.

ISM: Prealgebra

Chapter 8: Graphing and Introduction to Statistics and Probability

13. A whole circle contains 360°. 90° + 10° + 20° + 100° + 100° = 320° The unknown sector must contain 360° − 320° = 40°; C. 14. Three of the five possible outcomes is a red 3 marble. The probability is ; C. 5 15. None (0) of the five possible outcomes is a green 0 marble. The probability is = 0; A. 5

Data sets for Exercises 16 through 18: A. range = 10 − 10 = 0 10 + 10 + 10 + 10 + 10 50 = = 10 mean = 5 5 median = 10 B. range = 14 − 6 = 8 6 + 8 + 10 + 12 + 14 50 = = 10 mean = 5 5 median = 10 C. range = 12 − 8 = 4 8 + 9 + 10 + 11 + 12 50 = = 10 mean = 5 5 median = 10 16. Data set B has the greatest range. 17. The median of all three data sets is 10; D. 18. The mean of all three data sets is 10; D.

Data sets for Exercises 19 through 22: A. range = 50 − 10 = 40 10 + 20 + 30 + 30 + 40 + 50 180 mean = = = 30 6 6 30 + 30 median = = 30 2 mode = 30 B. range = 48 − 9 = 39 9 + 11 + 11 + 30 + 48 109 mean = = = 21.8 5 5 median = 11 mode = 11

C. range = 40 − 20 = 20 20 + 25 + 30 + 35 + 40 150 = = 30 mean = 5 5 median = 30 mode: none D. range = 70 − 50 = 20 50 + 55 + 60 + 65 + 70 300 mean = = = 60 5 5 median = 60 mode: none 19. Data sets A and C both have mean 30. 20. Data sets A and B each have one mode. 21. Data sets C and D both have range 20. 22. Data sets A and C both have median 30. Chapter 8 Test 1 dollar symbols for the second 2 week. Each dollar symbol corresponds to $50. 1 9 $450 4 ⋅ $50 = ⋅ $50 = = $225 2 2 2 $225 was collected during the second week.

1. There are 4

2. Week 3 has the greatest number of dollar symbols. So the most money was collected during the 3rd week. The 3rd week has 7 dollar symbols and each dollar symbol corresponds to $50, so 7 ⋅ $50 = $350 was collected during week 3. 3. There are a total of 22 dollar symbols and each dollar symbol corresponds to $50, so a total amount of 22 ⋅ $50 = $1100 was collected. 4. Look for the bars whose height is greater than 9. June, August, and September normally have more than 9 centimeters. 5. The shortest bar corresponds to February. The normal monthly rainfall in February in Chicago is 3 centimeters. 6. The bars corresponding to March and November have a height of 7. Thus, during March and November, 7 centimeters of precipitation normally occurs.

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ISM: Prealgebra

8. The point on the graph corresponding to 2020 is at about 1.4. Thus, the annual inflation rate in 2020 was about 1.4%. 9. The line graph is above the 3 level for 2021, 2022, and 2023. Thus the inflation rate was greater than 3% in 2021, 2022, and 2023. 10. Look for the years where the line graph is decreasing. During 2014−2015, 2017−2018, 2019−2020, 2021−2022, and 2022−2023 the inflation rate was decreasing. 11.

12.

number who prefer rock music 85 17 = = total number 200 40 17 The ratio is . 40

number who prefer country 62 31 = = number who prefer jazz 44 22

The ratio is

31 . 22

13. The sector corresponding to under 18 is 20%. 20% of 374 million = 0.20 ⋅ 374 million = 74.8 million About 75 million people are expected to be in the under 18 age group in 2040. 14. The sector corresponding 85 and older is 4%. 4% of 374 million = 0.04 ⋅ 374 million = 14.96 million. About 15 million people are expected to be in the 85 and older age group in 2040. 15. The height of the bar for 5'8"− 5'11" is 9. Thus, there are 9 students who are 5'8"− 5'11" tall. 16. Add the heights of the bars for 5'0"− 5'3" and 5' 4"− 5'7". There are 5 + 6 = 11 students who are 5'7" tall or shorter.

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17.

Chapter 8: Graphing and Introduction to Statistics and Probability d.

Class Interval (scores)

Tally

Class Frequency (number of students)

40−49



50−59



60−69



70−79

||||

80−89

|||| |||

90−99



22. There are 3 + 6 + 7 + 3 + 8 = 27 data items. a.

18.

19. range = greatest data value − smallest data value = 18 − 10 =8 20. range = greatest data value − smallest data value = 99 − 43 = 56 21. There are 3 + 1 + 2 + 8 + 8 = 22 data items. a.

90 ⋅ 3 + 91 ⋅1 + 92 ⋅ 2 + 93 ⋅ 8 + 94 ⋅ 8 22 2041 = 22 ≈ 92.8

mean =

n + 1 22 + 1 = = 11.5 2 2 The mean of the data items in the eleventh 93 + 93 and twelfth positions is = 93, so the 2 median is 93.

The data items with the greatest frequency are the mode, so the mode is 93 and 94.

range = greatest data value − smallest data value = 94 − 90 =4

15 ⋅ 3 + 20 ⋅ 6 + 25 ⋅ 7 + 30 ⋅ 3 + 35 ⋅ 8 27 710 = 27 ≈ 26.3

mean =

n + 1 27 + 1 = = 14 2 2 The data item in the 14th position is 25, so the median is 25.

The data item with the greatest frequency is the mode, so the mode is 35.

range = greatest data value − smallest data value = 35 − 15 = 20

23. Point A has coordinates (4, 0). 24. Point B has coordinates (0, −3). 25. Point C has coordinates (−3, 4). 26. Point D has coordinates (−2, −1). 27.

x = −6 y 0 = −6 y 0 −6 y = −6 −6 0= y (0, 0) x = −6y x = −6(1) x = −6 (−6, 1) x = −6 y 12 = −6 y 12 −6 y = −6 −6 −2 = y (12, −2)

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ISM: Prealgebra

Let x = −2. y + x = −4 y + (−2) = −4 y + (−2) + 2 = −4 + 2 y = −2 (−2, −2) Plot (0, −4), (−4, 0), and (−2, −2). Then draw a line through them.

28. y = 7x − 4 y=7⋅2−4 y = 14 − 4 y = 10 (2, 10) y = 7x − 4 y = 7(−1) − 4 y = −7 − 4 y = −11 (−1, −11) y = 7x − 4 y=7⋅0−4 y=0−4 y = −4 (0, −4)

30. y = −4 No matter what x-value we choose, y is always −4.

29. y + x = −4 Find any 3 ordered pair solutions. Let x = 0. y + x = −4 y + 0 = −4 y = −4 (0, −4) Let y = 0. y + x = −4 0 + x = −4 x = −4 (−4, 0)

324

−2

−4

31. y = 3x − 5 Find any 3 ordered pair solutions. Let x = 0. y = 3x − 5 y=3⋅0−5 y=0−5 y = −5 (0, −5)

ISM: Prealgebra

Chapter 8: Graphing and Introduction to Statistics and Probability

Let x = 1. y = 3x − 5 y=3⋅1−5 y=3−5 y = −2 (1, −2) Let x = 2. y = 3x − 5 y=3⋅2−5 y=6−5 y=1 (2, 1) Plot (0, −5), (1, −2), and (2, 1). Then draw a line through them.

32. x = 5 No matter what y-value we choose, x is always 5.

−2

1 33. y = − x 2 Find any 3 ordered pair solutions. Let x = 0. 1 y=− x 2 1 y = − ⋅0 2 y=0 (0, 0) Let x = −2. 1 y=− x 2 1 y = − ⋅ (−2) 2 y =1 (−2, 1) Let x = 2. 1 y=− x 2 1 y = − ⋅2 2 y = −1 (2, −1) Plot (0, 0), (−2, 1), and (2, −1). Then draw a line through them.

34. 3x − 2y = 12 Find any 3 ordered pair solutions. Let x = 0. 3x − 2 y = 12 3 ⋅ 0 − 2 y = 12 −2 y = 12 −2 y 12 = −2 −2 y = −6

(0, −6)

325

Chapter 8: Graphing and Introduction to Statistics and Probability

Let y = 0. 3x − 2 y = 12 3 x − 2 ⋅ 0 = 12 3x = 12 3x 12 = 3 3 x=4 (4, 0) Let x = 2. 3 x − 2 y = 12 3 ⋅ 2 − 2 y = 12 6 − 2 y = 12 6 − 6 − 2 y = 12 − 6 −2 y = 6 −2 y 6 = −2 −2 y = −3 (2, −3) Plot (0, −6), (4, 0), and (2, −3). Then draw a line through them.

ISM: Prealgebra

37. One of the ten possible outcomes is a 6. The 1 probability is . 10 38. Two of the ten possible outcomes are a 3 or a 4. 2 1 The probability is = . 10 5 Cumulative Review Chapters 1−8 1. 43 + [32 − (10 ÷ 2)] − 7 ⋅ 3 = 43 + [32 − 5] − 7 ⋅ 3 = 43 + (9 − 5) − 7 ⋅ 3 = 43 + 4 − 7 ⋅ 3 = 64 + 4 − 7 ⋅ 3 = 64 + 4 − 21 = 68 − 21 = 47

2. 7 2 + [53 − (6 ÷ 3)] + 4 ⋅ 2 = 7 2 + [53 − 2] + 4 ⋅ 2 = 7 2 + (125 − 2) + 4 ⋅ 2 = 7 2 + 123 + 4 ⋅ 2 = 49 + 123 + 4 ⋅ 2 = 49 + 123 + 8 = 172 + 8 = 180

Red Blue Green Yellow Red Blue Green Yellow Red Blue Green Yellow Red Blue Green Yellow

Red

Blue

35. Green

Yellow

16 outcomes H

36. T

4 outcomes

326

H T H

3. x − y = −3 − 9 = −3 + (−9) = −12 4. x − y = 7 − (−2) = 7 + 2 = 9 5. 3 y − 7 y = 12 (3 − 7) y = 12 −4 y = 12 −4 y 12 = −4 −4 y = −3 6. 2 x − 6 x = 24 (2 − 6) x = 24 −4 x = 24 −4 x 24 = −4 −4 x = −6

ISM: Prealgebra

10.

Chapter 8: Graphing and Introduction to Statistics and Probability 14.

7.400 − 0.073 7.327

15.

0.0531 16 3186 + 5310 0.8496

4 decimal places

7 a + =1 2 4 7 a 4  +  = 4(1) 2 4 7 a 4 ⋅ + 4 ⋅ = 4 ⋅1 2 4 14 + a = 4 14 − 14 + a = 4 − 14 a = −10

16.

0.147 0.2 0.0294

3 decimal places 1 decimal place 3 + 1 = 4 decimal places

1 3 3 +5 8

17. 115

8 24 9 +5 24 17 7 24 1 3 17 2 +5 = 7 3 8 24 2

2 5 3 +4 4

8 20 15 +4 20 3 3 23 = 7 +1 = 8 7 20 20 20 2 3 3 3 +4 =8 5 4 20 3

8 4 12. 2.8 = 2 = 2 10 5

3.500 − 0.068 3.432

7.327 + 0.073 7.400

4 decimal places

0.052 5.980 −5 75 230 −230 0

−5.98 ÷ 115 = −0.052

9 11. 5.9 = 5 10

13.

Check:

4 x +1 = 6 3 x  4 6  + 1 = 6   6  3 4 x 6 ⋅ + 6 ⋅1 = 6 ⋅ 6 3 x+6 =8 x +6−6 = 8−6 x=2

Check:

0.136 18. 205 27.880 −20 5 7 38 −6 15 1 230 −1 230 0

27.88 ÷ 205 = 0.136 19. (−1.3) 2 + 2.4 = 1.69 + 2.4 = 4.09 20. (−2.7)2 = (−2.7)(−2.7) = 7.29 21.

1 1 ⋅ 25 25 = = = 0.25 4 4 ⋅ 25 100

22.

3 3 ⋅125 375 = = = 0.375 8 8 ⋅125 1000

3.432 + 0.068 3.500

327

Chapter 8: Graphing and Introduction to Statistics and Probability

23.

24.

5( x − 0.36) = − x + 2.4 5 ⋅ x − 5 ⋅ 0.36 = − x + 2.4 5 x − 1.8 = − x + 2.4 5 x + x − 1.8 = − x + x + 2.4 6 x − 1.8 = 2.4 6 x − 1.8 + 1.8 = 2.4 + 1.8 6 x = 4.2 6 x 4.2 = 6 6 x = 0.7

0.53 30. 3 1.59 −1 5 09 −9 0

4(0.35 − x) = x − 7 4 ⋅ 0.35 − 4 ⋅ x = x − 7 1.4 − 4 x = x − 7 1.4 − 4 x − x = x − x − 7 1.4 − 5 x = −7 1.4 − 1.4 − 5 x = −7 − 1.4 −5 x = −8.4 −5 x −8.4 = −5 −5 x = 1.68

31.

25.

80 ≈ 8.944

26.

60 ≈ 7.746

27. The ratio of 21 to 29 is

$1.59 $0.53 = 3 ounces 1 ounce The unit rate is $0.53 per ounce.

32.

21 . 29

51 −3 = x 34 51 ⋅ x = 34 ⋅ (−3) 51x = −102 51x −102 = 51 51 x = −2 The solution is −2. 8 x = 5 10 8 ⋅10 = 5 ⋅ x 80 = 5 x 80 5 x = 5 5 16 = x The solution is 16.

7 28. The ratio of 7 to 15 is . 15

33.

12 feet 12 = 19 feet 19

22.5 29. 15 337.5 −30 37 −30 75 −7 5 0

34.

4 9

35. 4.6% = 4.6(0.01) = 0.046 36. 32% = 32(0.01) = 0.32 37. 0.74% = 0.74(0.01) = 0.0074 38. 2.7% = 2.7(0.01) = 0.027

337.5 miles 22.5 miles = 15 gallons 1 gallon The unit rate is 22.5 miles/gallon.

328

ISM: Prealgebra

39. 35% of 60 = 0.35 ⋅ 60 = 21 21 is 35% of 60. 40. 40% of 36 = 0.40 ⋅ 36 = 14.4 14.4 is 40% of 36.

ISM: Prealgebra

Chapter 8: Graphing and Introduction to Statistics and Probability

41. 20.8 = 40% ⋅ x 20.8 = 0.4 x 20.8 0.4 x = 0.4 0.4 52 = x 20.8 is 40% of 52. 42.

46.

9.5 = 25% ⋅ x 9.5 = 0.25 x 9.5 0.25 x = 0.25 0.25 38 = x 9.5 is 25% of 38.

47. The numbers are listed in order. Since there are an odd number of values, the median is the middle value, 57.

43. $310 ⋅ r = $26.35 310r = 26.35 310r 26.35 = 310 310 r = 0.085 or 8.5% The sales tax rate is 8.5%. 44. $2 ⋅ r = $0.13 2r = 0.13 2r 0.13 = 2 2 r = 0.065 or 6.5% The sales tax rate is 6.5%.  r 45. A = P  1 +   n

total $1600 + $128.60 = 12 months 12 $1728.60 = 12 = $144.05 The monthly payment is $144.05.

48. The numbers are listed in order. Since there are an even number of values, the median is 47 + 50 97 = = 48.5. 2 2 49. Two of the six possible outcomes are a 3 or a 4. 2 1 The probability is = . 6 3 50. Three of the six possible outcomes are even. The 3 1 probability is = . 6 2

n⋅t

 0.053  = $4000 ⋅  1 + 4   = $4000(1.01325)40 ≈ $6772.12

4⋅10

329

Chapter 9 Section 9.1 Practice Exercises 1. Figure (a) is part of a line with one endpoint, so → it is a ray. It is ray AB or AB . Figure (b) has two endpoints, so it is a line segment. It is line segment RS or RS . Figure (c) extends indefinitely in two directions, ↔ so it is a line. It is line EF or EF . Figure (d) has two rays with a common endpoint, so it is an angle. It is ∠HVT or ∠TVH or ∠V . 2. Two other ways to name ∠z are ∠RTS and ∠STR. 3. a.

∠R is an obtuse angle. It measures between 90° and 180°.

Vocabulary, Readiness & Video Check 9.1 1. A plane is a flat surface that extends indefinitely. 2. A point has no length, no width, and no height. 3. Space extends in all directions indefinitely.

∠N is a straight angle. It measures 180°.

4. A line is a set of points extending indefinitely in two directions.

∠M is an acute angle. It measures between 0° and 90°.

5. A ray is part of a line with one endpoint.

∠Q is a right angle. It measures 90°.

4. The complement of a 29° angle is an angle that measures 90° − 29° = 61°. 5. The supplement of a 67° angle is an angle that measures 180° − 67° = 113°. 6. a.

m∠y = m∠ADC − m∠BDC = 141° − 97° = 44°

m∠x = 79° − 51° = 28°

Since the measures of both ∠x and ∠y are between 0° and 90°, they are acute angles.

7. Since ∠a and the angle marked 109° are vertical angles, they have the same measure; so m∠a = 109°. Since ∠a and ∠b are adjacent angles, their measures have a sum of 180°. So m∠b = 180° − 109° = 71°. Since ∠b and ∠c are vertical angles, they have the same measure; so m∠c = 71°.

330

8. ∠w and ∠x are vertical angles. ∠w and ∠y are corresponding angles, as are ∠x and ∠d . So all of these angles have the same measure: m∠x = m∠y = m∠d = m∠w = 45°. ∠w and ∠a are adjacent angles, as are ∠w and ∠b, so m∠a = m∠b = 180° − 45° = 135°. ∠a and ∠c are corresponding angles so m∠c = m∠a = 135°. ∠c and ∠z are vertical angles, so m∠z = m∠c = 135°.

6. An angle is made up of two rays that share a common endpoint. The common endpoint is called the vertex. 7. A straight angle measures 180°. 8. A right angle measures 90°. 9. An acute angle measures between 0° and 90°. 10. An obtuse angle measures between 90° and 180°. 11. Parallel lines never meet and intersecting lines meet at a point. 12. Two intersecting lines are perpendicular if they form right angles when they intersect. 13. An angle can be measured in degrees. 14. A line that intersects two or more lines at different points is called a transversal. 15. When two lines intersect, four angles are formed. Two of these angles that are opposite each other are called vertical angles. 16. Two angles that share a common side are called adjacent angles.

ISM: Prealgebra

Chapter 9: Geometry and Measurement

30. 20° + 70° = 90°, so ∠CAD and ∠DAE are complementary. 63° + 27° = 90°, so ∠BAC and ∠EAF are complementary.

17. ∠WUV, ∠VUW, ∠U, ∠x 18. straight angle; 180° 19. 180° − 17° = 163°

32. 38° + 142° = 180°, so there are 4 pairs of supplementary angles: ∠NMX and ∠NMY , ∠NMX and ∠XMZ , ∠YMZ and ∠NMY , ∠YMZ and ∠XMZ .

20. intersect Exercise Set 9.1 2. The figure has two rays with a common endpoint. It is an angle, which can be named ∠GHI, ∠IHG, or ∠H.

34. m∠x = 150° − 48° = 102°

4. The figure has one endpoint and extends indefinitely in one direction, so it is a ray. It is → ray ST or ST .

38. ∠x and the angle marked 165° are supplementary, so m∠x = 180° − 165° = 15°. ∠y and the angle marked 165° are vertical angles, so m∠y = 165°. ∠x and ∠z are vertical angles, so m∠z = m∠x = 15°.

6. The figure extends indefinitely in two directions, ↔ so it is a line. It is line AB, line t, or AB . 8. The figure has two endpoints, so it is a line segment. It is line segment PQ or PQ 10. Two other ways to name ∠w are ∠APC and ∠CPA. 12. Two other ways to name ∠y are ∠MPQ and ∠QPM . 14. ∠H is an acute angle. It measures between 0° and 90°. 16. ∠T is an obtuse angle. It measures between 90° and 180°. 18. ∠M is a straight angle. It measures 180°. 20. ∠N is a right angle. It measures 90°. 22. The complement of an angle that measures 77° is an angle that measures 90° − 77° = 13°. 24. The supplement of an angle that measures 77° is an angle that measures 180° − 77° = 103°. 26. The complement of an angle that measures 22° is an angle that measures 90° − 22° = 68°. 28. The supplement of an angle that measures 130° is an angle that measures 180° − 130° = 50°.

36. m∠x = 36° + 12° = 48°

40. ∠x and the angle marked 44° are supplementary, so m∠x = 180° − 44° = 136°. ∠y and the angle marked 44° are vertical angles, so m∠y = 44°. ∠x and ∠z are vertical angles, so m∠z = m∠x = 136°. 42. ∠x and the angle marked 110° are vertical angles, so m∠x = 110°. ∠x and ∠y are corresponding angles, so m∠y = m∠x = 110°. ∠y and ∠z are vertical angles, so m∠z = m∠y = 110°. 44. ∠x and the angle marked 39° are supplementary, so m∠x = 180° − 39° = 141°. ∠y and the angle marked 39° are corresponding angles, so m∠y = 39°. ∠y and ∠z are supplementary, so m∠z = 180° − m∠y = 180° − 39° = 141°. 46. ∠y can also be named ∠CBD or ∠DBC. 48. ∠ABE can also be named ∠EBA. 50. m∠EBD = 45° 52. m∠CBA = 15°

331

Chapter 9: Geometry and Measurement 54. m∠EBC = m∠EBD + m∠DBC = 45° + 50° = 95°

Section 9.2 Practice Exercises

56. m∠ABE = m∠ABC + m∠CBD + m∠DBE = 15° + 50° + 45° = 110° 58.

60.

ISM: Prealgebra

7 1 7 1⋅ 2 7 2 5 − = − = − = 8 4 8 4⋅2 8 8 8

1. a.

Perimeter = 10 m + 10 m + 18 m + 18 m = 56 meters The perimeter is 56 meters.

Perimeter = 125 ft + 125 ft + 50 ft + 50 ft = 350 ft The perimeter is 350 feet.

2. P = 2 ⋅ l + 2 ⋅ w = 2 ⋅ 32 cm + 2 ⋅15 cm = 64 cm + 30 cm = 94 centimeters The perimeter is 94 centimeters.

7 1 7 4 7⋅4 7 1 ÷ = ⋅ = = or 3 8 4 8 1 2⋅4 2 2

1 1 10 5 62. 3 + 2 = + 3 2 3 2 10 ⋅ 2 5 ⋅ 3 = + 3⋅ 2 2⋅3 20 15 = + 6 6 35 5 = or 5 6 6

3. P = 4 ⋅ s = 4 ⋅ 4 ft = 16 feet The perimeter is 16 feet. 4. P = a + b + c = 6 cm + 10 cm + 8 cm = 24 centimeters The perimeter is 24 centimeters.

1 1 10 5 2 ⋅ 5 ⋅ 5 25 1 64. 3 ⋅ 2 = ⋅ = = or 8 3 2 3 2 3⋅ 2 3 3

5. Perimeter = 6 km + 4 km + 9 km + 4 km = 23 kilometers The perimeter is 23 kilometers.

66. The three red dots divide the circle into three parts. If the three dots are about the same number of degrees apart, they divide the 360° of the circle into three parts. 360° = 120° 3 The red dots are 120° degrees apart.

6. The unmarked horizontal side has length 20 m − 15 m = 5 m. The unmarked vertical side has length 31 m − 6 m = 25 m. P = 15 m + 31 m + 20 m + 6 m + 5 m + 25 m = 102 meters The perimeter is 102 meters.

68. The complement of an angle that measures 35° is an angle that measures 90° − 35° = 55°. 70. The complement of an angle that measures 53.13° is an angle that measures 90° − 53.13° = 36.87°.

8. C = π ⋅ d = π ⋅ 20 yd = 20π yd ≈ 62.8 yd The exact circumference of the watered region is 20π yards, which is approximately 62.8 yards.

72. False; answers may vary 74. True, since 5° + 175° = 180°. 76. The sides of a rectangle are parallel. Thus, AB and CD , as well as AC and BD are parallel. 78. answers may vary 80. no; answers may vary

332

7. P = 2 ⋅ l + 2 ⋅ w = 2 ⋅120 feet + 2 ⋅ 60 feet = 240 feet + 120 feet = 360 feet cost = $1.90 per foot ⋅ 360 feet = $684 The cost of the fencing is $684.

Vocabulary, Readiness & Video Check 9.2 1. The perimeter of a polygon is the sum of the lengths of its sides. 2. The distance around a circle is called the circumference.

ISM: Prealgebra

Chapter 9: Geometry and Measurement

3. The exact ratio of circumference to diameter is π. 4. The diameter of a circle is double its radius. 5. Both

22  22  (or 3.14) and 3.14  or  are 7  7 

approximations for π. 6. The radius of a circle is half its diameter. 7. Opposite sides of a rectangle have the same measure, so we can just find the sum of the measures of all four sides. 8. the perimeter of a circle

18. P = 8 ⋅ 12 in. = 96 in. The perimeter is 96 inches. 20. P = 2 ⋅ l + 2 ⋅ w = 2 ⋅ 70 ft + 2 ⋅ 21 ft = 140 ft + 42 ft = 182 ft 182 feet of fencing is needed. 22. The amount of fencing needed is 182 feet. 182 feet ⋅ $2 per foot = $364 The total cost of the fencing is $364.

Exercise Set 9.2 2. P = 2 ⋅ l + 2 ⋅ w = 2 ⋅14 m + 2 ⋅ 5 m = 28 m + 10 m = 38 m The perimeter is 38 meters.

24. a.

A pentagon has five sides.

b. P = 5 ⋅ 14 m = 70 m The perimeter is 70 meters.

4. P = 2 yd + 3 yd + 2 yd + 3 yd = 10 yd The perimeter is 10 yards. 6. P = a + b + c = 5 units + 10 units + 11 units = 26 units The perimeter is 26 units. 8. Sum the lengths of the sides. P = 10 m + 4 m + 10 m + 9 m + 20 m + 13 m = 66 m The perimeter is 66 meters. 10. All sides of a regular polygon have the same length, so the perimeter is the number of sides multiplied by the length of a side. P = 4 ⋅ 50 m = 200 m The perimeter is 200 meters. 12. All sides of a regular polygon have the same length, so the perimeter is the number of sides multiplied by the length of a side. P = 6 ⋅ 15 yd = 90 yd The perimeter is 90 yards. 14. P = a + b + c = 8 in. + 12 in. + 10 in. = 30 in. The perimeter is 30 inches.

16. P = 2 ⋅ l + 2 ⋅ w = 2 ⋅ 6 ft + 2 ⋅ 4 ft = 12 ft + 8 ft = 20 ft The perimeter of each batter’s box is 20 feet.

26. P = 4 ⋅ s = 4 ⋅ 3 in. = 12 in. The perimeter is 12 inches. 28. P = 2 ⋅ l + 2 ⋅ w = 2 ⋅ 85 ft + 2 ⋅ 70 ft = 170 ft + 140 ft = 310 ft 310 ft ⋅ $2.36 per foot = $731.60 The cost is $731.60. 30. The unmarked vertical side has length 13 in. − 6 in. = 7 in. The unmarked horizontal side has length 30 in. − 13 in. = 17 in. P = 13 in. + 13 in. + 30 in. + 7 in. + 17 in. + 6 in. = 86 in. The perimeter is 86 inches. 32. The unmarked vertical side has length (2 + 4 + 3) cm = 9 cm. The width of the figure in the vertical section of height 4 cm is 16 − 11 = 5 cm. So the unmarked horizontal side has length (9 − 5) cm = 4 cm. P = (16 + 2 + 11 + 4 + 4 + 3 + 9 + 9) cm = 58 cm The perimeter is 58 centimeters. 34. The unmarked vertical side has length (22 + 5) km = 27 km. The unmarked horizontal side has length (12 + 6) km = 18 km P = (18 + 27 + 6 + 5 + 12 + 22) km = 90 km The perimeter is 90 kilometers.

333

Chapter 9: Geometry and Measurement

ISM: Prealgebra

36. C = 2 ⋅ π ⋅ r = 2 ⋅ π ⋅ 2.5 in. = 5π in. ≈ 15.7 in. The circumference is exactly 5π inches or approximately 15.7 inches.

62. (72 ÷ 2) ⋅ 6 = 36 ⋅ 6 = 216

38. C = π ⋅ d = π ⋅ 50 ft = 50π ft ≈ 157 ft The circumference is exactly 50π feet or approximately 157 feet.

66. a.

The first age category that 12-year-old children fit into is “Under 13,” so the maximum width is 60 yards and the maximum length is 110 yards.

P = 2⋅l + 2⋅ w = 2 ⋅110 yd + 2 ⋅ 60 yd = 220 yd + 120 yd = 340 yd The perimeter of the field is 340 yards.

40. C = 2 ⋅ π ⋅ r = 2 ⋅ π ⋅ 10 yd = 20π yd ≈ 62.8 yd The circumference is exactly 20π yards or approximately 62.8 yards.

64. 41 ⋅ (23 − 8) = 41 ⋅ (8 − 8) = 41 ⋅ 0 = 4 ⋅ 0 = 0

42. C = 2 ⋅ π ⋅ r = 2 ⋅ π ⋅ 18 ft = 36π ft ≈ 113.04 ft The circumference is exactly 36π feet or approximately 113.04 feet. 44. C = π ⋅ d = π ⋅ 1640 ft = 1640π ft ≈ 5149.6 ft The circumference is exactly 1640π feet or approximately 5149.6 feet. 46. C = π ⋅ d = π ⋅ 150 ft = 150π ft ≈ 471 feet The circumference of the barn is 150π feet, or about 471 feet. 48. C = π ⋅ d  1 = π ⋅  5  in.  2 11 = π in. 2 11 22 2 ≈ ⋅ in. = 17 in. 2 7 7

68. The circle’s circumference is π ⋅ 7 in. ≈ 21.98 in. The square’s perimeter is 4 ⋅ 7 in. = 28 in. So the square has the greater distance around; b. 70. a.

Smaller circle: C = π ⋅ d = π ⋅ 16 m = 16π m ≈ 50.24 m Larger circle: C = π ⋅ d = π ⋅ 32 m = 32π m ≈ 100.48 m

b. Yes, when the diameter of a circle is doubled, the circumference is also doubled. 72. answers may vary

The distance around is about 17

2 inches or 7

17.29 inches. 50. P = (4.5 + 7 + 9) yd = 20.5 yd The perimeter is 20.5 yards. 52. C = π ⋅ d = π ⋅ 11 m = 11π m ≈ 34.54 m The circumference is 11π meters ≈ 34.54 meters. 54. P = 4 ⋅ 19 km = 76 km The perimeter is 76 kilometers. 56. The unmarked vertical side has length (40 − 9) mi = 31 mi. The unmarked horizontal side has length (44 − 9) mi = 35 mi. P = (44 + 31 + 9 + 9 + 35 + 40) mi = 168 mi The perimeter is 168 miles.

74. The length of the curved section at the top is half of the circumference of a circle of diameter 6 meters. 1 1 1 ⋅ C = ⋅ π ⋅ d = ⋅ π ⋅ 6 m = 3π m ≈ 9.4 meters 2 2 2 The total length of the straight sides is 3 ⋅ 6 m = 18 m. The perimeter is the sum of these. 9.4 m + 18 m = 27.4 m The perimeter of the figure is 27.4 meters. 76. The three linear sides have lengths of 7 feet, 5 feet, and 7 feet. The length of the curved side is half of the circumference of a circle with diameter 5 feet, or 1 1 π ⋅ d = π ⋅ 5 = 2.5π ≈ 7.9 feet 2 2 7 ft + 5 ft + 7 ft + 7.9 ft = 26.9 ft The perimeter of the window is 26.9 feet.

58. 25 − 3 ⋅ 7 = 25 − 21 = 4 60. 6 ⋅ (8 + 2) = 6 ⋅ 10 = 60 334

ISM: Prealgebra

Chapter 9: Geometry and Measurement

Section 9.3 Practice Exercises

4 3 4 1  πr = π ⋅  cm  3 3 2  4 1 = π ⋅ cu cm 3 8 1 = π cu cm 6 1 22 ≈ ⋅ cu cm 6 7 11 = cu cm 21 1 The volume is π cubic centimeter, which is 6 11 approximately cubic centimeter. 21

6. V =

1 1. A = bh 2 1 1 = ⋅12 in. ⋅ 8 in. 2 4 1 33 = ⋅12 in. ⋅ in. 2 4 4 ⋅ 3 ⋅ 33 = sq in. 2⋅4 1 = 49 sq in. 2 1 The area is 49 square inches. 2 1 (b + B)h 2 1 = (5 yd + 11 yd)(6.1 yd) 2 = 48.8 sq yd The area is 48.8 square yards.

2. A =

3. Split the rectangle into two pieces, a top rectangle with dimensions 12 m by 24 m, and a bottom rectangle with dimensions 6 m by 18 m. The area of the figure is the sum of the areas of these. A = 12 m ⋅ 24 m + 6 m ⋅18 m = 288 sq m + 108 sq m = 396 sq m The area is 396 square meters. 4. A = πr 2 = π ⋅ (7 cm) 2 = 49π sq cm ≈ 153.86 sq cm The area is 49π square centimeters, which is approximately 153.86 square centimeters.

5. V = lwh = 7 ft ⋅ 3 ft ⋅ 4 ft = 84 cu ft The volume of the box is 84 cubic feet. SA = 2lh + 2 wh + 2lw = 2(7 ft)(4 ft) + 2(3 ft)(4 ft) + 2(7 ft)(3 ft) = 56 sq ft + 24 sq ft + 42 sq ft = 122 sq ft The surface area of the box is 122 square feet.

1  SA = 4πr 2 = 4π ⋅  cm  2  1 = 4π ⋅ sq cm 4 = π sq cm 22 1 ≈ sq cm or 3 sq cm 7 7 The surface area is π square centimeters, which 1 is approximately 3 square centimeters. 7

7. V = πr 2 h = π ⋅ (5 in.) 2 ⋅ 9 in. = π ⋅ 25 sq in. ⋅ 9 in. = 225π cu in. ≈ 706.5 cu in. The volume is 225π cubic inches, which is approximately 706.5 cubic inches.

1 1 8. V = s 2 h = ⋅ (3 m)2 ⋅ 5.1 m 3 3 1 = ⋅ 9 sq m ⋅ 5.1 m 3 = 15.3 cu m The volume is 15.3 cubic meters. Vocabulary, Readiness & Vocabulary Check 9.3 1. The surface area of a polyhedron is the sum of the areas of its faces. 2. The measure of the amount of space inside a solid is its volume. 3. Area measures the amount of surface enclosed by a region.

335

Chapter 9: Geometry and Measurement 4. Volume is measured in cubic units. 5. Area is measured in square units. 6. Surface area is measured in square units. 7. We don’t have a formula for an L-shaped figure, so we divide it into two rectangles, use the formula to find the area of each, and then add these two areas. 8. For each example, the exact volume is found and an approximate volume is found. Exact answers are in terms of π, and approximate answers use an approximation for π. Exercise Set 9.3 2. A = l ⋅ w = 3.5 ft ⋅ 1.2 ft = 4.2 sq ft The area is 4.2 square feet.

1 ⋅b⋅h 2 1 1 = ⋅ 4 ft ⋅ 5 ft 2 2 1 9 = ⋅ ft ⋅ 5 ft 2 2 45 = sq ft 4 1 = 11 sq ft 4 1 The area is 11 square feet. 4

4. A =

1 ⋅b⋅h 2 1 = ⋅ 7 ft ⋅ 5 ft 2 35 = sq ft 2 1 = 17 sq ft 2 1 The area is 17 square feet. 2

6. A =

336

ISM: Prealgebra

8. A = πr 2

= π(5 cm)2 = 25π sq cm 25 22 = ⋅ sq cm 1 7 550 4 sq cm = 78 sq cm ≈ 7 7 The area is 25π square centimeters 4 ≈ 78 square centimeters. 7 10. A = b ⋅ h = 3 cm ⋅ 4.25 cm = 12.75 sq cm The area is 12.75 square centimeters.

1 (b + B) ⋅ h 2 1 1  =  5 in. + 8 in.  ⋅ 6 in. 2 2  1 27 in. ⋅ 6 in. = ⋅ 2 2 81 sq in. = 2 1 = 40 sq in. 2 1 The area is 40 square inches. 2

12. A =

1 (b + B) ⋅ h 2 1 = (10 ft + 5 ft) ⋅ 3 ft 2 1 = ⋅15 ft ⋅ 3 ft 2 = 22.5 sq ft The area is 22.5 square feet.

14. A =

16. A = b ⋅ h 3 cm 4 31 = 5 cm ⋅ cm 4 155 = sq cm 4 3 = 38 sq cm 4 3 The area is 38 square centimeters. 4 = 5 cm ⋅ 7

ISM: Prealgebra

Chapter 9: Geometry and Measurement

18. A = b ⋅ h = 6 m ⋅ 4 m = 24 sq m The area is 24 square meters.

30. V = s3 = (11 mi)3 = 1331 cu mi The volume is 1331 cubic meters.

20. The area of the trapezoid that forms the top of the figure is 1 1 (b + B) ⋅ h = (6 km + 10 km) ⋅ 4 km 2 2 = 32 sq km. The area of the rectangle that forms the bottom of the figure is l ⋅ w = 10 km ⋅ 5 km = 50 sq km. The area of the figure is the sum of these. A = 32 sq km + 50 sq km = 82 sq km The total area is 82 square kilometers. 22. The figure can be divided into two rectangles, one measuring 25 cm by 12 cm and one measuring 20 cm by 3 cm. The area of the figure is the sum of the areas of the two rectangles. A = 25 cm ⋅12 cm + 20 cm ⋅ 3 cm = 300 sq cm + 60 sq cm = 360 sq cm The total area is 360 square centimeters. 24. The top of the figure is a square with sides of length 5 in., so its area is s 2 = (5 in.) 2 = 25 sq in. The bottom of the figure is a parallelogram with area b ⋅ h = 5 in. ⋅ 4 in. = 20 sq in. The area of the figure is the sum of these. A = 25 sq in. + 20 sq in. = 45 sq in. The total area is 45 square inches.

26. r = d ÷ 2 = 5 m ÷ 2 = 2.5 m

A = πr 2 = π ⋅ (2.5 m) 2 = 6.25π sq m ≈ 19.625 sq m The area is 6.25π square meters ≈ 19.625 square meters. 28. V = l ⋅ w ⋅ h = 8 cm ⋅ 4 cm ⋅ 4 cm = 128 cu cm The volume is 128 cubic centimeters. SA = 2lh + 2 wh + 2lw = 2 ⋅ 8 cm ⋅ 4 cm + 2 ⋅ 4 cm ⋅ 4 cm + 2 ⋅ 8 cm ⋅ 4 cm = 64 sq cm + 32 sq cm + 64 sq cm = 160 sq cm The surface area is 160 square centimeters.

SA = 6 s 2 = 6(11 mi)2 = 6 ⋅121 sq mi = 726 sq mi The surface area is 726 square miles.

1 32. V = π ⋅ r 2 ⋅ h 3 2 1  3  = π ⋅ 1 in.  ⋅ 9 in. 3  4  147 = π cu in. 16 147 22 7 ≈ ⋅ cu in. = 28 cu in. 16 7 8 The volume is 147 7 π cubic inches ≈ 28 cubic inches. 16 8 SA = π ⋅ r r 2 + h 2 + πr 2 2

7 7  7  in.  in.  + (9 in.)2 + π  in.  4 4  4  7 49 = π 1345 sq in. + π sq in. 16 16 49  7 =  π 1345 + π  sq in. 16   16 ≈ 60.00 sq in. The surface area is 49   7  16 π 1345 + 16 π  square inches   ≈ 60.00 square inches.

= π⋅

4 π ⋅ r3 3 4 = π ⋅ (3 mi)3 3 = 36π cu mi 22 1 cu mi = 113 cu mi ≈ 36 ⋅ 7 7 The volume is 1 36π cubic miles ≈ 113 cubic miles. 7

34. V =

SA = 4πr 2 = 4π(3 mi)2 = 36π sq mi 22 1 ≈ 36 ⋅ sq mi = 113 sq mi 7 7 The surface area is 1 36π square miles ≈ 113 square miles. 7

337

Chapter 9: Geometry and Measurement

36. r =

1 1 ⋅ d = ⋅10 ft = 5 ft 2 2

V = π ⋅ r2 ⋅ h = π ⋅ (5 ft)2 ⋅ 6 ft = 150π cu ft 22 3 ≈ 150 ⋅ cu ft = 471 cu ft 7 7 The volume is 3 150π cubic feet ≈ 471 cubic feet. 7 1 1 38. V = ⋅ s 2 ⋅ h = ⋅ (7 m) 2 ⋅15 m = 245 cu m 3 3 The volume is 245 cubic meters.

1 40. V = π ⋅ r 2 ⋅ h 3 1 = π ⋅ (14 ft)2 ⋅15 ft 3 = 980π cu ft 22 cu ft = 3080 cu ft ≈ 980 ⋅ 7 The volume is 980π cubic feet ≈ 3080 cubic feet.

ISM: Prealgebra 50. Including the house, the land is in the shape of a trapezoid, whose area is 1 1 (b + B ) ⋅ h = (96 ft + 132 ft) ⋅ 48 ft 2 2 1 = ⋅ 228 ft ⋅ 48 ft 2 = 5472 sq ft. The house is a rectangle, with area l ⋅ w = 48 ft ⋅ 24 ft = 1152 sq ft. The area of the yard is the difference between these. A = 5472 sq ft − 1152 sq ft = 4320 sq ft The area of the yard is 4320 square feet. 52. V = π ⋅ r 2 ⋅ h = π ⋅ (3 ft) 2 ⋅ 8 ft = 72π cu ft The volume is 72π cubic feet. SA = 2πr ⋅ h + 2π ⋅ r 2 = 2π(3 ft)(8 ft) + 2π(3 ft) 2 = 66π sq ft The surface area is 66π square feet.

54. a.

42. V = s 3 = (5 ft)3 = 125 cu ft The volume is 125 cubic feet. SA = 6 s 2 = 6(5 ft) 2 = 6 ⋅ 25 sq ft = 150 sq ft The surface area is 150 square feet.

44. A = l ⋅ w = 52.7 m ⋅ 3.6 m = 189.72 sq m The area of the sign is 189.72 square meters. 46. A = l ⋅ w = 27.6 cm ⋅ 21.5 cm = 593.4 sq cm The area of a page is 593.4 square centimeters.

1 4 ⋅ ⋅ π ⋅ r3 2 3 2 = π ⋅ (10 in.)3 3 2000 = π cu in. 3 2000 22 5 cu in. = 2095 cu in. ≈ ⋅ 3 7 21 The volume is 2000 5 π cubic inches ≈ 2095 cubic inches. 3 12

48. V =

338

The top part of the wall is a triangle, with 1 1 area ⋅ b ⋅ h = ⋅ 24 ft ⋅ 4 ft = 48 sq ft. 2 2 The bottom part is a rectangle, with area l ⋅ w = 24 ft ⋅ 8 ft = 192 sq ft. The area of the wall is the sum of these. A = 48 sq ft + 192 sq ft = 240 sq ft The total area is 240 square feet.

b. Divide the area of the wall by the side area of each brick. 1 6 240 ÷ = 240 ⋅ = 1440 6 1 The number of bricks needed to brick the end of the building is 1440. 1 1 ⋅ d = ⋅ 6 in. = 3 in. 2 2 4 3 V = ⋅π⋅r 3 4 = π ⋅ (3 in.)3 3 = 36π cu in. ≈ 113.04 cu in. The volume is 36π cubic inches ≈ 113.04 cubic inches.

56. r =

ISM: Prealgebra

Chapter 9: Geometry and Measurement

58. r = d ÷ 2 = 2 cm ÷ 2 = 1 cm 2

A = πr = π ⋅ (1 cm) = π sq cm ≈ 3.14 sq cm

78. Gutters go on the edges of a house, so the situation involves perimeter.

The area of the watch face is π square centimeters, or about 3.14 square centimeters.

80. Baseboards go around the edges of a room, so the situation involves perimeter.

1 60. V = ⋅ π ⋅ r 2 ⋅ h 3 1 = π(3 km) 2 (3.5 km) 3 = 10.5π cu km  22  ≈ 10.5   cu km = 33 cu km  7  The volume is 10.5π cubic kilometers ≈ 33 cubic kilometers.

82. Fertilizer is spread throughout the surface of a yard, so the situation involves area.

62. A = l ⋅ w = 35 ft ⋅ 24 ft = 840 sq ft 840 square feet of insulation are needed. 4 ⋅ π ⋅ r3 3 4 = π ⋅ (2 cm)3 3 32 = π cu cm 3 32 ≈ ⋅ 3.14 cu cm 3 ≈ 33.49 cu cm The volume is 32 π cubic centimeters 3 ≈ 33.49 cubic centimeters.

84. This could be answered by dividing Utah into two rectangles, in two different ways. But it’s easier to regard the state as a 350 mile by 270 mile rectangle from which a 70 mile by 105 mile rectangle has been removed. A = 350 mi ⋅ 270 mi − 70 mi ⋅105 mi = 94,500 sq mi − 7350 sq mi = 87,150 sq mi The area of Utah is approximately 87,150 square miles. d 55 ft = = 27.5 ft 2 2 1 SA = ⋅ 4π r 2 2 1 = ⋅ 4(3.14)(27.5 ft)2 2 ≈ 4749.25 sq ft The surface area of the hemisphere is about 4749.25 sq ft.

64. V =

86. r =

88. answers may vary 90. r =

66. SA = 2lh + 2 wh + 2lw = 2(2 in.)(2.2 in.) + 2( 2 in.)(2.2 in.) + 2(2 in.)(2 in.) =8.8 sq in. + 8.8 sq in. + 8 sq in. = 25.6 sq in. The surface area is 25.6 square inches. 68. 7 2 = 7 ⋅ 7 = 49 70. 20 2 = 20 ⋅ 20 = 400 72. 52 + 32 = 5 ⋅ 5 + 3 ⋅ 3 = 25 + 9 = 34 74. 12 + 62 = 1 ⋅1 + 6 ⋅ 6 = 1 + 36 = 37

d 80 ft = = 40 ft 2 2

A = πr 2 = π ⋅ (40 ft)2 = 1600π sq ft ≈ 5024 sq ft The face of the tire has an area of 1600π square feet, or about 5024 square feet. 92. The semicircle at the top has radius 2.5 feet, so 1 1 its area is ⋅ πr 2 = ⋅ π ⋅ (2.5 ft)2 = 3.125π sq ft. 2 2 The rectangle at the bottom has area l ⋅ w = 7 ft ⋅ 5 ft = 35 sq ft. The total area is 3.125π sq ft + 35 sq ft ≈ 44.8 sq ft.

76. Grass seed is planted throughout the surface of a yard, so the situation involves area.

339

Chapter 9: Geometry and Measurement 94. First solid: V = lwh = (6 ft)(2 ft)(4 ft) = 48 cu ft SA = 2lh + 2 wh + 2lw = 2(6 ft)(4 ft) + 2(2 ft)(4 ft) + 2(6 ft)(2 ft) = 88 sq ft Second solid: V = lwh = (4 ft)(3 ft)(4 ft) = 48 cu ft SA = 2lh + 2 wh + 2lw = 2(4 ft)(4 ft) + 2(3 ft)(4 ft) + 2( 4 ft)(3 ft) = 80 sq ft no; answers may vary Mid-Chapter Review 1. The supplement of a 27° angle measures 180° − 27° = 153°. The complement of a 27° angle measures 90° − 27° = 63°. 2. ∠x and the angle marked 105° are supplementary angles, so m∠x = 180° − 105° = 75°. ∠y and the angle marked 105° are vertical angles, so m∠y = 105°.

∠z and the angle marked 105° are supplementary angles, so m∠z = 180° − 105° = 75°. 3. ∠x and the angle marked 52° are supplementary angles, so m∠x = 180° − 52° = 128°. ∠y and the angle marked 52° are corresponding angles, so m∠y = 52°. ∠z and ∠y are supplementary angles, so m∠z = 180° − m∠y = 180° − 52° = 128°. 4. The sum of the measures of the angles of a triangle is 180°. m∠x = 180° − 90° − 38° = 52° 5. d = 2 ⋅ r = 2 ⋅ 2.3 in. = 4.6 in.

1 ⋅d 2 1 1 = ⋅ 8 in. 2 2 1 17 = ⋅ in. 2 2 17 = in. 4 1 = 4 in. 4

6. r =

340

ISM: Prealgebra

7. P = 4 ⋅ s = 4 ⋅ 5 m = 20 m The perimeter is 20 meters. A = s 2 = (5 m)2 = 25 sq m The area is 25 square meters.

8. P = a + b + c = 4 ft + 5 ft + 3 ft = 12 ft The perimeter is 12 feet. 1 1 A = ⋅ b ⋅ h = ⋅ 3 ft ⋅ 4 ft = 6 sq ft 2 2 The area is 6 square feet. 9. C = 2 ⋅ π ⋅ r = 2 ⋅ π ⋅ 5 cm = 10π cm ≈ 31.4 cm The circumference is 10π centimeters ≈ 31.4 centimeters. A = π ⋅ r 2 = π(5 cm) 2 = 25π sq cm ≈ 78.5 sq cm

The area is 25π square centimeters ≈ 78.5 square centimeters. 10. P = 11 mi + 5 mi + 11 mi + 5 mi = 32 mi The perimeter is 32 miles. A = l ⋅ w = 11 mi ⋅ 4 mi = 44 sq mi The area is 44 square meters. 11. The unmarked horizontal side has length 17 cm − 8 cm = 9 cm. The unmarked vertical side has length 7 cm + 3 cm = 10 cm. P = (8 + 3 + 9 + 7 + 17 + 10) cm = 54 cm The perimeter is 54 centimeters. The figure is made up of two rectangles, one with dimensions 10 cm by 8 cm, the other with dimensions 7 cm by 9 cm. A = 10 cm ⋅ 8 cm + 7 cm ⋅ 9 cm = 143 sq cm The area is 143 square centimeters. 12. P = 2 ⋅ l + 2 ⋅ w = 2(17 ft) + 2(14 ft) = 34 ft + 28 ft = 62 ft The perimeter is 62 feet. A = l ⋅ w = 17 ft ⋅ 14 ft = 238 sq ft The area is 238 square feet. 13. P = 2 ⋅ l + 2 ⋅ w = 2(3500 ft) + 2(1400 ft) = 7000 ft + 2800 ft = 9800 ft The perimeter is 9800 feet. A = l ⋅ w = 3500 ft ⋅ 1400 ft = 4,900,000 sq ft The area is 4,900,000 square feet.

ISM: Prealgebra

Chapter 9: Geometry and Measurement

14. V = s 3 = (4 in.)3 = 64 cu in. The volume is 64 cubic inches.

4. 68 in. =

5 12 68 −60 8 Thus, 68 in. = 5 ft 8 in.

SA = 6 ⋅ s 2 = 6(4 in.)2 = 6 ⋅16 sq in. = 96 sq in. The surface area is 96 square inches.

15. V = l ⋅ w ⋅ h = 3 ft ⋅ 2 ft ⋅ 5.1 ft = 30.6 cu ft The volume is 30.6 cubic feet. SA = 2lh + 2 wh + 2lw = 2(3 ft)(5.1 ft) + 2(2 ft)(5.1 ft) + 2(3 ft)(2 ft) = 30.6 sq ft + 20.4 sq ft + 12 sq ft = 63 sq ft The surface area is 63 square feet.

5 yd 3 ft ⋅ = 15 ft 1 1 yd 5 yd 2 ft = 15 ft + 2 ft = 17 ft

5. 5 yd =

4 ft 8 in. + 8 ft 11 in. 12 ft 19 in. Since 19 inches is the same as 1 ft 7 in., we have 12 ft 19 in. = 12 ft + 1 ft 7 in. = 13 ft 7 in.

4 ft 7 in. × 4 16 ft 28 in. Since 28 in. is the same as 2 ft 4 in., we simplify as 16 ft 28 in. = 16 ft + 2 ft 4 in. = 18 ft 4 in.

5 ft 8 in. → 4 ft 20 in. − 1 ft 9 in. − 1 ft 9 in. 3 ft 11 in. The remaining board length is 3 ft 11 in.

1 1 16. V = s 2 h = (10 cm) 2 (12 cm) = 400 cu cm 3 3 The volume is 400 cubic centimeters.

1 1 3 ⋅ d = ⋅ 3 mi = mi 2 2 2 4 3 V = πr 3 3 4 3  = π  mi  3 2  9 = π cu mi 2 1 = 4 π cu mi 2 9 22 1 ≈ ⋅ cu mi = 14 cu mi 2 7 7 The volume is 1 1 4 π cubic miles ≈ 14 cubic miles. 2 7

17. r =

18. The complement of an angle that measures 55° is an angle that measures 90° − 55° = 35°. Section 9.4 Practice Exercises 6 ft 6 ft 12 in. 1. 6 ft = ⋅1 = ⋅ = 6 ⋅12 in. = 72 in. 1 1 1 ft

2. 8 yd =

8 yd 8 yd 3 ft ⋅1 = ⋅ = 8 ⋅ 3 ft = 24 ft 1 1 1 yd

3. 18 in. =

18 in. 1 ft 18 ⋅ = ft = 1.5 ft 1 12 in. 12

68 in. 1 ft 68 ⋅ = ft 1 12 in. 12

9. 2.5 m =

2.5 m 1000 mm ⋅ = 2500 mm 1 1m

10. 3500 m = 3.500 km or 3.5 km 11. 640 m = 0.64 km 2.10 km − 0.64 km 1.46 km

2.1 km = 2100 m 2100 m − 640 m 1460 m 12. 18.3 hm × 5 91.5 hm 13. 0.8 m = 80 cm 80 cm + 45 cm = 125 cm The scarf will be 125 cm or 1.25 m.

341

Chapter 9: Geometry and Measurement Vocabulary, Readiness & Video Check 9.4 1. The basic unit of length in the metric system is the meter. 2. The expression

1 foot is an example of a 12 inches

unit fraction.

ISM: Prealgebra

3.8 mi 5280 ft ⋅ 1 1 mi = 3.8 ⋅ 5280 ft = 20, 064 ft

12. 3.8 mi =

7216 yd 1 mi ⋅ 1 1760 yd 7216 = mi 1760 = 4.1 mi

14. 7216 yd =

3. A meter is slightly longer than a yard. 4. One foot equals 12 inches. 5. One yard equals 3 feet.

16. 129 in. =

6. One yard equals 36 inches. 7. One mile equals 5280 feet.

100 ft 1 yd 100 ⋅ = yd 1 3 ft 3 33 yd 1 ft 3 100 −9 10 −9 1

18. 100 ft =

8. feet; Feet are the original units and we want them to divide out. 9. Both mean addition; 5

129 in. 1 ft 129 ⋅ = ft = 10.75 ft 1 12 in. 12

2 2 = 5 + and 5 5

5 ft 2 in. = 5 ft + 2 in. 10. The sum of 21 yd 4 ft is correct, but is not in a good format since there is a yard in 4 feet. Convert 4 ft = 1 yd 1 ft and add again: 21 yd + 1 yd + 1 ft = 22 yd 1 ft. 11. Since the metric system is based on base 10, we just need to move the decimal point to convert from one unit to another. 12. 1.29 cm and 12.9 mm; these two different-unit lengths are equal. Exercise Set 9.4 2. 84 in. =

84 in. 1 ft 84 ⋅ = ft = 7 ft 1 12 in. 12

4. 18 yd =

18 yd 3 ft ⋅ = 18 ⋅ 3 ft = 54 ft 1 1 yd

36,960 ft 1 mi ⋅ 1 5280 ft 36,960 mi = 5280 = 7 mi

59 in. 1 ft 59 ⋅ = ft 1 12 in. 12 4 ft 11 in. 12 59 −48 11

20. 59 in. =

25, 000 ft 1 mi 25, 000 ⋅ = mi 1 5280 ft 5280 4 mi 3880 ft 5280 25, 000 − 21,120 3 880

22. 25, 000 ft =

6 ft 12 in. ⋅ + 10 in. 1 1 ft = 72 in. + 10 in. = 82 in.

24. 6 ft 10 in. =

6. 36,960 ft =

8. 12

1 12.5 ft 12 in. ft = ⋅ = 12.5 ⋅12 in. = 150 in. 2 1 1 ft

10. 25 ft = 342

25 ft 1 yd 25 1 ⋅ = yd = 8 yd 1 3 ft 3 3

26. 4 yd 1 ft =

4 yd 3 ft ⋅ + 1 ft = 12 ft + 1 ft = 13 ft 1 1 yd

28. 1 yd 2 ft =

1 yd 3 ft ⋅ + 2 ft = 3 ft + 2 ft = 5 ft 1 1 yd

5 ft =

5 ft 12 in. ⋅ = 5 ⋅12 in. = 60 in. 1 1 ft

ISM: Prealgebra

Chapter 9: Geometry and Measurement

30. 12 ft 7 in. + 9 ft 11 in. = 21 ft 18 in. = 21 ft + 1 ft 6 in. = 22 ft 6 in. 32. 16 yd 2 ft + 8 yd 2 ft = 24 yd 4 ft = 24 yd + 1 yd 1 ft = 25 yd 1 ft 34.

15 ft 5 in. − 8 ft 2 in. 7 ft 3 in.

36.

14 ft 8 in. → 13 ft 20 in. − 3 ft 11 in. − 3 ft 11 in. 10 ft 9 in.

38. 34 ft 6 in. ÷ 2 = 17 ft 3 in. 40.

15 yd 1 ft × 8 120 yd 8 ft = 120 yd + 2 yd 2 ft = 122 yd 2 ft

42. 46 m =

46 m 100 cm ⋅ = 4600 cm 1 1m

44. 14 mm =

14 mm 1 cm ⋅ = 1.4 cm 1 10 mm

46. 400 m =

400 m 1 km ⋅ = 0.4 km 1 1000 m

48. 6400 mm =

6400 mm 1m ⋅ = 6.4 m 1 1000 mm

50. 6400 cm =

6400 cm 1 m ⋅ = 64 m 1 100 cm

52. 0.95 km =

0.95 km 100, 000 cm ⋅ = 95, 000 cm 1 1 km

54. 5 km =

5 km 1000 m ⋅ = 5000 m 1 1 km

56. 4.6 cm =

4.6 cm 10 mm ⋅ = 46 mm 1 1 cm

58. 140.2 mm = 60. 0.2 m =

140.2 mm 1 dm ⋅ = 1.402 dm 1 100 mm

0.2 m 1000 mm ⋅ = 200 mm 1 1

343

Chapter 9: Geometry and Measurement 62.

14.10 cm + 3.96 cm 18.06 cm

64.

30 cm + 8.9 m

66.

45.3 m − 2.16 dam

68.

14 cm − 15 mm

ISM: Prealgebra

30 cm 0.3 m + 890 cm or + 8.9 m 920 cm 9.2 m 45.3 m 4.53 dam − 2.16 dam or − 21.6 m 23.7 m 2.37 dam 14.0 cm 140 mm − 1.5 cm or − 15 mm 12.5 cm 125 mm

70. 14.1 m × 4 = 56.4 m 72. 9.6 m ÷ 5 = 1.92 m

Yards

Feet

Inches

74.

Four-story building

792

76.

Ostrich height

108 Meters

Millimeters

Kilometers

Centimeters

78.

Height of grizzly bear

3000

0.003

300

80.

Golf ball diameter

0.046

0.000046

4.6

82.

Distance from Houston to Dallas

396,000

396,000,000

396

39,600,000

84.

1 mi 1400 ft + 1 mi 4000 ft 2 mi 5400 ft = 2 mi + 1 mi 120 ft = 3 mi 120 ft The probe is 3 mi 120 ft below the surface.

86.

1 mi → 2640 ft 2 − 1150 ft − 1150 ft 1490 ft At their narrowest points, the Grand Canyon of the Yellowstone is 1490 feet wider than the Black Canyon of the Gunnison.

88. 3.4 m + 5.8 m = 9.2 m 9.2 m 9.20 m − 8 cm − 0.08 m 9.12 m The length of the tied ropes is 9.12 m.

344

ISM: Prealgebra 90.

Chapter 9: Geometry and Measurement

14 cm − 22 mm

14.0 cm − 2.2 cm 11.8 cm The sediment is 11.8 cm thick now.

92. 1 yd 2 ft × 25 = 25 yd 50 ft = 25 yd + 16 yd 2 ft = 41 yd 2 ft The total length is 41 yd 2 ft.

114. answers may vary Section 9.5 Practice Exercises 6500 lb 1 ton ⋅ 1 2000 lb 6500 = tons 2000 13 1 = tons or 3 tons 4 4

12.5 94. 9 112.5 −9 22 −18 45 −4 5 0 12 whole sections are removed.

1. 6500 lb =

2. 72 oz =

4837 ft 1 yd 4837 1 ⋅ = yd = 1612 yd 1 3 ft 3 3 1 4837 feet is 1612 yards. 3

96. 4837 ft =

98. 0.86 =

112. answers may vary; for example, 7 m 100 cm 7m= ⋅ = 700 cm 1 1m 7 m 1000 mm 7m= ⋅ = 7000 mm 1 1m

1 lb 47 = lb 16 oz 16 2 lb 15 oz 16 47 −32 15 Thus, 47 oz = 2 lb 15 oz

3. 47 oz = 47 oz ⋅

86 43 = 100 50

47 100. = 0.47 100 3 3 5 15 = ⋅ = = 0.15 20 20 5 100

72 oz 1 lb 72 9 1 ⋅ = lb = lb or 4 lb 1 16 oz 16 2 2

8 tons 100 lb 7 tons 2100 lb − 5 tons 1200 lb → − 5 tons 1200 lb 2 tons 900 lb

104. Yes, a window measuring 1 meter by 0.5 meter is reasonable.

1 lb 6 oz 5. 4 5 lb 8 oz − 4 lb 1 lb = 16 oz 24 oz

106. No, a paper clip being 4 kilometers long is not reasonable.

102.

108. Yes, a model’s hair being 30 centimeters long is reasonable. 110.

45 ft 1 in. − 10 ft 11 in. Estimate: 45 ft − 11 ft = 34 ft

batch weight 5 lb 14 oz + container weight → + 6 oz 5 lb 20 oz total weight 5 lb 20 oz = 5 lb + 1 lb 4 oz = 6 lb 4 oz The total weight is 6 lb 4 oz.

7. 3.41 g =

3.41 g 1000 mg ⋅ = 3410 mg 1 1g

8. 56.2 cg = 56.2  cg = 0.562 g

345

Chapter 9: Geometry and Measurement 9. 3.1 dg = 0.31 g 2.50 g − 0.31 g 2.19 g

2.5 g = 25 dg 25.0 dg − 3.1 dg 21.9 dg

22.9 ≈ 23 550.0 kg −48 70 −48 22 0 −21 6 4 Each bag weighs about 23 kg.

ISM: Prealgebra

8. 90 oz =

90 oz 1 lb 90 45 5 ⋅ = lb = lb = 5 lb 1 16 oz 16 8 8

11, 000 lb 1 ton ⋅ 1 2000 lb 11, 000 = tons 2000 11 = tons 2 1 = 5 tons 2

10. 11, 000 lb =

10. 24

12. 9.5 lb =

Vocabulary, Readiness & Video Check 9.5 1. Mass is a measure of the amount of substance in an object. This measure does not change.

8.3 tons 2000 lb ⋅ 1 1 ton = 8.3 ⋅ 2000 lb = 16, 600 lb

14. 8.3 tons =

2. Weight is a measure of the pull of gravity. 3. The basic unit of mass in the metric system is the gram.

16. 9

4. One pound equals 16 ounces. 5. One ton equals 2000 pounds. 6. pounds; Pounds are the units we’re converting to. 7. We can’t subtract 9 oz from 4 oz, so we borrow 1 lb (= 16 oz) from 12 lb to add to the 4 oz: 12 lb 4 oz becomes 11 lb 20 oz.

51 oz 1 lb ⋅ 1 16 oz 51 = lb 16 = 3.1875 lb ≈ 3.2 lb

9. 18.50 dg

2. 5 lb =

20.

5 lb 16 oz ⋅ = 5 ⋅16 oz = 80 oz 1 1 lb 7 tons 2000 lb ⋅ 1 1 ton = 7 ⋅ 2000 lb = 14, 000 lb

4. 7 tons =

1 73 lb = lb 8 8 73 lb 16 oz = 8 ⋅ 1 1 lb 73 = ⋅16 oz 8 = 146 oz

18. 51 oz =

8. 3 places to the right; 4 g = 4000 mg

Exercise Set 9.5

9.5 lb 16 oz ⋅ = 9.5 ⋅16 oz = 152 oz 1 1 lb

1 oz 1 1 lb 1 1 1 oz = 4 ⋅ = ⋅ lb = lb 4 1 16 oz 4 16 64

22. 2

9 lb 1 9 16 oz 9 lb = lb = 4 ⋅ = ⋅16 oz = 36 oz 4 4 1 1 lb 4

24. 7 lb 6 oz = 7 ⋅16 oz + 6 oz = 112 oz + 6 oz = 118 oz

28, 000 lb 1 ton ⋅ 1 2000 lb 28, 000 = tons 2000 = 14 tons

6. 28, 000 lb =

346

ISM: Prealgebra

Chapter 9: Geometry and Measurement

100 oz 1 lb 100 ⋅ = lb 1 16 oz 16 6 lb 4 oz 16 100 − 96 4 100 oz = 6 lb 4 oz

26. 100 oz =

28. 6 lb 10 oz + 10 lb 8 oz = 16 lb 18 oz = 16 lb + 1 lb 2 oz = 17 lb 2 oz 30. 1 ton 1140 lb + 5 tons 1200 lb = 6 tons 2340 lb = 6 tons + 1 ton 340 lb = 7 tons 340 lb 32.

4 tons 850 lb − 1 ton 260 lb 3 tons 590 lb

34.

45 lb 6 oz − 26 lb 10 oz

44 lb 22 oz − 26 lb 10 oz 18 lb 12 oz

38. 5 tons 400 lb = 4 tons + 1 ton 400 lb = 4 tons + 2000 lb + 400 lb = 4 tons 2400 lb 4 2400 4 tons 2400 lb ÷ 4 = ton lb 4 4 = 1 ton 600 lb 820 g 1 kg 820 40. 820 g = ⋅ = kg = 0.82 kg 1 1000 g 1000 9 g 1000 mg ⋅ = 9 ⋅1000 mg = 9000 mg 1 1g

18 kg 1000 g 44. 18 kg = ⋅ = 18 ⋅1000 g = 18, 000 g 1 1 kg

46. 112 mg =

48. 4.9 g =

3.16 kg 1000 g ⋅ 1 1 kg = 3.16 ⋅1000 g = 3160 g

52. 3.16 kg =

4.26 cg 1 dag ⋅ 1 1000 cg 4.26 = dag 1000 = 0.00426 dag

54. 4.26 cg =

56.

36. 2 lb 5 oz × 5 = 10 lb 25 oz = 10 lb + 1 lb 9 oz = 11 lb 9 oz

42. 9 g =

16.23 g 1000 mg ⋅ 1 1g = 16.23 ⋅1000 mg = 16, 230 mg

50. 16.23 g =

112 mg 1g 112 ⋅ = g = 0.112 g 1 1000 mg 1000

4.9 g 1 kg 4.9 ⋅ = kg = 0.0049 kg 1 1000 g 1000

41.6 g + 9.8 g 51.4 g

58. 2.1 g + 153 mg = 2.1 g + 0.153 g = 2.253 g or 2.1 g + 153 mg = 2100 mg + 153 mg = 2253 mg 60. 6.13 g − 418 mg = 6130 mg − 418 mg = 5712 mg or 6.13 g − 418 mg = 6.13 g − 0.418 g = 5.712 g 62. 4 kg − 2410 g 4000 g 4.00 kg − 2.41 kg or − 2140 g 1590 g 1.59 kg 64.

4.8 kg × 9.3 44.64 kg

66. 8.25 g ÷ 6 =

8.25 g 6

1.375 6 8.250 −6 22 −1 8 45 −42 30 −30 0

347

Chapter 9: Geometry and Measurement

ISM: Prealgebra

Object

Tons

Pounds

Ounces

68.

Statue of Liberty⎯weight of steel

125

250,000

4,000,000

70.

A 12-inch cube of lithium (lightest metal)

or 0.016

512

2 125

Object

Grams

Kilograms

Milligrams

Centigrams

72.

Tablet of Topamax (epilepsy and migraine uses)

0.025

0.000025

2.5

74.

A golf ball

0.045

45,000

4500

76.

73 kg → 73.0 kg − 2800 g − 2.8 kg 70.2 kg His new weight is 70.2 kg.

78. 177 g ÷ 6 =

177 g 6

29.5 6 177.0 −12 57 −54 30 −3 0 0 Each serving weighs 29.5 g.

80.

3 tons 1500 lb + 2 tons 1200 lb 5 tons 2700 lb = 5 tons + 1 ton 700 lb = 6 tons 700 lb The total amount delivered was 6 tons 700 lb.

82.

22 lb 8 oz − 7 lb 12 oz

21 lb 24 oz − 7 lb 12 oz 14 lb 12 oz This is 14 lb 12 oz heavier.

84. 1900 g × 6 × 5 = 57,000 g 57, 000 g 1 kg ⋅ = 57 kg 1 1000 g The weight of 5 cartons is 57 kg. 86. 0.385 kg − 39 g = 0.385 kg − 0.039 kg = 0.346 kg 0.346 kg × 8 = 2.768 kg 8 boxes contain 2.768 kg of cocoa.

348

ISM: Prealgebra

Chapter 9: Geometry and Measurement

85 g = 42.5 g 2 Each package will weigh 42.5 g.

5 qt 1 gal 1 qt − 2 qt → − 2 qt 3 qt

1 lb 2 oz × 12 12 lb 24 oz = 12 lb + 1 lb 8 oz = 13 lb 8 oz Each carton weighs 3 lb 8 oz. 13 lb 8 oz × 3 39 lb 24 oz = 39 lb + 1 lb 8 oz = 40 lb 8 oz 3 cartons weigh 40 lb 8 oz.

15 gal 3 qt + 4 gal 3 qt 19 gal 6 qt = 19 gal + 1 gal 2 qt = 20 gal 2 qt The total amount of oil will be 20 gal 2 qt.

5. 2100 ml =

2100 ml 1L 2100 ⋅ = L = 2.1 L 1 1000 ml 1000

43 lb 2 oz − 3 lb 4 oz

6. 2.13 dal =

2.13 dal 10 L ⋅ = 2.13 ⋅10 L = 21.3 L 1 1 dal

88. 85 g ÷ 2 =

90.

92.

42 lb 18 oz − 3 lb 4 oz 39 lb 14 oz

39 lb 14 oz × 3 117 lb 42 oz = 117 lb + 2 lb 10 oz = 119 lb 10 oz 3 cartons contain 119 lb 10 oz of ham.

94.

3 3 20 60 = ⋅ = = 0.60 or 0.6 5 5 20 100

96.

3 3 625 1875 = ⋅ = = 0.1875 16 16 625 10, 000

98. No, a full grown cat weighing 15 g is not reasonable. 100. Yes, a staple weighing 15 mg is reasonable. 102. No, a car weighing 2000 mg is not reasonable. 104. answers may vary; for example, 2 tons or 64,000 oz 106. true

28.6 L × 85 143 0 2288 0 2431.0 L Thus, 2431 L can be pumped in 85 minutes.

Vocabulary, Readiness & Video Check 9.6 1. Units of capacity are generally used to measure liquids. 2. The basic unit of capacity in the metric system is the liter. 3. One cup equals 8 fluid ounces. 4. One quart equals 2 pints. 5. One pint equals 2 cups.

108. answers may vary

6. One quart equals 4 cups.

Section 9.6 Practice Exercises

7. One gallon equals 4 quarts.

43 pt 1 qt 43 1 ⋅ = qt = 21 qt 1. 43 pt = 1 2 pt 2 2

2. 26 qt =

7. 1250 ml = 1.250 L 2.9 L = 2900 ml 1.25 L 1250 ml + 2.9 L + 2900 ml 4.15 L 4150 ml The total is 4.15 L or 4150 ml.

26 qt 4 c ⋅ = 26 ⋅ 4 c = 104 c 1 1 qt

8. When using a unit fraction, we are not changing the amount, we are changing the unit. 9. We can’t subtract 3 qt from 0 qt, so we borrow 1 gal (= 4 qt) from 3 gal to get 2 gal 4 qt. 10. 3 places to the left; 5600 ml = 5.6 L 11. 0.45 dal

349

Chapter 9: Geometry and Measurement Exercise Set 9.6 2. 16 qt =

4. 9 pt =

6. 11 c =

9 pt 1 qt 9 1 ⋅ = qt = 4 qt 1 2 pt 2 2 11 c 1 pt 11 1 ⋅ = pt = 5 pt 1 2c 2 2

8. 18 pt = 9 qt =

10. 3 pt =

16 qt 1 gal 16 ⋅ = gal = 4 gal 1 4 qt 4

18 pt 1 qt 18 ⋅ = qt = 9 qt 1 2 pt 2

9 qt 1 gal 9 1 ⋅ = gal = 2 gal 1 4 qt 4 4 3 pt 2 c 8 fl oz ⋅ ⋅ = 3 ⋅ 2 ⋅ 8 fl oz = 48 fl oz 1 1 pt 1 c

20 c 1 pt 1 qt 1 gal ⋅ ⋅ ⋅ 1 2 c 2 pt 4 qt 20 gal = 2⋅2⋅4 5 = gal 4 1 = 1 gal 4

ISM: Prealgebra 22. 70 qt = 68 qt + 2 qt 68 qt 1 gal = ⋅ + 2 qt 1 4 qt 68 = gal + 2 qt 4 = 17 gal 2 qt 24. 29 pt = 28 pt + 1 pt 28 pt 1 qt = ⋅ + 1 pt 1 2 pt = 14 qt + 1 pt = 12 qt + 2 qt + 1 pt 12 qt 1 gal = ⋅ + 2 qt + 1 pt 1 4 qt = 3 gal + 2 qt + 1 pt = 3 gal 2 qt 1 pt 26. 3

12. 20 c =

14. 7 qt =

16. 6

28.

7 qt 2 pt 2 c ⋅ ⋅ = 7 ⋅ 2 ⋅ 2 c = 28 c 1 1 qt 1 pt

1 13 gal = gal 2 2 13 gal 4 qt = 2 ⋅ 1 1 gal 13 = ⋅ 4 qt 2 = 26 qt

2 pt − 1 pt 1 c

1 pt 2 c − 1 pt 1 c 1c

34. 3 qt 1 c − 1 c 4 fl oz 2 qt 1 pt 3 c − 1 c 4 fl oz

4 gal 4 qt ⋅ + 1 qt 1 1 gal = 4 ⋅ 4 qt + 1 qt = 16 qt + 1 qt = 17 qt

350

2 gal 2 qt + 9 gal 3 qt 11 gal 5 qt = 11 gal + 1 gal 1 qt = 12 gal 1 qt

30. 2 c 3 fl oz + 2 c 6 fl oz = 4 c 9 fl oz = 4 c + 1 c 1 fl oz = 5 c 1 fl oz 32.

18. 4 gal 1 qt =

20.

1 13 qt = qt 4 4 13 qt 2 pt 2 c = 4 ⋅ ⋅ 1 1 qt 1 pt 13 = ⋅2⋅2 c 4 = 13 c

1 pt 1 1 qt 1 1 1 pt = 2 ⋅ = ⋅ qt = qt 2 1 2 pt 2 2 4

2 qt 2 pt 1 c − 1 c 4 fl oz 2 qt 1 pt 2 c 8 fl oz 1 c 4 fl oz − 2 qt 1 pt 1 c 4 fl oz

36. 6 gal 1 pt × 2 = 12 gal 2 pt = 12 gal + 1 qt = 12 gal 1 qt

ISM: Prealgebra

Chapter 9: Geometry and Measurement

5 6 gal fl oz 2 2 1 = 2 gal 3 fl oz 2 1 = 2 gal + gal + 3 fl oz 2 = 2 gal 2 qt 3 fl oz

38. 5 gal 6 fl oz ÷ 2 =

40. 8 L =

8 L 1000 ml ⋅ = 8000 ml 1 1L

0.127 L 1 kl ⋅ 1 1000 L 0.127 = kl 1000 = 0.000127 kl

42. 0.127 L =

44. 1500 ml = 46. 1.7 L =

1500 ml 1L 1500 ⋅ = L = 1.5 L 1 1000 ml 1000

1.7 L 100 cl ⋅ = 1.7 ⋅100 cl = 170 cl 1 1L

48. 250 L =

250 L 1 kl 250 ⋅ = kl = 0.25 kl 1 1000 L 1000

50. 39 ml =

39 ml 1L 39 ⋅ = L = 0.039 L 1 1000 ml 1000

0.48 kl 1000 L ⋅ 1 1 kl = 0.48 ⋅1000 L = 480 L

52. 0.48 kl =

54. 1.9 L =

1.9 L 1000 ml ⋅ = 1.9 ⋅1000 ml = 1900 ml 1 1L

56. 18.5 L + 4.6 L = 23.1 L 58. 4.6 L = 4600 ml 4600 ml + 1600 ml 6200 ml

1600 ml = 1.6 L 4.6 L + 1.6 L 6.2 L

60.

4.8 L − 283 ml

4.800 L 4800 ml − 0.283 L or − 283 ml 4.517 L 4517 ml

62.

6850 ml − 0.3 L

6.85 L 6850 ml − 300 ml or − 0.30 L 6.55 L 6550 ml

351

Chapter 9: Geometry and Measurement

ISM: Prealgebra

64. 290 ml × 6 = 1740 ml 66. 5.4 L ÷ 3.6 =

5.4 L = 1.5 L 3.6

Capacity

Cups

Gallons

Quarts

Pints

68.

A dairy cow’s daily milk yield

4 34

70.

The amount of water needed in a punch recipe

1 8

1 2

72.

54.5 L − 3.8 L 50.7 L 50.7 L are needed to fill the tank. 2 1 L = L = 0.25 L 8 4 Each person (Avery plus 7 friends) will get 0.25 L.

74. 2 L ÷ 8 =

76.

1 qt 1 pt × 3 3 qt 3 pt = 3 qt + 1 qt 1 pt = 4 qt + 1 pt = 1 gal 1 pt The total amount of soup is 1 gal 1 pt. 3 c 8 fl oz ⋅ = 3 ⋅ 8 fl oz = 24 fl oz 1 1c 24 24 fl oz ÷ 6 = fl oz = 4 fl oz 6 Each dish has 4 fl oz.

78. 3 c =

40 cl 10 ml ⋅ − 40 ml 1 1 cl = 400 ml − 40 ml = 360 ml The excess amount of water is 360 ml.

80. 40 cl − 40 ml =

82.

75 3 ⋅ 25 3 = = 100 4 ⋅ 25 4

84.

56 14 ⋅ 4 14 = = 60 15 ⋅ 4 15

86.

18 9 ⋅ 2 9 = = 20 10 ⋅ 2 10

88. Yes, drinking 250 ml of milk with lunch is reasonable. 90. Yes, a gas tank of a car holding 60 L of gasoline is reasonable. 352

ISM: Prealgebra

Chapter 9: Geometry and Measurement

92. greater than; answers may vary 94. answers may vary

1.5 gal 4 qt 2 pt 2 c 8 fl oz ⋅ ⋅ ⋅ ⋅ 1 1 gal 1 qt 1 pt 1 c = 1.5 ⋅ 4 ⋅ 2 ⋅ 2 ⋅ 8 fl oz = 192 fl oz There are 192 fl oz in 1.5 gallons.

96. 1.5 gal =

98. A indicates 0.6 cc. 100. C indicates 1.9 cc. 102. A indicates 38 u or 0.38 cc. 104. C indicates 72 u or 0.72 cc. Section 9.7 Practice Exercises 1. 1.5 cm = 2. 8 oz ≈

1.5 cm 1 in. 1.5 ⋅ = in. ≈ 0.59 in. 1 2.54 cm 2.54

8 oz 28.35 g ⋅ = 8 ⋅ 28.35 g = 226.8 g 1 1 oz

237 ml 1 fl oz ⋅ 1 29.57 ml 237 = fl oz 29.57 ≈ 8 fl oz

3. 237 ml ≈

9 9 4. F = ⋅ C + 32 = ⋅ 60 + 32 = 108 + 32 = 140 5 5 Thus, 60°C is equivalent to 140°F.

5. F = 1.8 ⋅ C + 32 = 1.8 ⋅ 32 + 32 = 57.6 + 32 = 89.6 Therefore, 32°C is the same as 89.6°F. 5 5 5 6. C = ⋅ (F − 32) = ⋅ (68 − 32) = ⋅ (36) = 20 9 9 9 Therefore, 68°F is the same temperature as 20°C. 5 5 5 7. C = ⋅ (F − 32) = ⋅ (113 − 32) = ⋅ (81) = 45 9 9 9 Therefore, 113°F is 45°C.

353

Chapter 9: Geometry and Measurement

ISM: Prealgebra

5 8. C = ⋅ (F − 32) 9 5 = ⋅ (102.8 − 32) 9 5 = ⋅ (70.8) 9 = 39.3 Albert’s temperature is 39.3°C.

Vocabulary, Readiness & Video Check 9.7 1. 1 L ≈ 0.26 gal or 3.79 L ≈ 1 gal 2. The original stated example asked for the answer in grams, so a conversion from kg to g is still needed. 3. F = 1.8C + 32; 27 4. 77; 77°F = 25°C Exercise Set 9.7 2. 18 L ≈

18 L 1.06 qt ⋅ = 18 ⋅1.06 qt = 19.08 qt 1 1L

86 mi 1.61 km ⋅ 1 1 mi = 86 ⋅1.61 km = 138.46 km

4. 86 mi ≈

6. 100 kg ≈

8. 9.8 m ≈

100 kg 2.20 lb ⋅ = 100 ⋅ 2.2 lb = 220 lb 1 1 kg

9.8 m 3.28 ft ⋅ = 9.8 ⋅ 3.28 ft ≈ 32.14 ft 1 1m

150 ml 1 fl oz ⋅ 1 29.57 ml 150 = fl oz 29.57 ≈ 5.07 fl oz

10. 150 ml ≈

12. 15 oz ≈

15 oz 28.35 g ⋅ = 15 ⋅ 28.35 g = 425.25 g 1 1 oz

Meters

Yards

Centimeters

Feet

Inches

14.

Statue of Liberty Length of Nose

1.37

1.5

137

4.5

16.

Blue Whale

3300

108

1296

354

ISM: Prealgebra

Chapter 9: Geometry and Measurement

250 cm 1 in. ⋅ ≈ 98.43 in. 1 2.54 cm 98.43 in. 1 ft 98.43 in. = ⋅ ≈ 8.20 ft 1 12 in. The rings are approximately 98.43 inches or 8.20 feet from the floor.

9 lb 0.45 kg ⋅ = 9 ⋅ 0.45 kg = 4.05 kg 1 1 lb 4.05 kg 1000 g 4.05 kg = ⋅ = 4050 g 1 1 kg The organ weighs approximately 4050 g.

18. 250 cm =

34. 9 lb ≈

70 mph 1.61 km ⋅ = 112.7 kph 20. 70 mph ≈ 1 1 mi The speed limit is approximately 112.7 kilometers per hour.

36. 906 kph ≈

0.5 g 0.04 oz ⋅ = 0.02 oz 1 1g The tablets are approximately 0.02 ounce.

22. 500 mg = 0.5 g ≈

24. 26 m ≈

26 m 3.28 ft ⋅ = 85.28 ft 1 1m

26. 2.7 million kg 2, 700, 000 kg 2.20 lb 1 ton ≈ ⋅ ⋅ 1 1 kg 2000 lb = 2970 tons 5 ft 12 in. ⋅ + 11 in. 1 1 ft = 5 ⋅12 in. + 11 in. = 60 in. + 11 in. = 71 in. 71 in. 2.54 cm = ⋅ 1 1 in. = 71 ⋅ 2.54 cm = 180.34 cm Yes; the stamp is approximately correct.

28. 5 ft 11 in. =

18 ft 0.30 m ⋅ = 5.4 m 1 1 ft The nests are up to about 5.4 meters.

30. 18 ft ≈

84 cm 1 in. 32. 84 cm = ⋅ ≈ 33 in. 1 2.54 cm 33 in. = 24 in. + 9 in. 24 in. 1 ft = ⋅ + 9 in. 1 12 in. = 2 ft 9 in. The average two-year-old is approximately 2 ft 9 in.

906 kph 0.62 mi ⋅ ≈ 562 mph 1 1 km The speed is approximately 562 miles per hour.

303 km 0.62 mi ⋅ = 187.86 mi 1 1 km The diameter is approximately 187.86 mi.

38. 303 km ≈

2079 km 0.62 mi ⋅ ≈ 1289 mi 1 1 km The distance is approximately 1289 mi.

40. 2079 km ≈

24 = 4 doses 6 per day and 4 × 10 = 40 doses for 10 days. 12 ml × 40 = 480 ml 480 ml 1 fl oz 480 ml ≈ ⋅ ≈ 16.23 fl oz 1 29.57 ml 17 fluid ounces of medicine should be purchased.

42. One dose every 6 hours results in

44. A mile is longer than a kilometer; b. 46. A foot is shorter than a meter; a 48. A football field is 100 yards, which is about 90 m, since a yard is a little less than a meter; b 50. A 5-gallon gasoline can has a capacity of about 19 L (1 gal ≈ 3.79 L); a 52. The weight of a pill is about 200 mg; d 5 54. C = (F − 32) 9 5 = (86 − 32) 9 5 = (54) 9 = 30 86°F is 30°C. 5 5 5 56. C = (F − 32) = (140 − 32) = (108) = 60 9 9 9 140°F is 60°C.

355

Chapter 9: Geometry and Measurement

ISM: Prealgebra 76. F = 1.8C + 32 = 1.8(127) + 32 = 228.6 + 32 = 260.6 127°C is 260.6°F.

9 58. F = C + 32 5 9 = (80) + 32 5 = 144 + 32 = 176 80°C is 176°F. 9 9 60. F = C + 32 = (225) + 32 = 405 + 32 = 437 5 5 225°C is 437°F. 5 5 5 62. C = (F − 32) = (26 − 32) = ( −6) ≈ −3.3 9 9 9 26°F is −3.3°C.

5 78. C = (F − 32) 9 5 = (58.3 − 32) 9 5 = (26.3) 9 ≈ 14.6 58.3°F is 14.6°C.

80. 10 ÷ 2 + 9(8) = 5 + 9(8) = 5 + 72 = 77

5 5 5 64. C = (F − 32) = (43.4 − 32) = (11.4) ≈ 6.3 9 9 9 43.4°F is 6.3°C.

82. 5[(18 − 8) − 9] = 5[10 − 9] = 5(1) = 5

66. F = 1.8C + 32 = 1.8(75) + 32 = 135 + 32 = 167 75°C is 167°F.

86. Yes, an air temperature of 20°F on a Vermont ski slope in winter is reasonable.

68. F = 1.8C + 32 = 1.8(48.6) + 32 = 87.48 + 32 ≈ 119.5 48.6°C is 119.5°F.

90. No, water cooled to 32°C will not freeze.

88. No, a room temperature of 60°C does not feel chilly.

63 × 157 ≈ 1.66 3600 The BSA is approximately 1.66 sq m.

92. BSA =

5 70. C = (F − 32) 9 5 = (95 − 32) 9 5 = (63) 9 = 35 95°F is 35°C.

94. 26 in. =

356

26 in. 2.54 cm ⋅ = 66.04 cm 1 1 in.

13 × 66.04 ≈ 0.49 3600 The BSA is approximately 0.49 sq m. BSA =

69 in. 2.54 cm ⋅ = 175.26 cm 1 1 in. 172 lb 0.45 kg 172 lb ≈ ⋅ = 77.4 kg 1 1 lb

96. 69 in. =

72. F = 1.8C + 32 = 1.8(18) + 32 = 32.4 + 32 = 64.4 18°C is 64.4°F. 5 74. C = (F − 32) 9 5 = (66 − 32) 9 5 = (34) 9 ≈ 18.9 66°F is 18.9°C.

84. False; the boiling point of water is 212°F.

77.4 × 175.26 ≈ 1.94 3600 The BSA is approximately 1.94 sq m. BSA =

ISM: Prealgebra

Chapter 9: Geometry and Measurement

5 98. C = (F − 32) 9 5 = (9010 − 32) 9 5 = (8978) 9 ≈ 4988 9010°F is 4988°C.

16. The measure of the space of a solid is called its volume.

100. answers may vary

19. An angle whose measure is between 90° and 180° is called an obtuse angle.

17. When two lines intersect, four angles are formed. Two of these angles that are opposite each other are called vertical angles. 18. Two of the angles from Exercise 17 that share a common side are called adjacent angles.

Chapter 9 Vocabulary Check 1. Weight is a measure of the pull of gravity.

20. An angle that measures 90° is called a right angle.

2. Mass is a measure of the amount of substance in an object. This measure does not change.

21. An angle whose measure is between 0° and 90° is called an acute angle.

3. The basic unit of length in the metric system is the meter. 4. To convert from one unit of length to another, unit fractions may be used. 5. The gram is the basic unit of mass in the metric system. 6. The liter is the basic unit of capacity in the metric system.

22. Two angles that have a sum of 180° are called supplementary angles. 23. The surface area of a polyhedron is the sum of the areas of the faces of the polyhedron. Chapter 9 Review 1. ∠A is a right angle. It measures 90°. 2. ∠B is a straight angle. It measures 180°.

7. A line segment is a piece of a line with two endpoints.

3. ∠C is an acute angle. It measures between 0° and 90°.

8. Two angles that have a sum of 90° are called complementary angles.

4. ∠D is an obtuse angle. It measures between 90° and 180°.

9. A line is a set of points extending indefinitely in two directions.

5. The complement of a 25° angle has measure 90° − 25° = 65°.

10. The perimeter of a polygon is the distance around the polygon.

6. The supplement of a 105° angle has measure 180° − 105° = 75°.

11. An angle is made up of two rays that share the same endpoint. The common endpoint is called the vertex.

7. m∠x = 90° − 32° = 58°

12. Area measures the amount of surface of a region.

9. m∠x = 105° − 15° = 90°

13. A ray is a part of a line with one endpoint. A ray extends indefinitely in one direction.

10. m∠x = 45° − 20° = 25°

14. A line that intersects two or more lines at different points is called a transversal.

8. m∠x = 180° − 82° = 98°

11. 47° + 133° = 180°, so ∠a and ∠b are supplementary. So are ∠b and ∠c, ∠c and

∠d , and ∠d and ∠a.

15. An angle that measures 180° is called a straight angle.

357

Chapter 9: Geometry and Measurement

12. 47° + 43° = 90°, so ∠x and ∠w are complementary. Also, 58° + 32° = 90°, so ∠y and ∠z are complementary. 13. ∠x and the angle marked 100° are vertical angles, so m∠x = 100°. ∠x and ∠y are adjacent angles, so m∠y = 180° − 100° = 80°. ∠y and ∠z are vertical angles, so m∠z = m∠y = 80°. 14. ∠x and the angle marked 25° are adjacent angles, so m∠x = 180° − 25° = 155°. ∠x and ∠y are vertical angles, so m∠y = m∠x = 155°.

∠z and the angle marked 25° are vertical angles, so m∠z = 25°. 15. ∠x and the angle marked 53° are vertical angles, so m∠x = 53°. ∠x and ∠y are alternate interior angles, so m∠y = m∠x = 53°. ∠y and ∠z are adjacent angles, so m∠z = 180° − m∠y = 180° − 53° = 127°. 16. ∠x and the angle marked 42° are vertical angles, so m∠x = 42°. ∠x and ∠y are alternate interior angles, so m∠y = m∠x = 42°. ∠y and ∠z are adjacent angles, so m∠z = 180° − m∠y = 180° − 42° = 138°. 1 1 17. P = 23 m + 11 m + 23 m + 11 m = 69 m 2 2 The perimeter is 69 meters.

18. P = 11 cm + 7.6 cm + 12 cm = 30.6 cm The perimeter is 30.6 centimeters. 19. The unmarked vertical side has length 8 m − 5 m = 3 m. The unmarked horizontal side has length 10 m − 7 m = 3 m. P = (7 + 3 + 3 + 5 + 10 + 8) m = 36 m The perimeter is 36 meters. 20. The unmarked vertical side has length 5 ft + 4 ft + 11 ft = 20 ft. P = (22 + 20 + 22 + 11 + 3 + 4 + 3 + 5) ft = 90 ft The perimeter is 90 feet.

358

ISM: Prealgebra

21. P = 2 ⋅ l + 2 ⋅ w = 2 ⋅ 10 ft + 2 ⋅ 6 ft = 32 ft The perimeter is 32 feet. 22. P = 4 ⋅ s = 4 ⋅ 110 ft = 440 ft The perimeter is 440 feet. 23. C = π ⋅ d = π ⋅ 1.7 in. ≈ 3.14 ⋅ 1.7 in. = 5.338 in. The circumference is 5.338 inches. 24. C = 2 ⋅ π ⋅ r = 2 ⋅ π ⋅ 5 yd = π ⋅10 yd ≈ 3.14 ⋅10 yd = 31.4 yd The circumference is 31.4 yards.

1 ⋅ (b + B) ⋅ h 2 1 = ⋅ (12 ft + 36 ft) ⋅10 ft 2 1 = ⋅ 48 ft ⋅10 ft 2 = 240 sq ft The area is 240 square feet.

25. A =

26. A = b ⋅ h = 21 yd ⋅ 9 yd = 189 sq yd The area is 189 square yards. 27. A = l ⋅ w = 40 cm ⋅ 15 cm = 600 sq cm The area is 600 square centimeters. 28. A = s 2 = (9.1 m)2 = 82.81 sq m The area is 82.81 square meters. 29. A = π ⋅ r 2 = π ⋅ (7 ft)2 = 49π sq ft ≈ 153.86 sq ft

The area is 49π square feet ≈ 153.86 square feet. 30. A = π ⋅ r 2 = π(3 in.) 2 = 9π sq in. ≈ 28.26 sq in. The area is 9π square inches ≈ 28.26 square inches. 1 1 ⋅ b ⋅ h = ⋅ 34 in. ⋅ 7 in. = 119 sq in. 2 2 The area is 119 square inches.

31. A =

1 1 ⋅ b ⋅ h = ⋅ 20 m ⋅ 14 m = 140 sq m 2 2 The area is 140 square meters.

32. A =

ISM: Prealgebra

Chapter 9: Geometry and Measurement

33. The unmarked horizontal side has length 13 m − 3 m = 10 m. The unmarked vertical side has length 12 m − 4 m = 8 m. The area is the sum of the areas of the two rectangles. A = 12 m ⋅10 m + 8 m ⋅ 3 m = 120 sq m + 24 sq m = 144 sq m The area is 144 square meters. 34. The unmarked vertical side has length 30 cm − 5 cm = 25 cm. The unmarked horizontal side has length 60 cm − 35 cm = 25 cm. The area is the sum of the areas of the two rectangles. A = 30 cm ⋅ 25 cm + 35 cm ⋅ 25 cm = 1625 sq cm The area is 1625 square centimeters. 35. A = l ⋅ w = 36 ft ⋅ 12 ft = 432 sq ft The area of the driveway is 432 square feet. 36. A = 10 ft ⋅ 13 ft = 130 sq ft 130 square feet of carpet are needed. 37. V = s 3 3

 1  =  2 in.  2   3 5  =  in.  2  125 = cu in. 8 5 = 15 cu in. 8

The volume is 15

5 cubic inches. 8

SA = 6s 2 2

 1  = 6  2 in.   2  2 5  = 6  in.  2    25  = 6   sq in.  4  75 = sq in. 2 1 = 37 sq in. 2

The surface area is 37

38. V = l ⋅ w ⋅ h = 2 ft ⋅ 7 ft ⋅ 6 ft = 84 cu ft The volume is 84 cubic feet. SA = 2lh + 2wh + 2lw = 2 ⋅ 7 ft ⋅ 6 ft + 2 ⋅ 2 ft ⋅ 6 ft + 2 ⋅ 7 ft ⋅ 2 ft = 84 sq ft + 24 sq ft + 28 sq ft = 136 sq ft The surface area is 136 square feet. 39. V = π ⋅ r 2 ⋅ h = π ⋅ (20 cm)2 ⋅ 50 cm = 20, 000π cu cm ≈ 62,800 cu cm The volume is 20,000π cubic centimeters ≈ 62,800 cubic centimeters.

4 ⋅ π ⋅ r3 3 3 4 1  = ⋅ π ⋅  km  3 2  1 = π cu km 6 11 cu km ≈ 21 The volume is 1 11 π cubic kilometers ≈ cubic kilometers. 6 21

40. V =

1 41. V = ⋅ s 2 ⋅ h 3 1 = ⋅ (2 ft)2 ⋅ 2 ft 3 8 = cu ft 3 2 = 2 cu ft 3 The volume of the pyramid is 2

2 cubic feet. 3

42. V = π ⋅ r 2 ⋅ h

= π ⋅ (3.5 in.)2 ⋅ 8 in. = 98π cu in. ≈ 307.72 cu in. The volume of the can is about 307.72 cubic inches.

1 square inches. 2

359

Chapter 9: Geometry and Measurement 43. Find the volume of each drawer. V = l ⋅ w⋅h  1   1  2  =  2 ft  ⋅ 1 ft  ⋅  ft   2   2  3  5 3 2 = ⋅ ⋅ cu ft 2 2 3 5 = cu ft 2 The three drawers have volume 5 15 1 3⋅ = = 7 cubic feet. 2 2 2

ISM: Prealgebra

12.18 mm 1m ⋅ 1 1000 mm 12.18 m = 1000 = 0.01218 m

53. 12.18 mm =

2.31 m 1 km ⋅ 1 1000 m 2.31 km = 1000 = 0.00231 km

54. 2.31 m =

55.

44. r = d ÷ 2 = 1 ft ÷ 2 = 0.5 ft V = π ⋅ r 2 ⋅ h = π ⋅ (0.5 ft)2 ⋅ 2 ft = 0.5π cu ft

The volume of the canister is 0.5π cubic feet or 1 π cubic feet. 2 45. 108 in. = 46. 72 ft =

72 ft 1 yd 72 ⋅ = yd = 24 yd 1 3 ft 3

47. 1.5 mi =

48.

108 in. 1 ft 108 ⋅ = ft = 9 ft 1 12 in. 12

1.5 mi 5280 ft ⋅ = 1.5 ⋅ 5280 ft = 7920 ft 1 1 mi

1 yd 1 3 ft 12 in. 1 yd = 2 ⋅ ⋅ = ⋅ 3 ⋅12 in. = 18 in. 2 1 1 yd 1 ft 2

49. 52 ft = 51 ft + 1 ft 51 ft 1 yd = ⋅ + 1 ft 1 3 ft 51 yd + 1 ft = 3 = 17 yd 1 ft 50. 46 in. = 36 in. + 10 in. 36 in. 1 ft = ⋅ + 10 in. 1 12 in. 36 ft + 10 in. = 12 = 3 ft 10 in. 51. 42 m =

42 m 100 cm ⋅ = 42 ⋅100 cm = 4200 cm 1 1m

52. 82 cm = 360

82 cm 10 mm ⋅ = 82 ⋅10 mm = 820 mm 1 1 cm

4 yd 2 ft + 16 yd 2 ft 20 yd 4 ft = 20 yd + 1 yd 1 ft = 21 yd 1 ft

56. 7 ft 4 in. ÷ 2 = (6 ft + 1 ft 4 in.) ÷ 2 = (6 ft + 16 in.) ÷ 2 6 16 = ft + in. 2 2 = 3 ft 8 in. 57. 8 cm = 80 mm 80 mm or + 15 mm 95 mm

15 mm = 1.5 cm 8.0 cm + 1.5 cm 9.5 cm

58. 4 m = 400 cm 400 cm or − 126 cm 274 cm

126 cm = 1.26 m 4.00 m − 1.26 m 2.74 m

59.

332 yd 4 ft − 163 yd 2 ft 169 yd 2 ft The amount of material that remains is 169 yd 2 ft.

60.

5 ft 2 in. × 50 250 ft 100 in. = 250 ft + 96 in. + 4 in. = 250 ft + 8 ft 4 in. = 258 ft 4 in. The sashes require 258 ft 4 in. of material.

333 yd 1 ft − 163 yd 2 ft

61. 1235 km × 2 2470 km 2470 km = 617.5 km 4 Each must drive 617.5 km. 2470 km ÷ 4 =

ISM: Prealgebra

62.

Chapter 9: Geometry and Measurement

0.8 m × 0.3 m 0.24 sq m

0.8 m × 30 cm

The area is 0.24 sq m. 66 oz 1 lb 66 33 1 63. 66 oz = ⋅ = lb = lb = 4 lb 1 16 oz 16 8 8

2.3 tons 2000 lb ⋅ 1 1 ton = 2.3 ⋅ 2000 lb = 4600 lb

64. 2.3 tons =

73.

74. 4.8 kg = 4800 g 4800 g − 4200 g or 600 g 75.

65. 52 oz = 48 oz + 4 oz 48 oz 1 lb = ⋅ + 4 oz 1 16 oz 48 = lb + 4 oz 16 = 3 lb 4 oz

4.3 mg × 5 21.5 mg 4200 g = 4.2 kg 4.8 kg − 4.2 kg 0.6 kg

1 lb 12 oz + 2 lb 8 oz 3 lb 20 oz = 3 lb + 1 lb 4 oz = 4 lb 4 oz The total weight was 4 lb 4 oz.

38 300 tons lb 4 4 1 = 9 tons 75 lb 2 1 = 9 tons + ton + 75 lb 2 = 9 tons + 1000 lb + 75 lb = 9 tons 1075 lb They each receive 9 tons 1075 lb.

76. 38 tons 300 lb ÷ 4 =

66. 10,300 lb = 10, 000 lb + 300 lb 10, 000 lb 1 ton = ⋅ + 300 lb 1 2000 lb 10, 000 = tons + 300 lb 2000 = 5 tons 300 lb

77. 28 pt =

28 pt 1 qt 28 ⋅ = qt = 14 qt 1 2 pt 2

67. 27 mg =

27 mg 1g 27 ⋅ = g = 0.027 g 1 1000 mg 1000

78. 40 fl oz =

68. 40 kg =

40 kg 1000 g ⋅ = 40 ⋅1000 g = 40, 000 g 1 1 kg

79. 3 qt 1 pt =

69. 2.1 hg =

2.1 hg 10 dag ⋅ = 2.1 ⋅10 dag = 21 dag 1 1 hg 0.03 mg 1 dg ⋅ 1 100 mg 0.03 = dg 100 = 0.0003 dg

3 qt 2 pt ⋅ + 1 pt 1 1 qt = 3 ⋅ 2 pt + 1 pt = 6 pt + 1 pt = 7 pt

80. 18 qt =

70. 0.03 mg =

71.

6 lb 5 oz − 2 lb 12 oz

72.

8 lb 6 oz × 4 32 lb 24 oz = 32 lb + 1 lb 8 oz = 33 lb 8 oz

5 lb 21 oz − 2 lb 12 oz 3 lb 9 oz

40 fl oz 1 c 40 ⋅ = c=5c 1 8 fl oz 8

18 qt 2 pt 2 c ⋅ ⋅ = 18 ⋅ 2 ⋅ 2 c = 72 c 1 1 qt 1 pt

81. 9 pt = 8 pt + 1 pt 8 pt 1 qt = ⋅ + 1 pt 1 2 pt 8 = qt + 1 pt 2 = 4 qt 1 pt

361

Chapter 9: Geometry and Measurement 82. 15 qt = 12 qt + 3 qt 12 qt 1 gal = ⋅ + 3 qt 1 4 qt 12 = gal + 3 qt 4 = 3 gal 3 qt

93. 85 ml × 8 × 16 = 10,880 ml 10,880 ml 1L 10,880 ⋅ = L = 10.88 L 1 1000 ml 1000 There are 10.88 L of polish in 8 boxes. 94. 6 L + 1300 ml + 2.6 L = 6 L + 1.3 L + 2.6 L = 9.9 L Since 9.9 L is less than 10 L, yes it will fit.

3.8 L 1000 ml ⋅ 1 1L = 3.8 ⋅1000 ml = 3800 ml

83. 3.8 L =

84. 14 hl =

11.5 yd 1 m ⋅ ≈ 10.55 m 1 1.09 yd

17.5 L 0.26 gal ⋅ = 4.55 gal 1 1L

2.45 ml 1L ⋅ = 0.00245 L 1 1000 ml

98. 7.8 L ≈

7.8 L 1.06 qt ⋅ = 8.268 qt 1 1L

99. 15 oz ≈

15 oz 28.35 g ⋅ = 425.25 g 1 1 oz

100. 23 lb ≈

23 lb 0.45 kg ⋅ = 10.35 kg 1 1 lb

1 qt 1 pt + 3 qt 1 pt 4 qt 2 pt = 4 qt + 1 qt = 1 gal 1 qt

89. 0.946 L = 946 ml 946 ml or − 210 ml 736 ml

210 ml = 0.21 L 0.946 L − 0.210 L 0.736 L

90. 6.1 L = 6100 ml 6100 ml + 9400 ml or 15,500 ml

9400 ml = 9.4 L 6.1 L + 9.4 L 15.5 L

3 gal 6 qt − 1 gal 3 qt 2 gal 3 qt There are 2 gal 3 qt of tea remaining. 4 gal 2 qt − 1 gal 3 qt

92. 1 c 4 fl oz ÷ 2 = (8 fl oz + 4 fl oz) ÷ 2 = 12 fl oz ÷ 2 = 6 fl oz Use 6 fl oz of stock for half of a recipe.

362

96. 11.5 yd ≈

97. 17.5 L ≈

88. 3 gal 2 qt × 2 6 gal 4 qt = 6 gal + 1 gal = 7 gal

91.

7 m 3.28 ft ⋅ = 22.96 ft 1 1m

30.6 L 100 cl ⋅ = 30.6 ⋅100 cl = 3060 cl 1 1L

86. 2.45 ml = 87.

95. 7 m ≈

14 hl 1 kl 14 ⋅ = kl = 1.4 kl 1 10 hl 10

85. 30.6 L =

ISM: Prealgebra

101. 1.2 mm × 50 = 60 mm 60 mm 1 cm 60 mm = ⋅ = 6 cm 1 10 mm 6 cm 1 in. 6 cm = ⋅ ≈ 2.36 in. 1 2.54 cm The height of the stack is approximately 2.36 in. 102. 25 × 5.5 mm = 137.5 mm 137.5 mm 1 cm 137.5 mm = ⋅ = 13.75 cm 1 10 mm 13.75 cm 1 in. ⋅ ≈ 5.4 in. 13.75 cm = 1 2.54 cm The total thickness is approximately 5.4 in. 103. F = 1.8C + 32 = 1.8(42) + 32 = 75.6 + 32 = 107.6 42°C is 107.6°F. 104. F = 1.8C + 32 = 1.8(160) + 32 = 288 + 32 = 320 160°C is 320°F.

ISM: Prealgebra

Chapter 9: Geometry and Measurement

5 5 5 105. C = (F − 32) = (41.3 − 32) = (9.3) ≈ 5.2 9 9 9 41.3°F is 5.2°C. 5 5 5 106. C = (F − 32) = (80 − 32) = (48) ≈ 26.7 9 9 9 80°F is 26.7°C. 5 5 5 107. C = (F − 32) = (35 − 32) = (3) ≈ 1.7 9 9 9 35°F is 1.7°C.

108. F = 1.8C + 32 = 1.8(165) + 32 = 297 + 32 = 329

165°C is 329°F. 109. The supplement of a 72° angle is an angle that measures 180° − 72° = 108°. 110. The complement of a 1° angle is an angle that measures 90° − 1° = 89°. 111. ∠x and the angle marked 85° are adjacent angles, so m∠x = 180° − 85° = 95°. 112. Let ∠y be the angle corresponding to ∠x at the bottom intersection. Then ∠y and the angle marked 123° are adjacent angles, so m∠x = m∠y = 180° − 123° = 57°. 113. P = 7 in. + 11.2 in. + 9.1 in. = 27.3 in. The perimeter is 27.3 inches. 114. The unmarked horizontal side has length 40 ft − 22 ft − 11 ft = 7 ft. P = (22 + 15 + 7 + 15 + 11 + 42 + 40 + 42) ft = 194 ft The perimeter is 194 feet. 115. The unmarked horizontal side has length 43 m − 13 m = 30 m. The unmarked vertical side has length 42 m − 14 m = 28 m. The area is the sum of the areas of the two rectangles. A = 28 m ⋅13 m + 42 m ⋅ 30 m = 364 sq m + 1260 sq m = 1624 sq m The area is 1624 square meters. 2

116. A = π ⋅ r = π ⋅ (3 m) = 9π sq m ≈ 28.26 sq m The area is 9π square meters ≈ 28.26 square meters.

1 117. V = ⋅ π ⋅ r 2 ⋅ h 3 2 1  1  = ⋅ π ⋅  5 in.  ⋅12 in. 3  4  2

1  21  = ⋅π⋅ in.  ⋅12 cu in. 3  4  441 = π cu in. 4 441 22 1 cu in. = 346 cu in. ≈ ⋅ 4 7 2 1 The volume is 346 cubic inches. 2 118. V = l ⋅ w ⋅ h = 5 in. ⋅ 4 in. ⋅ 7 in. = 140 cu in. The volume is 140 cubic inches. SA = 2lh + 2 wh + 2lw = 2 ⋅ 7 in. ⋅ 5 in. + 2 ⋅ 4 in. ⋅ 5 in. + 2 ⋅ 7 in. ⋅ 4 in. = 70 sq in. + 40 sq in. + 56 sq in. = 166 sq in. The surface area is 166 square inches. 119. 6.25 ft =

6.25 ft 12 in. ⋅ = 75 in. 1 1 ft

120. 8200 lb = 8000 lb + 200 lb 8000 lb 1 ton = ⋅ + 200 lb 1 2000 lb = 4 tons 200 lb 121. 5 m =

5 m 100 cm ⋅ = 500 cm 1 1m 286 mm 1 km ⋅ 1 1,000,000 mm = 0.000286 km

122. 286 mm =

123. 1400 mg =

1400 mg 1g ⋅ = 1.4 g 1 1000 mg

124. 6.75 gal =

6.75 gal 4 qt ⋅ = 27 qt 1 1 gal

125. F = 1.8C + 32 = 1.8(86) + 32 = 154.8 + 32 = 186.8

86°C is 186.8°F. 5 5 5 126. C = (F − 32) = (51.8 − 32) = (19.8) = 11 9 9 9 51.8°F is 11°C.

363

Chapter 9: Geometry and Measurement 127. 9.3 km = 9300 m 9300 m − 183 m or 9117 m

183 m = 0.183 km 9.300 km − 0.183 km 9.117 km

128. 35 L = 35,000 ml 35, 000 ml + 700 ml 35, 700 ml

700 ml = 0.7 L 35.0 L + 0.7 L 35.7 L

129.

130.

10. When parallel lines are cut by a transversal, the measures of corresponding angles are equal and the measures of alternate interior angles are equal. Thus, ∠a has the same measure as ∠d (vertical angles), ∠b (corresponding angles), and ∠c (vertical angle with ∠b). Thus, ∠a and ∠c have the same measure; B. 11. Area is measured in square units; B.

3 gal 3 qt + 4 gal 2 qt 7 gal 5 qt = 7 gal + 1 gal 1 qt = 8 gal 1 qt

12. Volume is measured in cubic units; C.

3.2 kg × 4 12.8 kg

14. The amount of material needed for a tablecloth is a calculation of area; B.

Chapter 9 Getting Ready for the Test 1. A line is a set of points extending indefinitely in two directions; D. 2. A line segment is a piece of a line with two endpoints; B. 3. A ray is a part of a line with one endpoint; E. 4. A right angle is an angle that measures 90°; A. 5. An acute angle is an angle that measures between 0° and 90°; F. 6. An obtuse angle is an angle that measures between 90° and 180°; C. 7. When two lines intersect, adjacent angles are supplementary (have a sum of 180°). ∠a and ∠d are adjacent angles, so choice B is correct. 8. When two lines intersect, vertical angles (angles that are opposite each other) have the same measure. ∠b and ∠d re vertical angles, so choice C is correct. 9. When parallel lines are cut by a transversal, the measures of corresponding angles are equal and the measures of alternate interior angles are equal. Thus, ∠a has the same measure as ∠d (vertical angles), ∠b (corresponding angles), and ∠c (vertical angle with ∠b). ∠a and ∠f are adjacent angles, so they are supplementary. ∠e and ∠f are corresponding angles, so they have the same measure. Thus ∠a and ∠e are supplementary (have a sum of 180°); A. 364

ISM: Prealgebra

13. Perimeter and circumference are measured in units; A and D.

15. The amount of trim needed to go around the edge of a tablecloth is a calculation of perimeter or circumference; A or D. 16. The amount of soil needed to fill a hole in the ground is a calculation of volume; C. Chapter 9 Test 1. The complement of an angle that measures 78° is an angle that measures 90° − 78° = 12°. 2. The supplement of a 124° angle is an angle that measures 180° − 124° = 56°. 3. m∠x = 90° − 40° = 50° 4. ∠x and the angle marked 62° are adjacent angles, so m∠x = 180° − 62° = 118°. ∠y and the angle marked 62° are vertical angles, so m∠y = 62°. ∠x and ∠z are vertical angles, so m∠z = m∠x = 118°. 5. ∠x and the angle marked 73° are vertical angles, so m∠x = 73°. ∠x and ∠y are alternate interior angles, so m∠y = m∠x = 73°. ∠x and ∠z are corresponding angles, so m∠z = m∠x = 73°. 6. d = 2 ⋅ r = 2 ⋅ 3.1 m = 6.2 m 7. r = d ÷ 2 = 20 in. ÷ 2 = 10 in.

ISM: Prealgebra

Chapter 9: Geometry and Measurement

8. Circumference: C = 2⋅π⋅r = 2 ⋅ π ⋅ 9 in. = 18π in. ≈ 56.52 in. The circumference is 18π inches ≈ 56.52 inches. Area:

A = πr 2 = π(9 in.) 2 = 81π sq in. ≈ 254.34 sq in. The area is 81π square inches ≈ 254.34 square inches. 9. P = 2 ⋅ l + 2 ⋅ w = 2(7 yd) + 2(5.3 yd) = 14 yd + 10.6 yd = 24.6 yd The perimeter is 24.6 yards. A = l ⋅ w = 7 yd ⋅ 5.3 yd = 37.1 sq yd The area is 37.1 square yards. 10. The unmarked vertical side has length 11 in. − 7 in. = 4 in. The unmarked horizontal side has length 23 in. − 6 in. = 17 in. P = (6 + 4 + 17 + 7 + 23 + 11) in. = 68 in. The perimeter is 68 inches. Extending the unmarked vertical side downward divides the region into two rectangles. The region’s area is the sum of the areas of these: A = 11 in. ⋅ 6 in. + 7 in. ⋅17 in. = 66 sq in. + 119 sq in. = 185 sq in. The area is 185 square inches. 11. V = π ⋅ r 2 ⋅ h = π ⋅ (2 in.)2 ⋅ 5 in. = 20π cu in. 22 6 cu in. = 62 cu in. ≈ 20 ⋅ 7 7 6 The volume is 62 cubic inches. 7

15. P = 2 ⋅ l + 2 ⋅ w = 2 ⋅18 ft + 2 ⋅13 ft = 36 ft + 26 ft = 62 ft

cost = P ⋅ $1.87 per ft = 62 ft ⋅ $1.87 per ft = $115.94 62 feet of baseboard are needed, at a total cost of $115.94. 23 280 −24 40 −36 4 280 inches = 23 ft 4 in.

16. 12

17. 2

2 1 gal 4 qt 1 1 gal = 2 ⋅ = 2 ⋅ 4 qt = 10 qt 2 1 1 gal 2

18. 30 oz =

30 oz 1 lb 30 15 7 ⋅ = lb = lb = 1 lb 1 16 oz 16 8 8

2.8 tons 2000 lb ⋅ 1 1 ton = 2.8 ⋅ 2000 lb = 5600 lb

19. 2.8 tons =

38 pt 1 qt 1 gal ⋅ ⋅ 1 2 pt 4 qt 38 = gal 8 19 = gal 4 3 = 4 gal 4

20. 38 pt =

21. 40 mg =

40 mg 1g 40 ⋅ = g = 0.04 g 1 1000 mg 1000

22. 2.4 kg =

2.4 kg 1000 g ⋅ = 2.4 ⋅1000 g = 2400 g 1 1 kg

13. P = 4 ⋅ s = 4 ⋅ 4 in. = 16 in. The perimeter of the frame is 16 inches.

23. 3.6 cm =

3.6 cm 10 mm ⋅ = 3.6 ⋅10 mm = 36 mm 1 1 cm

14. V = l ⋅ w ⋅ h = 3 ft ⋅ 3 ft ⋅ 2 ft = 18 cu ft 18 cubic feet of soil are needed.

24. 4.3 dg =

4.3 dg 1 g 4.3 ⋅ = g = 0.43 g 1 10 dg 10

12. V = l ⋅ w ⋅ h = 5 ft ⋅ 3 ft ⋅ 2 ft = 30 cu ft The volume is 30 cubic feet.

365

Chapter 9: Geometry and Measurement

25. 0.83 L = 26.

27.

0.83 L 1000 ml ⋅ = 0.83 ⋅1000 = 830 ml 1 1L

3 qt 1 pt + 2 qt 1 pt 5 qt 2 pt = 4 qt + 1 qt + 2 pt = 1 gal + 1 qt + 1 qt = 1 gal 2 qt

ISM: Prealgebra

35.

36. 88 m + 340 cm = 88 m + 3.40 m = 91.4 m The span is 91.4 meters 37.

8 lb 6 oz 7 lb 22 oz − 4 lb 9 oz → − 4 lb 9 oz 3 lb 13 oz

3a − 6 = a + 4 3a − a − 6 = a − a + 4 2a − 6 = 4 2a − 6 + 6 = 4 + 6 2a = 10 2a 10 = 2 2 a=5

2 x + 1 = 3x − 5 2 x − 2 x + 1 = 3x − 2 x − 5 1= x −5 1+ 5 = x − 5 + 5 6=x

29. 5 gal 2 qt ÷ 2 = 4 gal 6 qt ÷ 2 4 6 = gal qt 2 2 = 2 gal 3 qt 30. 8 cm = 80 mm 80 mm − 14 mm or 66 mm

14 mm = 1.4 cm 8.0 cm − 1.4 cm 6.6 cm

31. 1.8 km = 1800 m 1800 m + 456 m or 2256 m

456 m = 0.456 km 1.800 km + 0.456 km 2.256 km

33. F = 1.8C + 32 = 1.8(12.6) + 32 = 22.68 + 32 ≈ 54.7 12.6°C is 54.7°F 2 8.4 2 34. 8.4 m ⋅ = ⋅ m = 5.6 m 3 1 3 The trees will be 5.6 m tall.

366

2 ft 9 in. × 6 12 ft 54 in. = 12 ft + 4 ft 6 in. = 16 ft 6 in. Thus, 16 ft 6 in. of material is needed.

Cumulative Review Chapters 1−9

28. 2 ft 9 in. × 3 = 6 ft 27 in. = 6 ft + 2 ft 3 in. = 8 ft 3 in.

5 32. C = (F − 32) 9 5 = (84 − 32) 9 5 = (52) 9 ≈ 28.9 84°F is 28.9°C.

19 gal 4 qt − 15 gal 1 qt 4 gal 3 qt Thus, 4 gal 3 qt remains in the container. 20 gal − 15 gal 1 qt

2 2 2 2 24 16 2 = ⋅ ⋅ ⋅ = =   5 5 5 5 54 625 5

3. a.

 1  1  1  1  −  =  −  −  =  4  4  4  16

1  1   1  1  1   −  =  −  −  −  = − 3 3 3 3 27      

4. a.

3 3 9 3   = ⋅ = 7 7 49 7

2 5

4 5

1 +1 2

2 5

8 10

5 10 13 3 3 8 = 8 +1 = 9 10 10 10

ISM: Prealgebra

1 3 2 4 5 +3 2

Chapter 9: Geometry and Measurement

14.

5 15 6 4 15 +3 11 9 15 2

7. 11.1x − 6.3 + 8.9 x − 4.6 = 11.1x + 8.9 x − 6.3 − 4.6 = 20 x − 10.9

15. Use a 2 + b 2 = c 2 where a = b = 300.

3002 + 3002 = c 2 90, 000 + 90, 000 = c 2 180, 000 = c 2 180, 000 = c 424 ≈ c The length of the diagonal is approximately 424 feet.

8. 2.5 y + 3.7 − 1.3 y − 1.9 = 2.5 y − 1.3 y + 3.7 − 1.9 = 1.2 y + 1.8 9.

10.

5.68 + (0.9) 2 ÷ 100 5.68 + 0.81 ÷ 100 = 0.2 0.2 5.68 + 0.0081 = 0.2 5.6881 = 0.2 = 28.4405

16. Use a 2 + b 2 = c 2 where a = 200 and b = 125.

2002 + 1252 = c 2 40, 000 + 15, 625 = c 2 55, 625 = c 2 55, 625 = c 236 ≈ c The length of the diagonal is approximately 236 feet.

0.12 + 0.96 1.08 = = 2.16 0.5 0.5

0.77 ... 11. 9 7.00 −63 70 −63 7

0.4 x − 9.3 = 2.7 0.4 x − 9.3 + 9.3 = 2.7 + 9.3 0.4 x = 12 0.4 x 12 = 0.4 0.4 x = 30

17. a. b.

7 Thus 0.7 = . 9

12. 5

0.4 2.0 −2 0 0

18. a.

2 Thus 0.43 > . 5

13.

0.5 y + 2.3 = 1.65 0.5 y + 2.3 − 2.3 = 1.65 − 2.3 0.5 y = −0.65 0.5 y −0.65 = 0.5 0.5 y = −1.3

width 5 feet 5 = = length 7 feet 7

P = 2⋅l + 2⋅ w = 2(7 feet) + 2(5 feet) = 14 feet + 10 feet = 24 feet length 7 feet 7 = = perimeter 24 feet 24 P = 4s = 4(9 in.) = 36 in. side 9 inches 9 1 = = = perimeter 36 inches 36 4 A = s 2 = (9 in.)2 = 81 sq in. perimeter 36 inches 36 4 = = = area 81 sq inches 81 9

19.

2160 dollars 180 dollars = 12 weeks 1 week

367

Chapter 9: Geometry and Measurement

20.

21.

22.

8 chaperones 1 chaperone = 40 students 5 students

27.

1.6 x = 1.1 0.3 1.6 ⋅ 0.3 = 1.1 ⋅ x 0.48 = 1.1x 0.48 1.1x = 1.1 1.1 0.44 ≈ x

28.

2.4 0.7 = 3.5 x 2.4 ⋅ x = 3.5 ⋅ 0.7 2.4 x = 2.45 2.4 x 2.45 = 2.4 2.4 x ≈ 1.02

23. Let x be the dose for a 140-lb person. 4 cc x cc = 25 lb 140 lb x 4 = 25 140 4 ⋅140 = 25 ⋅ x 560 = 25 x 560 25 x = 25 25 22.4 = x The dose is 22.4 cc. 24. Let x be the amount for 5 pie crusts. 3c xc = 2 crusts 5 crusts 3 x = 2 5 3⋅5 = 2 ⋅ x 15 = 2 x 15 2 x = 2 2 7.5 = x 5 pie crusts require 7.5 cups of flour. 25.

17 = 17% 100 17% of the people surveyed drive blue cars.

38 26. = 38% 100 38% of the shoppers used only cash.

368

ISM: Prealgebra

1 13 = 6 % ⋅ x 2 13 = 0.065 x 13 0.065 x = 0.065 0.065 200 = x 1 13 is 6 % of 200. 2 1 54 = 4 % ⋅ x 2 54 = 0.045 x 54 0.045 x = 0.045 0.045 1200 = x 1 54 is 4 % of 1200. 2

29. x = 30% ⋅ 9 x = 0.3 ⋅ 9 x = 2.7 2.7 is 30% of 9. 30. x = 42% ⋅ 30 x = 0.42 ⋅ 30 x = 12.6 12.6 is 42% of 30.

amount of increase original amount 45 − 34 = 34 11 = 34 ≈ 0.32 The scholarship applications increased by 32%.

31. percent of increase =

amount of increase original amount 19 − 15 = 15 4 = 15 ≈ 0.27 The price of the paint increased by 27%.

32. percent of increase =

33. sales tax = 85.50 ⋅ 0.075 = 6.4125 The sales tax is $6.41. 85.50 + 6.41 = 91.91 The total price is $91.91.

ISM: Prealgebra

Chapter 9: Geometry and Measurement 40. The five numbers in order are: 60, 72, 83, 89, 95. The median is the middle number, 83.

34. sales tax = 375 ⋅ 0.08 = 30 The sales tax is $30. 375 + 30 = 405 The total price is $405. 35. Point A is 4 units left of the y-axis and 2 units above the x-axis. The coordinates are (−4, 2). Point B is 1 unit right of the y-axis and 2 units above the x-axis. The coordinates are (1, 2). Point C is on the y-axis and 1 unit above the x-axis. The coordinates are (0, 1). Point D is 3 units left of the y-axis and on the x-axis. The coordinates are (−3, 0). Point E is 5 units right of the y-axis and 4 units below the x-axis. The coordinates are (5, −4). 36. Point A is 2 units right of the y-axis and 3 units below the x-axis. The coordinates are (2, −3). Point B is 5 units left of the y-axis and on the x-axis. The coordinates are (−5, 0). Point C is on the y-axis and 4 units above the x-axis. The coordinates are (0, 4). Point D is 3 units left of the y-axis and 2 units below the x-axis. The coordinates are (−3, −2). 37. No matter what x-value we choose, y is always 4.

41. There is 1 red marble and 1 + 1 + 2 = 4 total 1 marbles. The probability is . 4 42. There are 2 nickels and 2 + 2 + 3 = 7 total coins. 2 The probability is . 7 43. The complement of a 48° angle is an angle that has measure 90° − 48° = 42°. 44. The supplement of a 137° angle is an angle that has measure 180° − 137° = 43°. 45. 8 ft =

8 ft 12 in. ⋅ = 8 ⋅12 in. = 96 in. 1 1 ft

46. 7 yd =

7 yd 3 ft ⋅ = 7 ⋅ 3 ft = 21 ft 1 1 yd

47. A = πr 2 = π ⋅ 32 = 9π ≈ 9 ⋅ 3.14 = 28.26 The area is exactly 9π square feet or approximately 28.26 square feet. 48. A = πr 2 = π ⋅ 2 2 = 4π ≈ 4 ⋅ 3.14 = 12.56 The area is exactly 4π square miles or approximately 12.56 square miles.

38. No matter what x-value we choose, y is always −2.

5 5 5 49. C = (F − 32) = (59 − 32) = (27) = 15 9 9 9 59°F is 15°C. 5 5 5 50. C = (F − 32) = (86 − 32) = (54) = 30 9 9 9 86°F is 30°C.

39. The seven numbers are listed in order. The median is the middle number, 57.

369

Chapter 10 Section 10.1 Practice Exercises 1. (3 y + 7) + (−9 y − 14) = (3 y − 9 y) + (7 − 14) = (−6 y) + (−7) = −6 y − 7 2. ( x 2 − 4 x − 3) + (5 x 2 − 6 x )

The value of 2 y3 + y2 − 6 when y = 3 is 57. 11. −16t 2 + 530 = −16(1)2 + 530 = −16 + 530 = 514 The height of the object at 1 second is 514 feet.

= x 2 + 5x 2 − 4 x − 6 x − 3 = 6 x 2 − 10 x − 3 3. (− z 2 − 4.2 z + 11) + (9z 2 − 1.9z + 6.3)

= − z 2 + 9z 2 − 4.2 z − 1.9z + 11 + 6.3 = 8z 2 − 6.1z + 17.3 4.

10. 2 y3 + y2 − 6 = 2(3)3 + (3)2 − 6 = 2(27) + 9 − 6 = 54 + 9 − 6 = 57

−16t 2 + 530 = −16(4)2 + 530 = −16(16) + 530 = −256 + 530 = 274 The height of the object at 4 seconds is 274 feet.

Vocabulary, Readiness & Video Check 10.1

x 2 − x + 1.1 + − 8 x 2 − x − 6.7

1. The addends of an algebraic expression are the terms of the expression.

−7 x − 2 x − 5.6

5. −(3 y2 + y − 2) = −1(3 y2 + y − 2)

= −1(3 y2 ) + (−1)( y) + (−1)(−2) = −3 y 2 − y + 2

2. A polynomial with exactly one term is called a monomial. 3. A polynomial with exactly two terms is called a binomial.

6. (9b + 8) − (11b − 20) = (9b + 8) + (−11b + 20) = 9b − 11b + 8 + 20 = −2b + 28

4. A polynomial with exactly three terms is called a trinomial.

7. (11x 2 + 7 x + 2) − (15 x 2 + 4 x )

6. To subtract two polynomials, change the signs of the terms of the second polynomial; then add.

= (11x + 7 x + 2) + (−15 x − 4 x ) = 11x 2 − 15 x 2 + 7 x − 4 x + 2

7. terms where everything is the same except for the numerical coefficient

= −4 x + 3 x + 2

8. We need to be careful how we set up the subtraction when translating the word statement since order matters.

8. (−3 y 2 + 5 y) − (−7 y 2 + y − 4) = (−3 y2 + 5 y) + (7 y 2 − y + 4) = −3 y 2 + 7 y 2 + 5 y − y + 4

9. 2; −5

= 4 y2 + 4 y + 4

5. To add polynomials, combine like terms.

Exercise Set 10.1 2

−4 x + 20 x + 17

− (3 x 2 − 12 x )

−3 x 2 + 12 x −7 x 2 + 32 x + 17

2. (9 y − 16) + (−43 y + 16) = 9 y − 43 y − 16 + 16 = −34 y 4. (8a2 + 5a − 9) + (5a2 − 11a + 6)

= 8a2 + 5a2 + 5a − 11a − 9 + 6 = 13a2 − 6a − 3 370

ISM: Prealgebra

Chapter 10: Exponents and Polynomials

28. (−2a2 − 5a) + (6a2 − 2a + 9)

6. (5 x 2 − 6) + (−3 x 2 + 17 x − 2)

= 5 x 2 − 3 x 2 + 17 x − 6 − 2

= −2a2 + 6a2 − 5a − 2a + 9

= 2 x 2 + 17 x − 8

= 4 a 2 − 7a + 9

8. (−12.7z 3 − 14 z) + (−8.9z 3 + 12 z + 2) 3

= −12.7z − 8.9z − 14 z + 12 z + 2 = −21.6 z 3 − 2 z + 2

30. (20 x − 0.8) + ( x + 1.2) = 20 x + x − 0.8 + 1.2 = 21x + 0.4 32. (t + 9) − (−2t + 6) = (t + 9) + (2t − 6) = t + 2t + 9 − 6 = 3t + 3

10. −(4 y − 12) = −1(4 y − 12) = (−1)(4 y) + (−1)(−12) = −4 y + 12 12. −(−2 x 2 − x + 1) = −1(−2 x 2 − x + 1)

= −1(−2 x 2 ) + (−1)(− x ) + (−1)(1) = 2x2 + x − 1

34. (8.6 x + 4) − (9.7 x − 93) = (8.6 x + 4) + (−9.7 x + 93) = 8.6 x − 9.7 x + 4 + 93 = −1.1x + 97 36. (35 x 2 + x − 5) − (17 x 2 − x + 5)

= (35 x 2 + x − 5) + (−17 x 2 + x − 5)

14. (3b + 5) − (−2b + 9) = (3b + 5) + (2b − 9) = 3b + 2b + 5 − 9 = 5b − 4 16. (−9z 2 + 6 z + 2) − (3z 2 + 1) 2

= 35 x 2 − 17 x 2 + x + x − 5 − 5 = 18 x 2 + 2 x − 10 38. (3z 2 − 8z + 5) + (−3z 3 − 5z 2 − 2 z − 4)

= −3z 3 + 3z 2 − 5z 2 − 8z − 2 z + 5 − 4

= (−9z + 6 z + 2) + (−3z − 1) 2

= −3z 3 − 2 z 2 − 10 z + 1

= −9z − 3z + 6 z + 2 − 1 = −12 z 2 + 6 z + 1 3

40.

+ 9 x 2 − x + 14

18. (11x + 15 x − 9) − (− x + 10 x − 9) 3

= (11x + 15 x − 9) + ( x − 10 x + 9) 3

7 x 2 + 2 x + 14

= 11x + x − 10 x + 15 x − 9 + 9 = 12 x 3 − 10 x 2 + 15 x

42.

−7 a + 7

−7a + 7

− (4 a + 6 a + 1)

+ − 4a2 − 6 a − 1

20. 2

−2 x 2 + 3 x

3 y 10 7   −  8 y2 − y  10   −5 y2 +

−4a2 − 13a + 6

22.

3 y 10 7 + − 8 y2 + y 10 −5 y 2 +

−13 y2

12 x 2 − 3 x − 12

44. −5x − 7 = −5(2) − 7 = −10 − 7 = −17

− (16 x 2 − x + 1)

+ − 16 x 2 + x − 1

46. 5 x 2 + 4 x − 100 = 5(2)2 + 4(2) − 100 = 5(4) + 8 − 100 = 20 + 8 − 100 = −72

−4 x 2 − 2 x − 13

24. (14 x + 2) + ( −7 x − 1) = 14 x − 7 x + 2 − 1 = 7x + 1 26. (6 z − 3) − (8z + 5) = (6 z − 3) + ( −8z − 5) = 6 z − 8z − 3 − 5 = −2 z − 8

48.

7 x3 7(2)3 −x+5= −2+5 14 14 56 = −2+5 14 = 4−2+5 =7

371

Chapter 10: Exponents and Polynomials

50. −5x − 6 = −5(5) − 6 = −25 − 6 = −31 52. x 3 = 53 = 125 54. 4 x 2 − 5 x + 10 = 4(5)2 − 5(5) + 10 = 4(25) − 25 + 10 = 100 − 25 + 10 = 85

ISM: Prealgebra

66. 1053 − 16t 2 = 1053 − 16(6)2 = 1053 − 16(36) = 1053 − 576 = 477 After 6 seconds, the height of the object is 477 feet. 68. 2027 corresponds to x = 12. 4.1x 2 + 3.4 x + 304 = 4.1(12) 2 + 3.4(12) + 304 = 4.1(144) + 40.8 + 304 = 590.4 + 40.8 + 304 = 935.2 If cellular service continues to grow at this rate, we can expect 935.2 thousand or 935,200 cell phone antennas in the United States in 2027.

56. 16t 2 = 16(8)2 = 16(64) = 1024 The building is 1024 feet tall. 58. 3000 + 20 x = 3000 + (20)(100) = 3000 + 2000 = 5000 The cost for 100 file cabinets is $5000. 60. a.

The height of the bar for 2030 is about 385. The graph predicts that the size of the market for wearable technology in 2030 will be $385 billion or $385,000,000,000.

b. 2030 corresponds to x = 6.

72. 43 = 4 ⋅ 4 ⋅ 4 = 64 74. y ⋅ y ⋅ y ⋅ y ⋅ y = y5 76. 5 ⋅ 5 ⋅ 5 ⋅ b ⋅ b = 53 b 2

2.2 x 2 + 21.8 x + 180 = 2.2(6)2 + 21.8(6) + 180 = 2.2(36) + 130.8 + 180 = 79.2 + 130.8 + 180 = 390 The polynomial model predicts that the size of the market for wearable technology in 2030 will be $385 billion or $385,000,000,000. c.

70. (−2)5 = (−2)(−2)(−2)(−2)( −2) = −32

78. P = ( x + 1) + ( x 2 − 6) + (3 x − 10) + (5 x 2 + 2 x )

= x 2 + 5 x 2 + x + 3 x + 2 x + 1 − 6 − 10 = 6 x 2 + 6 x − 15 The perimeter is (6 x 2 + 6 x − 15) meters. 80. ( x 2 − 7 x + 6) − ( x 2 + 2) = ( x 2 − 7 x + 6) + (− x 2 − 2)

Yes, the answers are about the same, as they should be; answers may vary.

= x2 − x2 − 7x + 6 − 2 = −7 x + 4 The missing length is (−7x + 4) units.

62. 2025 corresponds to x = 8. 7.2 x 2 + 78.6 x + 522 = 7.2(8) 2 + 78.6(8) + 522 = 7.2(64) + 628.8 + 522 = 460.8 + 628.8 + 522 = 1611.6 The number of connected wearable devices worldwide in 2025 is predicted to be 1611.6 million or 1,611,600,000.

64. 867 − 16t 2 = 867 − 16(7) 2 = 867 − 16(49) = 867 − 784 = 83 The second climber is 83 feet above the ground.

372

82.

y2 +

4y −

+ 8y −

9 y2 +

2y +

Since 1y 2 + 8 y 2 = 9 y 2 , 4y − 2y = 2y, and −3 + 10 = 7, the missing numbers are 1, 2, and 10.

(1y 2 + 4 y − 3) + (8 y 2 − 2y + 10) = 9 y 2 + 2 y + 7 84. 3b3 + 4b2 − 100 = 3(−2.5)3 + 4(−2.5)2 − 100 = −121.875

ISM: Prealgebra

Chapter 10: Exponents and Polynomials

Section 10.2 Practice Exercises

4. b ⋅ b 4 = b1+ 4 = b5

1. z 5 ⋅ z 6 = z 5+ 6 = z11

6. 8r 2 ⋅ 2r15 = (8 ⋅ 2)(r 2 ⋅ r15 ) = 16r17

2. 8 y5 ⋅ 4 y9 = (8 ⋅ 4)( y5 ⋅ y9 ) = 32 y5+ 9 = 32 y14 3. (−4r 6 s 2 )(−3r 2 s 5 ) = (−4 ⋅ −3)(r 6 ⋅ r 2 )(s 2 ⋅ s 5 )

= 12r

6+2 2+5

8 7

= 12r s

8. −9 y ⋅ 3 y = (−9 ⋅ 3)( y ⋅ y) = −27 y2 10. 4 y ⋅ 3 y ⋅ 5 y = (4 ⋅ 3 ⋅ 5)( y ⋅ y ⋅ y) = 60 y3 12. b ⋅ 7b10 ⋅ 5b8 = (7 ⋅ 5)(b1 ⋅ b10 ⋅ b8 ) = 35b19

4. 11y5 ⋅ 3 y2 ⋅ y = (11 ⋅ 3)( y5 ⋅ y2 ⋅ y1 ) = 33 y8

14. (−2 xy 4 )(−6 x 3 y 7 ) = (−2 ⋅ −6)( x1 ⋅ x 3 )( y 4 ⋅ y 7 )

= 12 x 4 y11

5. ( z 3 )6 = z 3⋅6 = z18 6. ( z 4 )5 ⋅ ( z 3 ) 7 = ( z 20 )( z 21 ) = z 20 + 21 = z 41 7. (3b) 4 = 34 ⋅ b 4 = 81b 4

9. (2 x 2 y 4 )4 (3 x 6 y 9 )2 2 4

4 4

= 36 a 5 b15

18. ( y 4 )7 = y 4⋅7 = y28

8. (4 x 2 y6 )3 = 43 ( x 2 )3 ( y6 )3 = 64 x 6 y18

16. (3a3 b 6 )(12 a 2 b 9 ) = (3 ⋅ 12)( a3 ⋅ a2 )(b 6 ⋅ b 9 )

6 2

20. (a9 )3 = a9⋅3 = a27 22. ( x 2 )9 ⋅ ( x 5 )3 = x 2⋅9 ⋅ x 5⋅3

9 2

= 2 (x ) (y ) ⋅3 (x ) (y )

= x18 ⋅ x15

= 16 x 8 y16 ⋅ 9 x12 y18

= x18+15

= (16 ⋅ 9)( x 8 ⋅ x12 )( y16 ⋅ y18 )

= x 33

= 144 x 20 y34

Vocabulary, Readiness & Video Check 10.2

24. (2 y)5 = 25 ⋅ y5 = 32 y5

1. In 7 x 2 , the 2 is called the exponent.

26. ( x 7 y 4 )8 = ( x 7 )8 ( y 4 )8 = x 7⋅8 ⋅ y 4⋅8 = x 56 y32

2. To simplify x 4 ⋅ x 3 , we add the exponents.

28. (8a5b7 )2 = 82 (a5 )2 (b7 )2

= 64a5⋅2 b7⋅2

3. To simplify ( x 4 )3 , we multiply the exponents. 4. The expression (6 x )2 simplifies to 36 x 2 . 2

5. x ⋅ x = x

= 64a10 b14 30. (−2 x )(5 x 2 ) 4 = (−2 x ) ⋅ 54 ( x 2 )4

= −2 x ⋅ 625 x 8

= (−2 ⋅ 625)( x1 ⋅ x 8 )

6. Note whether you’re multiplying (product property) or dealing with the power of a power (power property).

= −1250 x 9 32. (2 xy)4 (3 x 4 y3 )3 = (2 4 x 4 y 4 )[33 ( x 4 )3 ( y3 )3 ] = (16 x 4 y 4 )(27 x12 y 9 )

7. 3 and a

= (16 ⋅ 27)( x 4 ⋅ x12 )( y 4 ⋅ y 9 )

Exercise Set 10.2 2. y 4 ⋅ y 7 = y 4+ 7 = y11

= 432 x16 y13

34. 4(y + 2) = 4 ⋅ y + 4 ⋅ 2 = 4y + 8

373

Chapter 10: Exponents and Polynomials

36. −3(8r + 3s) = −3 ⋅ 8r + (−3)(3s) = −24r − 9s

ISM: Prealgebra

6. (2a3 − 6a2 + 11) − (6a3 + 6a2 + 11)

= (2a3 − 6a2 + 11) + (−6a3 − 6a2 − 11)

38. 5( a + 7b − 3) = 5 ⋅ a + 5 ⋅ 7b − 5 ⋅ 3 = 5a + 35b − 15

= 2a3 − 6a3 − 6a2 − 6a2 + 11 − 11 = −4a3 − 12a2

40. area = lw = (9 y 2 )(9 y) 2

7. (4.5 x 2 + 8.1x ) + (2.8 x 2 − 12.3 x − 5.3)

= (9 ⋅ 9)( y ⋅ y ) = 81y

= 4.5 x 2 + 2.8 x 2 + 8.1x − 12.3 x − 5.3

= 7.3 x 2 − 4.2 x − 5.3

The area is 81y square centimeters. 42. Area = bh

8. (1.2 y2 − 3.6 y) + (0.6 y2 + 1.2 y − 5.6)

= 1.2 y2 + 0.6 y2 − 3.6 y + 1.2 y − 5.6

= (50 y15 )(30 y12 ) 15

= (50 ⋅ 30)( y = 1500 y

= 1.8 y2 − 2.4 y − 5.6

⋅y )

The area is 1500y27 square feet. 44. (5 x14 y 6 )7 (3 x 20 y19 )5 7

14 7

6 7

20 5

⋅ 243 x

100 95

100

19 5

= 5 (x ) (y ) ⋅ 3 (x ) (y ) 98 42

= 78,125 x y

= (78,125 ⋅ 243)( x

⋅x

10.

8x + 1 − (2 x − 6)

5 x 2 + 2 x − 10 − (3 x

− x + 2)

5 x 2 + 2 x − 10 + − 3x 2 + x − 2 2 x 2 + 3 x − 12

)( y 42 ⋅ y 95 )

= 18, 984,375 x198 y137

8x + 1 + − 2x + 6 6x + 7

11. 2x − 7 = 2(3) − 7 = 6 − 7 = −1

46. (4.6 a14 )4 = 4.6 4 (a14 ) 4 = 447.7456 a56

12. x 2 + 5 x + 2 = 32 + 5(3) + 2 = 9 + 15 + 2 = 26

48. ( a 20 b10 c5 )5 ⋅ ( a 9 b12 )3 = a100 b50 c 25 ⋅ a 27 b36

13. x 9 ⋅ x11 = x 9 +11 = x 20

= a127 b86 c 25

Mid-Chapter Review 1. (3 x + 5) + (− x − 8) = 3 x − x + 5 − 8 = 2 x − 3 2. (15 y − 7) + (5 y − 4) = 15 y + 5 y − 7 − 4 = 20 y − 11 3. (7 x + 1) − (−3 x − 2) = (7 x + 1) + (3 x + 2) = 7x + 3x + 1 + 2 = 10 x + 3 4. (14 y − 6) − (19 y − 2) = (14 y − 6) + (−19 y + 2) = 14 y − 19 y − 6 + 2 = −5 y − 4 5. (a4 + 5a) − (3a4 − 3a2 − 4a) 4

= a + 5a − 3a + 3a + 4a = a4 − 3a4 + 3a2 + 5a + 4a 4

= −2a + 3a + 9a 374

14. x 5 ⋅ x 5 = x 5+ 5 = x10 15. y3 ⋅ y = y3+1 = y 4 16. a ⋅ a10 = a1+10 = a11 17. ( x 7 )11 = x 7⋅11 = x 77 18. ( x 6 )6 = x 6⋅6 = x 36 19. ( x 3 ) 4 ⋅ ( x 5 )6 = x 3⋅4 ⋅ x 5⋅6 = x12 ⋅ x 30 = x 42 20. ( y2 )9 ⋅ ( y3 )3 = y2⋅9 ⋅ y3⋅3 = y18 ⋅ y9 = y27 21. (5 x )3 = 53 x 3 = 125 x 3 22. (2 y)5 = 25 y5 = 32 y5

ISM: Prealgebra

Chapter 10: Exponents and Polynomials

23. (−6 xy 2 )(2 xy 5 ) = (−6 ⋅ 2)( x ⋅ x )( y2 ⋅ y5 ) 2 7

= −12 x y

3. (b + 3)(b + 5) = b(b + 5) + 3(b + 5) = b ⋅ b + b ⋅ 5 + 3⋅ b + 3⋅ 5

= b 2 + 5b + 3b + 15 = b 2 + 8b + 15

24. (−4 a2 b3 )( −3ab) = ( −4 ⋅ −3)( a2 ⋅ a1 )(b3 ⋅ b1 ) = 12 a3 b 4

25. ( y11z13 )3 = ( y11 )3 ( z13 )3 = y11⋅3 z13⋅3 = y33 z 39

4. (7 x − 1)(5 x + 4) = 7 x(5 x + 4) − 1(5 x + 4) = 7 x ⋅ 5x + 7 x ⋅ 4 − 1⋅ 5 x − 1⋅ 4

= 35 x 2 + 28 x − 5 x − 4 = 35 x 2 + 23x − 4

26. (a5 b12 )4 = (a5 ) 4 (b12 )4 = a5⋅4 b12⋅4 = a20 b 48 27. (10 x 2 y)2 (3 y) = 10 2 ( x 2 )2 y 2 ⋅ 3 y = 100 x 4 y 2 ⋅ 3 y = (100 ⋅ 3) x 4 ( y 2 ⋅ y1 )

5. (6 y − 1)2 = (6 y − 1)(6 y − 1) = 6 y (6 y − 1) − 1(6 y − 1) = 6 y ⋅ 6 y + 6 y (−1) − 1 ⋅ 6 y − 1(−1) = 36 y 2 − 6 y − 6 y + 1

4 3

= 300 x y

= 36 y 2 − 12 y + 1

28. (8 y3 z )2 (2 z 5 ) = 82 ( y3 )2 z 2 ⋅ 2 z 5

6. (10 x − 7)(2 x + 3) = 10 x ⋅ 2 x + 10 x ⋅ 3 + ( −7)(2 x) + (−7)(3)

= 64 y 6 z 2 ⋅ 2 z 5

= 20 x 2 + 30 x − 14 x − 21

= (64 ⋅ 2) y 6 ( z 2 ⋅ z 5 )

= 20 x 2 + 16 x − 21

= 128 y 6 z 7

29. (2a5b)4 (3a9 b4 )2 = 24 (a5 )4 b4 ⋅ 32 (a9 )2 (b4 )2

= 16a20 b4 ⋅ 9a18b8

7. (3 x + 2) 2 = (3 x + 2)(3 x + 2) = 3x ⋅ 3x + 3x ⋅ 2 + 2 ⋅ 3x + 2 ⋅ 2 = 9 x2 + 6 x + 6 x + 4

= (16 ⋅ 9)(a20 ⋅ a18 )(b4 ⋅ b8 )

= 9 x 2 + 12 x + 4

= 144a38b12 30. (5 x 4 y 6 )3 ( x 2 y 2 )5 = 53 ( x 4 )3 ( y 6 )3 ⋅ ( x 2 )5 ( y 2 )5

8. (2 x + 5)( x 2 + 4 x − 1) = 2 x( x 2 + 4 x − 1) + 5( x 2 + 4 x − 1)

= 125 x12 y18 ⋅ x10 y10

= 2 x ⋅ x 2 + 2 x ⋅ 4 x + 2 x(−1) + 5 ⋅ x 2 + 5 ⋅ 4 x + 5(−1)

= 125( x12 ⋅ x10 )( y18 ⋅ y10 )

= 2 x3 + 8 x 2 − 2 x + 5 x 2 + 20 x − 5

= 125 x 22 y 28

= 2 x3 + 13 x 2 + 18 x − 5

31. 2030 corresponds to x = 10.

x 2 + 3x − 2 3x + 4

−2.1x 2 + 54.5 x + 210.3 = −2.1(10)2 + 54.5(10) + 210.3 = −2.1(100) + 545 + 210.3 = −210 + 545 + 210.3 = 545.3 The amount spent on mobile advertising is predicted to be $545.3 billion in 2030. Section 10.3 Practice Exercises 1. 4 y (8 y 2 + 5) = 4 y ⋅ 8 y 2 + 4 y ⋅ 5 = 32 y3 + 20 y

4 x 2 + 12 x − 8 3 x3 + 9 x 2 3

3 x + 13x

−6 + 6x − 8

Vocabulary, Readiness & Video Check 10.3 1. The monomial is multiplied by each term in the trinomial.

2. 3r (8r 2 − r + 11) = 3r ⋅ 8r 2 − 3r ⋅ r + 3r ⋅11 = 24r 3 − 3r 2 + 33r

375

Chapter 10: Exponents and Polynomials

ISM: Prealgebra

2. Three times; in the first step (2x − 6) is distributed to (x + 4), and then in the second step, 2x is distributed to (x + 4) and −6 is distributed to (x + 4). 3. to make the point that the power property applies only to products and not to sums, so we cannot apply the power property to a binomial squared 4. binomial times binomial 5. No; it is a binomial times a trinomial, and FOIL can only be used to multiply a binomial and a binomial. Exercise Set 10.3 2. 4 y (10 y 3 + 2 y ) = 4 y ⋅10 y 3 + 4 y ⋅ 2 y

= 40 y 4 + 8 y 2 4. −4b( −2b 2 − 5b + 8) = ( −4b)( −2b 2 ) − ( −4b)(5b) + ( −4b)(8) = 8b3 + 20b 2 − 32b

6. 6 z 2 ( −3 z 2 − z + 4) = (6 z 2 )(−3 z 2 ) − (6 z 2 )( z ) + (6 z 2 )(4) = −18 z 4 − 6 z 3 + 24 z 2

8. ( y + 5)( y + 9) = y ( y + 9) + 5( y + 9) = y ⋅ y + y ⋅9 + 5⋅ y + 5⋅9 = y 2 + 9 y + 5 y + 45 = y 2 + 14 y + 45

10. (7 z + 1)( z − 6) = 7 z ( z − 6) + 1( z − 6) = 7 z ⋅ z − 7 z ⋅ 6 + 1⋅ z − 1⋅ 6

= 7 z 2 − 42 z + z − 6 = 7 z 2 − 41z − 6 12. (8b − 3)2 = (8b − 3)(8b − 3) = 8b(8b − 3) − 3(8b − 3) = 8b ⋅ 8b + 8b(−3) − 3 ⋅ 8b − 3(−3) = 64b 2 − 24b − 24b + 9 = 64b 2 − 48b + 9

14. ( y + 4)( y 2 + 8 y − 2) = y ( y 2 + 8 y − 2) + 4( y 2 + 8 y − 2) = y ⋅ y 2 + y ⋅ 8 y + y (−2) + 4 ⋅ y 2 + 4 ⋅ 8 y + 4(−2) = y 3 + 8 y 2 − 2 y + 4 y 2 + 32 y − 8 = y 3 + 12 y 2 + 30 y − 8

16. (9 z − 2)(2 z 2 + z + 1) = 9 z (2 z 2 + z + 1) − 2(2 z 2 + z + 1)

= 9 z ⋅ 2 z 2 + 9 z ⋅ z + 9 z ⋅1 + (−2)(2 z 2 ) + (−2)( z ) + (−2)(1) = 18 z 3 + 9 z 2 + 9 z − 4 z 2 − 2 z − 2 = 18 z 3 + 5 z 2 + 7 z − 2

376

ISM: Prealgebra

Chapter 10: Exponents and Polynomials

18. ( y 2 − 2 y + 5)( y 3 + 2 + y ) = y 2 ( y 3 + 2 + y ) − 2 y ( y3 + 2 + y ) + 5( y 3 + 2 + y ) = y 2 ⋅ y 3 + y 2 ⋅ 2 + y 2 ⋅ y + (−2 y )( y 3 ) + (−2 y )(2) + (−2 y )( y ) + 5 ⋅ y 3 + 5 ⋅ 2 + 5 ⋅ y = y 5 + 2 y 2 + y 3 − 2 y 4 − 4 y − 2 y 2 + 5 y 3 + 10 + 5 y = y 5 − 2 y 4 + 6 y 3 + y + 10

20. 5 x(4 x 2 + 5) = 5 x ⋅ 4 x 2 + 5 x ⋅ 5 = 20 x3 + 25 x 22. 3 z 3 (4 z 4 − 2 z + z 3 ) = 3 z 3 ⋅ 4 z 4 + 3 z 3 ( −2 z ) + 3 z 3 ⋅ z 3 = 12 z 7 − 6 z 4 + 3 z 6

24. ( y + 7)( y − 7) = y ( y − 7) + 7( y − 7) = y ⋅ y + y ⋅ (−7) + 7 ⋅ y + 7(−7) = y 2 − 7 y + 7 y − 49 = y 2 − 49

26. (6s + 1)(3s − 1) = 6s (3s − 1) + 1(3s − 1) = 6s ⋅ 3s + 6s(−1) + 1(3s) + 1(−1)

= 18s 2 − 6 s + 3s − 1 = 18s 2 − 3s − 1 28. ( x + 3)2 = ( x + 3)( x + 3) = x( x + 3) + 3( x + 3) = x 2 + 3x + 3x + 9 = x2 + 6 x + 9 7  3 3   7  3   30.  a −   a +  = a  a +  +  −   a +  10   10  10   10   10    3 7 21 2 = a + a− a− 10 10 100 4 21 2 =a − a− 10 100 21 2 2 =a − a− 5 100

32. (9 y − 1)( y 2 + 3 y − 5) = 9 y ( y 2 + 3 y − 5) + (−1)( y 2 + 3 y − 5) = 9 y 3 + 27 y 2 − 45 y − y 2 − 3 y + 5 = 9 y 3 + 26 y 2 − 48 y + 5

34. (5 x + 9) 2 = (5 x + 9)(5 x + 9) = 5 x(5 x + 9) + 9(5 x + 9) = 25 x 2 + 45 x + 45 x + 81 = 25 x 2 + 90 x + 81

377

Chapter 10: Exponents and Polynomials

36. (4a − 3) 2 = (4a − 3)(4a − 3) = 4a (4a − 3) + (−3)(4a − 3) = 16a 2 − 12a − 12a + 9 = 16a 2 − 24a + 9

38. (3 y 2 + 2)(5 y 2 − y + 2) = 3 y 2 (5 y 2 − y + 2) + 2(5 y 2 − y + 2) = 15 y 4 − 3 y 3 + 6 y 2 + 10 y 2 − 2 y + 4 = 15 y 4 − 3 y 3 + 16 y 2 − 2 y + 4

40. (a 4 + a 2 + 1)(a 4 + a 2 − 1) = a 4 (a 4 + a 2 − 1) + a 2 (a 4 + a 2 − 1) + 1(a 4 + a 2 − 1)

= a8 + a 6 − a 4 + a 6 + a 4 − a 2 + a 4 + a 2 − 1 = a8 + 2a 6 + a 4 − 1 2b 2 − 4b + 3

42.

b2 − b + 2

4b 2 − 8b + 6 −2b3 + 4b 2 − 3b 4

2b − 4b3 + 3b 2 2b 4 − 6b3 + 11b 2 − 11b + 6

44. 48 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 = 2 4 ⋅ 3 46. 36 = 2 ⋅ 2 ⋅ 3 ⋅ 3 = 2 2 ⋅ 32 48. 300 = 2 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ 5 = 2 2 ⋅ 3 ⋅ 52 50. (2 x + 11)(2 x + 11) = 2 x(2 x + 11) + 11(2 x + 11)

= 4 x 2 + 22 x + 22 x + 121 = 4 x 2 + 44 x + 121 The area is (4 x 2 + 44 x + 121) square centimeters. 52. (3x + 5)(3x − 5) − (2 x)(2 x) = 3x(3x − 5) + 5(3 x − 5) − 4 x 2

= 9 x 2 − 15 x + 15 x − 25 − 4 x 2 = 5 x 2 − 25 The area of the shaded figure is (5 x 2 − 25) square miles.

378

ISM: Prealgebra

Chapter 10: Exponents and Polynomials

Section 10.4 Practice Exercises

Exercise Set 10.4

1. 42 = 2 ⋅ 3 ⋅ 7 28 = 2 ⋅ 2 ⋅ 7 ↓ ↓ 2 ⋅ 7 The GCF is 2 ⋅ 7 = 14.

2. 36 = 2 ⋅ 2 ⋅ 3 ⋅ 3 20 = 2 ⋅ 2 ⋅ 5 GCF = 2 ⋅ 2 = 4

2. The GCF of z 7 , z8 , and z = z1 is z1 = z since 1 is the smallest exponent to which z is raised. 3. The GCF of 6, 3, and 15 is 3.

The GCF of a 4 , a5 , and a 2 is a 2 . The GCF of 6a 4 , 3a5 , and 15a 2 is 3a 2 . 4. The GCF of 10 y 7 and 5 y9 is 5 y 7 .

10 y 7 + 5 y9 = 5 y 7 ⋅ 2 + 5 y 7 ⋅ y 2 = 5 y 7 (2 + y 2 ) 4 z 2 − 12 z + 2 = 2 ⋅ 2 z 2 − 2 ⋅ 6 z + 2 ⋅1

8. 30 = 2 ⋅ 3 ⋅ 5 50 = 2 ⋅ 5 ⋅ 5 200 = 2 ⋅ 2 ⋅ 2 ⋅ 5 ⋅ 5 GCF = 2 ⋅ 5 = 10

x = x1 x5 = x1 ⋅ x 4

= 2(2 z 2 − 6 z + 1) 6. −3 y − 9 y + 15 x

6. 18 = 2 ⋅ 3 ⋅ 3 24 = 2 ⋅ 2 ⋅ 2 ⋅ 3 60 = 2 ⋅ 2 ⋅ 3 ⋅ 5 GCF = 2 ⋅ 3 = 6

10. x3 = x1 ⋅ x 2

5. The GCF of the terms is 2.

4. 96 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 45 = 3 ⋅ 3 ⋅ 5 GCF = 3

GCF = x1 = x

12. b6 = b4 ⋅ b2

= 3 ⋅ − y 2 + 3 ⋅ −3 y + 3 ⋅ 5 x 2 = 3(− y 2 − 3 y + 5 x 2 ) or − 3( y 2 + 3 y − 5 x 2 )

Vocabulary, Readiness & Vocabulary Check 10.4 1. In −3 ⋅ x 4 = −3 x 4 , the −3 and the x 4 are each 4

called a factor and −3x is called a product. 2. The greatest common factor (GCF) of a list of integers is the largest integer that is a factor of all integers in the list. 3. The GCF of a list of variables raised to powers in the variable raised to the smallest exponent in the list. 4. Factoring is the process of writing an expression as a product. 5. Factor means to write as a product.

b6 = b 4 ⋅ b 2 b4 = b4 GCF = b 4 14. a5b3 = a5 ⋅ b ⋅ b2

a 5b 2 = a 5 ⋅ b ⋅ b a 5b = a 5 ⋅ b GCF = a5 ⋅ b = a5b 16. 9 z 6 = 3 ⋅ 3 ⋅ z 3 ⋅ z 3

4 z5 = 2 ⋅ 2 ⋅ z3 ⋅ z 2 2 z3 = 2 ⋅ z3 GCF = z 3 18. 6 y 7 = 2 ⋅ 3 ⋅ y 5 ⋅ y 2 9 y 6 = 3 ⋅ 3 ⋅ y 5 ⋅ y1 15 y 5 = 3 ⋅ 5 ⋅ y 5

6. 2; z 3 ; 2 z 3 7. Find the GCF of the terms of the binomial and then factor it out.

GCF = 3 y 5

379

Chapter 10: Exponents and Polynomials

ISM: Prealgebra

20. 2 x 2 = 2 ⋅ x ⋅ x 18x = 2 ⋅ 3 ⋅ 3 ⋅ x GCF = 2x

38. a.

area on the right = 2 ⋅ 3 x 2 = 6 x 2 total area = 3 x 2 (5 x + 2) or 15 x 3 + 6 x 2

2 x 2 + 18 x = 2 x ⋅ x + 2 x ⋅ 9 = 2 x( x + 9)

b. answers may vary

22. 21 y 5 = y 5 ⋅ 21

40. answers may vary

y10 = y 5 ⋅ y 5 GCF = y 5 5

area on the left = 5 x ⋅ 3 x 2 = 15 x 3

42. Let x = 2 and z = −7.

21y + y 3

24. 9b = 9b ⋅ b

= y ⋅ 21 + y ⋅ y = y (21 + y )

( xy + z ) x = (2 y + (−7))2 = (2 y − 7) 2 = (2 y − 7)(2 y − 7) = 2 y(2 y − 7) − 7(2 y − 7)

54b 2 = 9b ⋅ 6b 9b = 9b ⋅ 1 GCF = 9b 3

= 4 y 2 − 14 y − 14 y + 49

9b − 54b + 9b = 9b ⋅ b − 9b ⋅ 6b + 9b ⋅1 = 9b(b2 − 6b + 1)

1. Factoring is the process of writing an expression as a product.

4 y5 = y5 ⋅ 4 GCF = y 5

y10 + 4 y5 = y5 ⋅ y5 + y5 ⋅ 4 = y5 ( y5 + 4) 28. −20x = −2 ⋅ 10x or 2 ⋅ −10x 4 x 2 = −2 ⋅ −2 x 2 or 2 ⋅ 2x 2 −2 = −2 ⋅ 1 or 2 ⋅ −1 GCF = −2 or 2

4. A polynomial with exactly 3 terms is called a trinomial.

= (−2)(10 x) + (−2)(−2 x 2 ) + (−2)(1) = −2(10 x − 2 x 2 + 1) or 2(−10 x + 2 x 2 − 1) 2

30. 25 z = 5 z ⋅ 5 z 20 z 2 = 5 z 2 ⋅ 4

25 z 3 − 20 z 2 = 5 z 2 ⋅ 5 z − 5 z 2 ⋅ 4 = 5 z 2 (5 z − 4)

36. 380

65 5 ⋅ 13 13 = = 100 5 ⋅ 20 20

5. A polynomial with exactly 2 terms is called a binomial. 6. A polynomial with exactly 1 term is called a monomial. 7. Monomials, binomials, and trinomials are all examples of polynomials.

GCF = 5z 2

34. 65% =

2. The greatest common factor of a list of terms is the product of all common factors. 3. The FOIL method may be used when multiplying two binomials.

−20 x + 4 x 2 − 2

32. 45% ⋅ 265 = x 0.45 ⋅ 265 = x 119.25 = x 45% of 265 is 119.25.

44. answers may vary Chapter 10 Vocabulary Check

26. y10 = y 5 ⋅ y 5

= 4 y 2 − 28 y + 49

8. In 5 x3 , the 3 is called an exponent. Chapter 10 Review 1. (2b + 7) + (8b − 10) = 2b + 8b + 7 − 10 = 10 b − 3 2. (7s − 6) + (14 s − 9) = 7s + 14 s − 6 − 9 = 21s − 15

3 3 100% 300 = ⋅ = % = 75% 4 4 1 4

ISM: Prealgebra

Chapter 10: Exponents and Polynomials

3. (3 x + 0.2) − (4 x − 2.6) = (3 x + 0.2) + (−4 x + 2.6) = 3 x − 4 x + 0.2 + 2.6 = − x + 2.8 4. (10 y − 6) − (11y + 6) = (10 y − 6) + (−11y − 6) = 10 y − 11y − 6 − 6 = − y − 12

16. (−3 x 2 y)(5 xy 4 ) = −3 ⋅ 5 x 2+1 y1+ 4 = −15 x 3 y5 17. (a5 )7 = a5⋅7 = a35 18. ( x 2 )4 ⋅ ( x10 )2 = x 2⋅4 ⋅ x10⋅2

= x 8 ⋅ x 20 = x 8+20

5. (4 z 2 + 6 z − 1) + (5z − 5) = 4 z 2 + 6 z + 5z − 1 − 5

= x 28

= 4 z 2 + 11z − 6 3

19. (9b)2 = 92 ⋅ b2 = 81b2

6. (17a + 11a + a) + (14a − a)

= 17a3 + 11a2 + 14a2 + a − a 3

= 17a + 25a

20. (a4 b2 c)5 = (a4 )5 ⋅ (b2 )5 ⋅ (c)5

= a4⋅5 ⋅ b2⋅5 ⋅ c 5 = a20 b10 c5

1  1  7.  9 y 2 − y +  −  20 y2 −  2  4  1  1  2 =  9 y − y +  +  −20 y 2 +  2  4  1 1 2 2 = 9 y − 20 y − y + + 2 4 3 2 = −11y − y + 4 2

8. −

21. (7 x )(2 x 5 )3 = 7 x ⋅ 23 ( x 5 )3

= 7 x ⋅ 8 x 5⋅3 = 7 x ⋅ 8 x15 = 56 x16

22. (3 x 6 y 5 )3 (2 x 6 y 5 )2 = 33 ( x 6 )3 (55 )3 ⋅ 22 ( x 6 )2 ( y 5 )2

x − 6x +1 ( x − 2)

x − 6x +1 −x+2

= 27 x18 y15 ⋅ 4 x12 y10 = 108 x 30 y25

x − 7x + 3

23. A = s 2 = (9a7 )(9a 7 ) = 9 ⋅ 9a 7+ 7 = 81a14

9. 5 x 2 = 5(3)2 = 5 ⋅ 9 = 45

The area is 81a14 square miles.

10. 2 − 7x = 2 − 7(3) = 2 − 21 = −19 11. (3x + 16) + (10x − 2) + (3x + 16) + (10x − 2) = 3x + 10x + 3x + 10x + 16 − 2 + 16 − 2 = 26x + 28 The perimeter is (26x + 28) feet. 12. Perimeter 2

= (4 x + 1) + (4 x + 1) + (4 x + 1) + (4 x + 1) = 4x2 + 4 x2 + 4x2 + 4x2 + 1 + 1 + 1 + 1 = 16 x 2 + 4

The perimeter is (16 x 2 + 4) meters.

24. A = lw = 3 x 4 ⋅ 9 x = 3 ⋅ 9 x 4 +1 = 27 x 5

The area is 27x 5 square inches. 25. 2 a(5a2 − 6) = 2a ⋅ 5a2 − 2a ⋅ 6 = 10 a3 − 12 a 26. −3 y2 ( y2 − 2 y + 1)

= −3 y 2 ⋅ y2 − (−3 y2 )(2 y) + (−3 y2 )(1) = −3 y 4 + 6 y3 − 3 y 2 27. ( x + 2)( x + 6) = x ⋅ x + x ⋅ 6 + 2 ⋅ x + 2 ⋅ 6 = x 2 + 6 x + 2 x + 12 = x 2 + 8 x + 12

13. x10 ⋅ x14 = x10 +14 = x 24 14. y ⋅ y6 = y1+6 = y 7

28. (3 x − 1)(5 x − 9) = 3 x ⋅ 5 x + 3 x (−9) − 1 ⋅ 5 x − 1(−9) = 15 x 2 − 27 x − 5 x + 9 = 15 x 2 − 32 x + 9

15. 4 z 2 ⋅ 6 z 5 = 4 ⋅ 6z 2 + 5 = 24 z 7

381

Chapter 10: Exponents and Polynomials

29. ( y − 5)2 = ( y − 5)( y − 5) = y ⋅ y + y(−5) − 5 ⋅ y − 5( −5) = y 2 − 5 y − 5 y + 25 = y 2 − 10 y + 25 2

30. (7a + 1) = (7a + 1)(7a + 1) = 7a ⋅ 7a + 7a ⋅ 1 + 1 ⋅ 7a + 1 ⋅ 1 2

= 49a + 7a + 7a + 1 = 49a + 14 a + 1

= x ( x 2 − 2 x + 3) + 1( x 2 − 2 x + 3) 2

= x − 2 x + 3x + x − 2 x + 3 3

= x − x + x +3 32. (4 y 2 − 3)(2 y 2 + y + 1) = 4 y 2 (2 y 2 + y + 1) − 3(2 y 2 + y + 1) 3

x2 = x2 x10 = x 2 ⋅ x 8

y7 = y7 y7 = y7 GCF = y 7

39. x 3 = x 2 ⋅ x

40. y10 = y 7 ⋅ y3

31. ( x + 1)( x 2 − 2 x + 3) 2

38. 10 = 2 ⋅ 5 20 = 2 ⋅ 2 ⋅ 5 25 = 5 ⋅ 5 GCF = 5

GCF = x 2

ISM: Prealgebra

= 8y + 4 y + 4 y − 6 y − 3y − 3 = 8 y 4 + 4 y3 − 2 y 2 − 3 y − 3

41. xy 2 = x ⋅ y ⋅ y xy = x ⋅ y x 3 y3 = x ⋅ x 2 ⋅ y ⋅ y 2 GCF = xy

42. a5b4 = a5 ⋅ b2 ⋅ b2

a 6 b3 = a 5 ⋅ a ⋅ b 2 ⋅ b

33. (3z 2 + 2 z + 1)( z 2 + z + 1)

= 3z 2 ( z 2 + z + 1) + 2 z ( z 2 + z + 1) + 1( z 2 + z + 1)

a 7 b2 = a5 ⋅ a2 ⋅ b2

= 3z 4 + 3z 3 + 3z 2 + 2 z 3 + 2 z 2 + 2 z + z 2 + z + 1

GCF = a 5 b 2

= 3z 4 + 5z 3 + 6 z 2 + 3z + 1 34. (a + 6)(a2 − a + 1) = a(a2 − a + 1) + 6(a2 − a + 1)

= a3 − a 2 + a + 6 a 2 − 6 a + 6 = a3 + 5a2 − 5a + 6 The area is (a3 + 5a2 − 5a + 6) square centimeters. 35. 20 = 2 ⋅ 2 ⋅ 5 35 = 5⋅ 7 GCF = 5

43. 5a3 = 5 ⋅ a ⋅ a 2 10a = 2 ⋅ 5 ⋅ a 20 a 4 = 2 ⋅ 2 ⋅ 5 ⋅ a ⋅ a3 GCF = 5a 44. 12 y2 z = 4 y2 z ⋅ 3

20 y2 z = 4 y2 z ⋅ 5 24 y5 z = 4 y2 z ⋅ 6 y3

GCF = 4y2 z

36. 12 = 2 ⋅ 2 ⋅ 3 32 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 GCF = 2 ⋅ 2 = 4

45. 2 x 2 = 2 x ⋅ x 12x = 2x ⋅ 6 GCF = 2x

37. 24 = 2 ⋅ 2 ⋅ 2 ⋅ 3 30 = 2 ⋅ 3 ⋅ 5 60 = 2 ⋅ 2 ⋅ 3 ⋅ 5 GCF = 2 ⋅ 3 = 6

46. 6 a 2 = 6 a ⋅ a 12a = 6a ⋅ 2 GCF = 6a

2 x 2 + 12 x = 2 x ⋅ x + 2 x ⋅ 6 = 2 x ( x + 6)

6 a2 − 12 a = 6 a ⋅ a − 6 a ⋅ 2 = 6 a(a − 2)

382

ISM: Prealgebra

Chapter 10: Exponents and Polynomials

47. 6 y 4 = y 4 ⋅ 6

59. (3 x + 4)2 = (3 x + 4)(3 x + 4)

y6 = y 4 ⋅ y2

= 9 x 2 + 12 x + 12 x + 16

GCF = y 4 6 y 4 − y 6 = y 4 ⋅ 6 − y 4 ⋅ y 2 = y 4 (6 − y 2 )

61. 28 = 2 ⋅ 2 ⋅ 7 32 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 40 = 2 ⋅ 2 ⋅ 2 ⋅ 5 GCF = 2 ⋅ 2 = 4

7 x 2 + 14 x + 7 = 7 ⋅ x 2 + 7 ⋅ 2 x + 7 ⋅ 1 = 7( x 2 + 2 x + 1)

62. 5z 5 = 5 ⋅ z 4 ⋅ z

49. 5a7 = a3 ⋅ 5a4

12 z8 = 2 ⋅ 2 ⋅ 3 ⋅ z 4 ⋅ z 4

a4 = a3 ⋅ a1

3z 4 = 3 ⋅ z 4

a3 = a3 ⋅ 1

GCF = z 4

5a7 − a 4 + a3 = a3 ⋅ 5a4 − a3 ⋅ a + a3 ⋅ 1

50. 10 y = 10 y ⋅ y 10y = 10y ⋅ 1 GCF = 10y

GCF = z 7

z 9 + 4 z 7 = z 7 ⋅ z 2 + z 7 ⋅ 4 = z 7 ( z 2 + 4) 5

10 y − 10 y = 10 y ⋅ y − 10 y ⋅ 1 = 10 y( y − 1) 51.

6 x5 = x5 ⋅ 6 x12 − 6 x 5 = x 5 ⋅ x 7 − x 5 ⋅ 6 = x 5 ( x 7 − 6)

65. 15a 4 = 15a 4 ⋅ 1 45a5 = 15a 4 ⋅ 3a

8y − 5 + − 12 y + 3 −4 y − 2

8y − 5 − (12 y − 3)

GCF = 15a 4 15a 4 + 45a5 = 15a 4 ⋅ 1 + 15a 4 ⋅ 3 = 15a 4 (1 + 3a)

53. x 5 ⋅ x16 = x 5+16 = x 21

66. 16 z 5 = 8z 5 ⋅ 2 24 z 8 = 8z 5 ⋅ 3z 3

54. y8 ⋅ y = y8+1 = y9

GCF = 8z 5 16 z 5 − 24 z 8 = 8z 5 ⋅ 2 − 8z 5 ⋅ 3z 3 = 8z 5 (2 − 3z 3 )

55. (a3b 5 c)6 = ( a3 )6 (b5 )6 c 6 = a18 b30 c 6

Chapter 10 Getting Ready for the Test

56. (9 x 2 ) ⋅ (3 x 2 )2 = (9 x 2 )(9 x 4 ) = 81x 6 3

64. x12 = x 5 ⋅ x 7 GCF = x 5

z 2 − 5z + 8 + 6z − 4 z2 + z + 4

52.

63. z 9 = z 7 ⋅ z 2 4z 7 = z 7 ⋅ 4

= a3 (5a 4 − a + 1) 6

60. (6 z + 5)( z − 2) = 6 z 2 − 12 z + 5z − 10 = 6 z 2 − 7 z − 10

48. 7 x 2 = 7 ⋅ x 2 14x = 7 ⋅ 2x 7=7⋅1 GCF = 7

GCF = a

= 9 x 2 + 24 x + 16

57. 3a(4 a − 5) = 3a ⋅ 4 a + 3a(−5) = 12 a − 15a 58. ( x + 4)( x + 5) = x 2 + 5 x + 4 x + 20 = x 2 + 9 x + 20

1. ( x 6 ) 2 = x 6⋅2 = x12

To simplify ( x6 )2 , the exponents are multiplied; C.

383

Chapter 10: Exponents and Polynomials

2. x 6 ⋅ x 2 = x 6 + 2 = x8

To simplify x 6 ⋅ x 2 , the exponents are added; A. 3. (3x + 1) + (x − 2) = 3x + x + 1 − 2 = 4x − 1 The operation is addition; A. 4. (3 x + 1) − ( x − 2) = 3 x + 1 + (− x + 2) = 3x − x + 1 + 2 = 2x + 3 The operation is subtraction; B. 5. (3 x + 1)( x − 2) = 3 x( x − 2) + 1( x − 2) = 3x2 − 6 x + x − 2 = 3x2 − 5 x − 2 The operation is multiplication; C.

6. −3x − 4 = −3(2) − 4 = −6 − 4 = −10 The correct choice is B. 2

ISM: Prealgebra

7. 300 − 16 x = 300 − 16(3) = 300 − 16(9) = 300 − 144 = 156 The correct choice is D.

− (8a2 + a)

−2a2 + a + 1

5. x 2 − 6 x + 1 = (8)2 − 6(8) + 1 = 64 − 48 + 1 = 17 6. y3 ⋅ y11 = y3+11 = y14 7. ( y3 )11 = y3⋅11 = y33 8. (2 x 2 ) 4 = 2 4 ⋅ ( x 2 ) 4 = 16 ⋅ x 2⋅4 = 16 x 8 9. (6a3 )(−2a7 ) = (6)(−2)(a3 ⋅ a7 ) = −12a10

= p 42 +12 = p54

9. −(4 x 2 + 7 x − 9) = −1(4 x 2 + 7 x − 9)

= −1(4 x 2 ) + (−1)(7 x) − (−1)(9) = −4 x 2 − 7 x + 9 The correct choice is D.

11. (3a4 b)2 (2ba4 )3 = (32 a4⋅2 b2 )(23 b3 a 4⋅3 )

= 9a8b2 ⋅ 8b3 a12 = 9 ⋅ 8a8+12 b2 +3 = 72a20 b5 12. 5 x (2 x 2 + 1.3) = 5 x ⋅ 2 x 2 + 5 x ⋅ 1.3 = 10 x 3 + 6.5 x 13. −2 y ( y 3 + 6 y 2 − 4) = −2 y ⋅ y 3 − 2 y ⋅ 6 y 2 − 2 y ⋅ (−4) = −2 y 4 − 12 y 3 + 8 y

Chapter 10 Test 1. (11x − 3) + (4 x − 1) = 11x + 4 x − 3 − 1 = 15 x − 4 2. (11x − 3) − (4 x − 1) = (11x − 3) + (−4 x + 1) = 11x − 4 x − 3 + 1 = 7x − 2

14. ( x − 3)( x + 2) = x ( x + 2) − 3( x + 2) = x ⋅ x + x ⋅2 − 3⋅ x − 3⋅2 = x2 + 2 x − 3x − 6 = x2 − x − 6

15. (5 x + 2)2 = (5 x + 2)(5 x + 2) = 5 x (5 x + 2) + 2(5 x + 2) = 5x ⋅ 5x + 5x ⋅ 2 + 2 ⋅ 5x + 2 ⋅ 2 = 25 x 2 + 10 x + 10 x + 4

= 3.4 y + 2 y − 3

384

+ − 8a2 − a

= p 42 ⋅ p12

8. (2 x + 3) − ( x − 2) = (2 x + 3) + (− x + 2) = 2x − x + 3 + 2 = x+5 The correct choice is C.

= 1.3 y 2 + 2.1y 2 + 5 y − 3 y − 3

6a2 + 2 a + 1

10. ( p6 ) 7 ( p2 )6 = p6⋅7 ⋅ p2⋅6

3. (1.3 y 2 + 5 y) + (2.1y2 − 3 y − 3)

6 a2 + 2 a + 1

= 25 x 2 + 20 x + 4

ISM: Prealgebra

Chapter 10: Exponents and Polynomials

16. (a + 2)(a2 − 2 a + 4)

23. 7 x 6 = x 3 ⋅ 7 x 3

= a(a2 − 2 a + 4) + 2(a2 − 2a + 4)

6 x 4 = x3 ⋅ 6 x

= a ⋅ a2 + a(−2a) + a ⋅ 4 + 2 ⋅ a2 + 2(−2a) + 2 ⋅ 4

x 3 = x 3 ⋅1

= a3 − 2 a 2 + 4 a + 2 a 2 − 4 a + 8

GCF = x 3

= a3 + 8

7 x 6 − 6 x 4 + x3 = x3 ⋅ 7 x3 − x3 ⋅ 6 x + x3 ⋅1

17. Area: ( x + 7)(5 x − 2) = x(5 x − 2) + 7(5 x − 2)

= 5 x 2 − 2 x + 35 x − 14 = 5 x 2 + 33x − 14 The area is (5 x2 + 33x − 14) square inches. Perimeter: 2(2x) + 2(5x − 2) = 4x + 10x − 4 = 14x − 4 The perimeter is (14x − 4) inches. 18. 45 = 3 ⋅ 3 ⋅ 5 60 = 2 ⋅ 2 ⋅ 3 ⋅ 5 GCF = 3 ⋅ 5 = 15

= x 3 (7 x 3 − 6 x + 1) Cumulative Review Chapters 1−10 1. Area = 380 ⋅ 280 = 106,400 The area of Colorado is 106,400 square miles. 2. 21 × 7 = 147 There are 147 pecan trees. 3. 1 + (−10) + (−8) + 9 = −9 + (−8) + 9 = −17 + 9 = −8 4. −2 + (−7) + 3 + (−4) = −9 + 3 + (−4) = −6 + (−4) = −10

19. 6 y3 = 2 ⋅ 3 ⋅ y3 9 y 5 = 3 ⋅ 3 ⋅ y3 ⋅ y 2 18 y 4 = 2 ⋅ 3 ⋅ 3 ⋅ y3 ⋅ y

5. 8 − 15 = 8 + (−15) = −7

GCF = 3 y3

6. 4 − 7 = 4 + (−7) = −3

20. 3 y = 3 y ⋅ y 15y = 3y ⋅ 5 GCF = 3y

7. −4 − (−5) = −4 + 5 = 1

3 y2 − 15 y = 3 y ⋅ y − 3 y ⋅ 5 = 3 y( y − 5) 21. 10a 2 = 2a ⋅ 5a 12a = 2a ⋅ 6 GCF = 2a

8. 3 − (−2) = 3 + 2 = 5 9. x = −60 + 4 + 10 = −56 + 10 = −46 10. x = −12 + 3 + 7 = −9 + 7 = −2 11.

17 − 7 x + 3 = −3 x + 21 − 3 x 20 − 7 x = −6 x + 21 20 − 7 x + 7 x = −6 x + 21 + 7 x 20 = x + 21 20 − 21 = x + 21 − 21 −1 = x

12.

20 − 6 x + 4 = −2 x + 18 + 2 x 24 − 6 x = 18 24 − 6 x − 24 = 18 − 24 −6 x = −6 −6 x −6 = −6 −6 x =1

10a + 12a = 2a ⋅ 5a + 2a ⋅ 6 = 2a(5a + 6) 22. 6 x 2 = 6 ⋅ x 2 12x = 6 ⋅ 2x 30 = 6 ⋅ 5 GCF = 6

6 x 2 − 12 x − 30 = 6 ⋅ x 2 − 6 ⋅ 2 x − 6 ⋅ 5 = 6( x 2 − 2 x − 5)

385

Chapter 10: Exponents and Polynomials

13.

2 x 3 x 2 x 2 3 x 3 4 x 9 x 13 x + = ⋅ + ⋅ = + = 15 10 15 2 10 3 30 30 30

14.

5 9 5 2 9 10 9 1 − = ⋅ − = − = 7 y 14 y 7 y 2 14 y 14 y 14 y 14 y

15. To round 736.2359 to the nearest tenth, notice that the digit in the hundredths place is 3. Since this digit is less than 5, we do not add 1 to the digit in the tenths place. 736.2359 rounded to the nearest tenth is 736.2. 16. To round 328.174 to the nearest tenth, notice that the digit in the hundredths place is 7. Since this digit is at least 5, we add 1 to the digit in the tenths place. 328.174 rounded to the nearest tenth is 328.2. 17.

23.850 + 1.604 25.454

18.

12.762 + 4.290 17.052

19.

3.7 y = −3.33 3.7( −9) 0 − 3.33 −33.3 = −3.33 False

ISM: Prealgebra

3.142 ≈ 3.14 27. 7 22.000 −21 10 −7 30 − 28 20 − 14 6 22 ≈ 3.14 7

28. 19

37 ≈ 1.947 19

No, −9 is not a solution. 20.

21.

2.8 x = 16.8 2.8(6) 0 16.8 16.8 = 16.8 True Yes, 6 is a solution.

818 = 0.818 1000

23.

0.12 = 0.012 10

24.

5.03 = 0.0503 100

29.

1 1 1 1 1 1 = since   = ⋅ = . 36 6 6 6 36 6

30.

4 2 2 2 4 2 = since   = ⋅ = . 25 5 5 5 25 5

786.1 = 0.7861 1000

22.

1.9473 ≈ 1.947 37.0000 −19 18 0 −17 1 90 −76 140 −133 70 −57 13

31. Let x be the height of the tree. 6 x = 9 69 6 ⋅ 69 = 9 x 414 = 9 x 414 9 x = 9 9 46 = x The height of the tree is 46 feet.

25. −2x + 5 = −2(3.8) + 5 = −7.6 + 5 = −2.6 26. 6x − 1 = 6(−2.1) − 1 = −12.6 − 1 = −13.6 386

ISM: Prealgebra

Chapter 10: Exponents and Polynomials

32. Let x be the height of the hydrant. 1 x = 2 6 1⋅ 6 = 2 ⋅ x 6 = 2x 6 2x = 2 2 3= x The height of the fire hydrant is 3 feet.

39. simple interest = P ⋅ R ⋅ T 8 12 2 = $2400 ⋅ 0.10 ⋅ 3 = $160 = $2400 ⋅ 10% ⋅

The interest is $160. 40. simple interest = P ⋅ R ⋅ T 10 12 5 = $1000 ⋅ 0.03 ⋅ 6 = $25

33. 1.2 = 30% ⋅ x

= $1000 ⋅ 3% ⋅

34. 9 = 45% ⋅ x 35. x ⋅ 50 = 8 50 x = 8 50 x 8 = 50 50 x = 0.16 x = 16%

The interest is $25. 41. 47% + 11% = 58%

58% of visitors came from Mexico or Canada.

16% of 50 is 8.

42. 2% + 2% + 13% = 17%

17% of visitors came from India, the United

36. x ⋅ 16 = 4 16 x = 4 16 x 4 = 16 16 x = 0.25 x = 25%

Kingdom, or South America & Caribbean. 43. P = 2l + 2w = 2(11) + 2(3) = 22 + 6 = 28

The perimeter is 28 inches.

25% of 16 is 4. 37.

44. P = 6 + 8 + 11 = 25

The perimeter is 25 feet.

31 = 4% ⋅ x 31 = 0.04 x 31 0.04 x = 0.04 0.04 775 = x

45. A = bh = 1.5 ⋅ 3.4 = 5.1

The area is 5.1 square miles.

There are 775 freshmen.

46. A =

38. 2% ⋅ x = 29 0.02 x = 29 0.02 x 29 = 0.02 0.02 x = 1450

1 1 bh = (17)(8) = 68 2 2

The area of the triangle is 68 square inches. 47.

8 tons 1000 lb − 3 tons 1350 lb

48.

5 tons 700 lb × 3 15 tons 2100 lb = 15 tons + 1 ton 100 lb =16 tons 100 lb

There are 1450 apples in the shipment.

7 tons 3000 lb − 3 tons 1350 lb 4 tons 1650 lb

387

Chapter 10: Exponents and Polynomials

ISM: Prealgebra 53. ( x + 2)( x + 3) = x( x + 3) + 2( x + 3) = x ⋅ x + x ⋅3+ 2⋅ x + 2⋅3

3210 ml 1L ⋅ 1 1000 ml 3210 = L 1000 = 3.21 L

49. 3210 ml =

= x2 + 3x + 2 x + 6 = x2 + 5x + 6

4321 cl 1 L 4321 50. 4321 cl = L = 43.21 L ⋅ = 1 100 cl 100

54. (2 x + 5)( x + 7) = 2 x( x + 7) + 5( x + 7) = 2x ⋅ x + 2x ⋅ 7 + 5 ⋅ x + 5 ⋅ 7

51. (3x − 1) + (−6x + 2) = 3x − 6x − 1 + 2 = −3x + 1 52. (7 a + 4) − (3a − 8) = (7 a + 4) + (−3a + 8) = 7 a − 3a + 4 + 8 = 4a + 12

388

= 2 x 2 + 14 x + 5 x + 35 = 2 x 2 + 19 x + 35

Appendices Appendix B Practice Exercises 1.

y10 y6 511 8

10 −6

14. y −6 ⋅ y 3 ⋅ y −4 = y −6+3 ⋅ y −4 = y −3 ⋅ y −4

= y −3+ ( −4) = y −7 1 = y7

= 511−8 = 53 = 125

15. (a 6b −4 )(a −3b8 ) = a 6+ ( −3) ⋅ b −4+8 = a3b 4

a 4 b11 12a 4b11 = 12 ⋅ ⋅ ab a1 b1 = 12 ⋅ (a 4−1 ) ⋅ (b11−1 )

16. (3 y 9 z10 )(2 y3 z −12 ) = 3 ⋅ 2 ⋅ y 9+3 ⋅ z10+ ( −12)

= 6 ⋅ y12 ⋅ z −2

= 12a3b10

4. 6 = 1 5. (−8)0 = 1

7. 7 y 0 = 7 ⋅ y 0 = 7 ⋅1 = 7 8. 5

1 = = 2 25 5

9. 5 x −2 = 5 ⋅

1 x

5 x

1 1 3 4 7 10. 4−1 + 3−1 = + = + = 4 3 12 12 12

6 11.   7 12.

13.

x x

−4

−2

6−2 7 −2

x1 x

Appendix B Exercise Set

6. −80 = −1 ⋅ 80 = −1 ⋅ 1 = −1

−2

6 y12

−4

6−2 1 1 7 2 7 2 49 ⋅ = ⋅ = = 1 7 −2 62 1 62 36

= x1− ( −4) = x5

= y −4−6 = y −10 =

y10 y

= y10−9 = y1 = y

57 54

= 57 − 4 = 53 = 125

x8 y 6 xy 5

9a 4b7

x8 y 6 ⋅ = x8−1 ⋅ y 6−5 = x7 y1 = x7 y 1 5 x y

9 a 4 b7 ⋅ ⋅ 27 ab 2 27 a1 b 2 1 = ⋅ a 4−1 ⋅ b7 − 2 3 1 3 5 = a b 3 a 3b 5 = 3 =

10. 230 = 1 1 y10

12. 4 y 0 = 4 ⋅ y 0 = 4 ⋅1 = 4 14. −20 = −1 ⋅ 20 = −1 ⋅1 = −1 16. (−2)0 = 1

389

Appendices

18. 6−2 =

20. 5 y

−4

ISM: Prealgebra

62 = 5⋅

24.

26.

28.

1 q −5

y y −3 x −4 x −1

34.

1 y

−6

1 1

= x −3+ ( −2)

= x −5 1 = x5

1 4

1 1 4 1 5 + = + = 4 16 16 16 16

46. (2 x10 y −3 )(9 x 4 y −7 ) = 2 ⋅ 9 ⋅ x10+ 4 ⋅ y −3+ ( −7) = 18 ⋅ x14 ⋅ y −10 1 = 18 x14 ⋅ y10

= q5

= y1−( −3) = y 4

y −3

= x −4− ( −1) = x −3 =

30. 4−2 − 4−3 =

3 32.    x

= x −3 ⋅ x −2

22. 4−1 + 4−2 =

44. x8 ⋅ x −11 ⋅ x −2 = x8+ ( −11) ⋅ x −2

1 36

−3

1 4

3−3 x

−3

−

1 3

48.

1 1 4 1 3 − = − = 16 64 64 64 64

3−3 1 1 x3 x3 ⋅ = ⋅ = 1 x −3 33 1 27

= y6

50.

52.

y19

y10

= y19−10 = y 9

y10

x11 y 7 x11 y 7 = ⋅ = x11−1 ⋅ y 7 −1 = x10 y 6 1 1 xy x y z4 12

= z 4−12 = z −8 =

1 z8

54. 5 y 0 = 5 ⋅ y 0 = 5 ⋅1 = 5

36. z 4 ⋅ z −5 ⋅ z 3 = z 4+ ( −5) ⋅ z 3 = z −1 ⋅ z 3 = z −1+3 = z 2 38. (a −20b8 )(a 22b −4 ) = a −20+ 22 ⋅ b8+ ( −4) = a 2b 4

56. 7−2 =

1 7

58. 9 y −7 = 9 ⋅

40. y −6 ⋅ y −3 ⋅ y 2 = y −6+ ( −3) ⋅ y 2 = y −9 ⋅ y 2 =y

−9 + 2

42. (4 x9 )(6 x −13 ) = 4 ⋅ 6 ⋅ x9+ ( −13)

1 49

1 y

60. 7−1 + 14−1 =

−7

=y 1 = y7

62.

r −15 r

−4

1 1

9 y7

1 1

1 1 2 1 3 + = + = 7 14 14 14 14

= r −15−( −4) = r −11 =

1 r11

64. y −9 ⋅ y 6 ⋅ y −9 = y −9+ 6 ⋅ y −9 = y −3 ⋅ y −9

= 36 ⋅ x −4 1 = 36 ⋅ x4 36 = x4

390

18 x14

= y −3+ ( −9) = y −12 1 = y12

ISM: Prealgebra

Appendices

14. 4.8 × 10 −6 = 0.0000048

66. ( x −4 y 5 )( x13 y −14 ) = x −4+13 ⋅ y 5+ ( −14) = x9 ⋅ y −9 1 = x9 ⋅ y9 = 16 −3

16. 9.07 ×1010 = 90, 700, 000, 000 18. 6.02214199 × 1023 = 602, 214,199, 000, 000, 000, 000, 000

x9 y9

−6 −3

16 + ( −6) −3+ ( −3)

68. (4m n )(11m n ) = 4 ⋅11 ⋅ m 10

= 44 ⋅ m ⋅ n 1 = 44m10 ⋅ n6 10 44m = n6

−6

760,000 = 7.6 ×105

0.00035 = 3.5 × 10 −4

2. a. b. 3. a.

9.062 × 10

−4

= 3.95 × 1011

26. (2.5 ×106 )(2 ×10−6 ) = 2.5 ⋅ 2 ⋅106+ ( −6)

= 5 × 100 =5 28. (5 × 106 )(4 × 10−8 ) = 5 ⋅ 4 ⋅106+ ( −8)

= 20 × 10−2 = 0.2

= 0.0009062

8.002 ×106 = 8,002, 000 7

−9

(8 ×10 )(3 ×10 ) = 8 ⋅ 3 ⋅10 ⋅10 = 24 ×10 = 0.24

22. 1.58 ×1011 = 158,000, 000, 000 24. The height of the bar appears to be about 395. 395 billion = 395(1, 000, 000, 000) = 395, 000, 000, 000

Appendix C Practice Exercises 1. a.

20. 721, 000, 000,000 = 7.21×1011

8 × 104

8 = × 104−( −3) −3 2 2 × 10

= 4 × 107 = 40, 000, 000

−9

30.

−2

32.

25 × 10−4 5 × 10

−9

25 × 10−4−( −9) 5

= 5 × 105 = 500, 000

0.4 × 105 11

0.2 × 10

0.4 × 105−11 0.2

= 2 × 10−6 = 0.000002

34. (9.460 ×1012 )(10, 000) = (9.460 ×1012 )(1.0 ×104 )

Appendix C Exercise Set

= 9.460 ⋅1.0 ⋅1012+ 4

2. 9,300, 000, 000 = 9.3 ×10 4. 0.00000017 = 1.7 × 10 −7

= 9.460 ×1016 Light travels 9.46 × 1016 kilometers in 10,000 years.

6. 0.00194 = 1.94 × 10−3 8. 700,000 = 7.0 ×105 10. 13, 000, 000, 000 = 1.3 ×1010 12. 9.056 × 10−4 = 0.0009056

391

Practice Final Exam 1. 23 ⋅ 52 = 2 ⋅ 2 ⋅ 2 ⋅ 5 ⋅ 5 = 200

12. −

2. 16 + 9 ÷ 3 ⋅ 4 − 7 = 16 + 3 ⋅ 4 − 7 = 16 + 12 − 7 = 28 − 7 = 21 3. 18 − 24 = 18 + (−24) = −6 4. 5 ⋅ (−20) = −100

13.

49 = 7 because 7 2 = 49.

14.

7. 0 ÷ 49 = 0 15.

8. 62 ÷ 0 is undefined.

10.

11.

8 2 −8 − 2 −10 2⋅5 2 − = = =− =− 15 y 15 y 15 y 15 y 3⋅5⋅ y 3y

11 3 5 11 ⋅ 2 3 ⋅ 3 5 − + = − + 12 8 24 12 ⋅ 2 8 ⋅ 3 24 22 9 5 = − + 24 24 24 22 − 9 + 5 = 24 18 = 24 3⋅ 6 = 4⋅6 3 = 4

11 11 3 −2 11 8 16 11 18

19 −2

6. (−5)3 − 24 ÷ (−3) = −125 − 24 ÷ (−3) = −125 − (−8) = −125 + 8 = −117

9. −

16 3 16 12 ÷− = − ⋅− 3 12 3 3 16 ⋅12 = 3⋅3 16 ⋅ 3 ⋅ 4 = 3⋅3 64 1 = or 21 3 3

3 11

0.23 + 1.63 1.86 = = −6.2 −0.3 −0.3

10.2 × 4.01 102 40 800 40.902

1 decimal place 2 decimal places

1 + 2 = 3 decimal places

16. 0.6% = 0.6(0.01) = 0.006 17. 6.1 = 6.1(100%) = 610% 18.

3 3 100 300 = ⋅ %= % = 37.5% 8 8 1 8

19. 0.345 = 20. −

3a 16 3a ⋅16 3⋅ a ⋅8⋅ 2 1 1 ⋅ = = = = 8 6a 3 8 ⋅ 6a 3 8 ⋅ 2 ⋅ 3 ⋅ a ⋅ a ⋅ a a ⋅ a a 2

345 5 ⋅ 69 69 = = 1000 5 ⋅ 200 200

13 1 ⋅13 1 1⋅ 5 5 =− =− =− = − = −0.5 26 2 ⋅13 2 2⋅5 10

21. 34.8923 rounded to the nearest tenth is 34.9. 22. 5( x3 − 2) = 5(23 − 2) = 5(8 − 2) = 5(6) = 30 23. 10 − y 2 = 10 − (−3)2 = 10 − 9 = 1

392

ISM: Prealgebra

Practice Final Exam

1 7 ÷3 2 8 1 31 = ÷ 2 8 1 8 = ⋅ 2 31 1⋅ 8 = 2 ⋅ 31 1⋅ 2 ⋅ 4 = 2 ⋅ 31 4 = 31

24. x ÷ y =

30.

25. −(3 z + 2) − 5 z − 18 = −1(3z + 2) − 5 z − 18 = −1 ⋅ 3z + (−1) ⋅ 2 − 5 z − 18 = −3z − 2 − 5 z − 18 = −3z − 5 z − 2 − 18 = −8 z − 20

31.

26. perimeter = 3(5x + 5) = 3 ⋅ 5x + 3 ⋅ 5 = 15x + 15 The perimeter is (15x + 15) inches. 27.

28.

n =4 −7 n −7 ⋅ = −7 ⋅ 4 −7 −7 ⋅ n = −7 ⋅ 4 −7 n = −28

−4 x + 7 = 15 −4 x + 7 − 7 = 15 − 7 −4 x = 8 −4 x 8 = −4 −4 x = −2

29. −4( x − 11) − 34 = 10 − 12 −4 x + 44 − 34 = 10 − 12 −4 x + 10 = −2 −4 x + 10 − 10 = −2 − 10 −4 x = −12 −4 x −12 = −4 −4 x=3

32.

x 24 +x=− 5 5 x   24  5 + x  = 5 −  5   5  x 5 ⋅ + 5 ⋅ x = −24 5 x + 5 x = −24 6 x = −24 6 x −24 = 6 6 x = −4 2( x + 5.7) = 6 x − 3.4 2 x + 11.4 = 6 x − 3.4 2 x + 11.4 − 11.4 = 6 x − 3.4 − 11.4 2 x = 6 x − 14.8 2 x − 6 x = 6 x − 6 x − 14.8 −4 x = −14.8 −4 x −14.8 = −4 −4 x = 3.7 8 11 = x 6 8 ⋅ 6 = x ⋅11 48 = 11x 48 11x = 11 11 48 =x 11 4 4 =x 11

33. Perimeter = (20 + 10 + 20 + 10) yards = 60 yards Area = (length)(width) = (20 yards)(10 yards) = 200 square yards 34. average =

−12 + (−13) + 0 + 9 −16 = = −4 4 4

35. The difference of three times a number and five times the same number is 4 translates to 3x − 5 x = 4 −2 x = 4 −2 x 4 = −2 −2 x = −2 The number is −2.

393

Practice Final Exam

ISM: Prealgebra

3 258 43 258 4 43 ⋅ 6 ⋅ 4 = ÷ = ⋅ = = 24 4 1 4 1 43 1 ⋅ 43 Expect to travel 24 miles on 1 gallon of gas.

36. 258 ÷ 10

37. Let x be the number of children participating in the event. Since the number of adults participating in the event is 112 more than the number of children, the number of adults is x + 112. Since the total number of participants in the event is 600, the sum of x and x + 112 is 600. x + x + 112 = 600 2 x + 112 = 600 2 x + 112 − 112 = 600 − 112 2 x = 488 2 x 488 = 2 2 x = 244 244 children participated in the event. 38. Let x be the number of grams. grams →10 x ← grams = pounds →15 80 ← pounds 10 ⋅ 80 = 15 ⋅ x 800 = 15 x 800 15 x = 15 15 1 53 = x 3 The standard dose for an 80-pound dog is 1 53 grams. 3 39. Amount of discount = 15% ⋅ $120 = 0.15 ⋅ $120 = $18 Sale price = $120 − $18 = $102 The amount of the discount is $18; the sale price is $102. 40. y + x = −4 Find any 3 ordered-pair solutions. Let x = 0. y + x = −4 y + 0 = −4 y = −4 (0, −4) Let y = 0. y + x = −4 0 + x = −4 x = −4 (−4, 0) 394

Let x = −2. y + x = −4 y + (−2) = −4 y + (−2) + 2 = −4 + 2 y = −2 (−2, −2) Plot (0, −4), (−4, 0), and (−2, −2). Then draw the line through them.

41. y = 3x − 5 Find any 3 ordered-pair solutions. Let x = 0. y = 3x − 5 y=3⋅0−5 y=0−5 y = −5 (0, −5) Let x = 1. y = 3x − 5 y=3⋅1−5 y=3−5 y = −2 (1, −2) Let x = 2. y = 3x − 5 y=3⋅2−5 y=6−5 y=1 (2, 1) Plot (0, −5), (1, −2), and (2, 1). Then draw the line through them.

ISM: Prealgebra

Practice Final Exam

42. y = −4 No matter what x-value we choose, y is always −4.

−2

−4

49. The complement of an angle that measures 78° is an angle that measures 90° − 78° = 12°. 50. ∠x and the angle marked 73° are vertical angles, so m∠x = 73°. ∠x and ∠y are alternate interior angles, so m∠y = m∠x = 73°. ∠x and ∠z are corresponding angles, so m∠z = m∠x = 73°. 51. The unmarked vertical side has length 11 in. − 7 in. = 4 in. The unmarked horizontal side has length 23 in. − 6 in. = 17 in. P = (6 + 4 + 17 + 7 + 23 + 11) in. = 68 in. Extending the unmarked vertical side downward divides the region into two rectangles. The region’s area is the sum of the areas of these: A = 11 in. ⋅ 6 in. + 7 in. ⋅17 in. = 66 sq in. + 119 sq in. = 185 sq in.

43. (11x − 3) + (4 x − 1) = 11x + 4 x − 3 − 1 = 15 x − 4 44.

6 a2 + 2 a + 1 − (8a2 + a)

6 a2 + 2 a + 1 + − 8a 2 − a −2 a 2 + a + 1

45. (6a3 )(−2a 7 ) = (6)(−2)(a3 ⋅ a 7 ) = −12a10

52. Circumference: C = 2⋅π⋅r = 2 ⋅ π ⋅ 9 in. = 18π in. ≈ 56.52 in. Area:

A = πr 2 = π(9 in.) 2 = 81π sq in. ≈ 254.34 sq in.

46. (3a 4 b)2 (2ba 4 )3 = (32 a 4⋅2 b2 )(23 b3 a 4⋅3 )

= 9a8 b2 ⋅ 8b3 a12 = 9 ⋅ 8a8+12 b2 +3 20 5

= 72 a b

47. ( x − 3)( x + 2) = x ( x + 2) − 3( x + 2) = x ⋅ x + x ⋅2 − 3⋅ x − 3⋅2

53. 2

2 1 gal 4 qt 1 1 gal = 2 ⋅ = 2 ⋅ 4 qt = 10 qt 2 1 1 gal 2

54. 2.4 kg =

2.4 kg 1000 g ⋅ = 2.4 ⋅1000 g = 2400 g 1 1 kg

395

= x 2 + 2 x − 3x − 6 = x2 − x − 6

48. 3 y 2 = 3 y ⋅ y 15y = 3y ⋅ 5 GCF = 3y 3 y 2 − 15 y = 3 y ⋅ y − 3 y ⋅ 5 = 3 y( y − 5)

INSTRUCTOR’S MANUAL WITH TESTS AND MINI-LECTURES EMILY KEATON

PREALGEBRA NINTH EDITION

Elayn Martin-Gay

Table of Contents Mini-Lectures (M) Chapter 1....................................................................................................................................... 1 Chapter 2....................................................................................................................................... 9 Chapter 3..................................................................................................................................... 15 Chapter 4..................................................................................................................................... 19 Chapter 5..................................................................................................................................... 27 Chapter 6..................................................................................................................................... 34 Chapter 7..................................................................................................................................... 39 Chapter 8..................................................................................................................................... 45 Chapter 9..................................................................................................................................... 50 Chapter 10................................................................................................................................... 57 Mini-Lecture Graphing Answers ................................................................................................ 61

Additional Exercises (E) Chapter 1....................................................................................................................................... 1 Chapter 2..................................................................................................................................... 12 Chapter 3..................................................................................................................................... 19 Chapter 4..................................................................................................................................... 24 Chapter 5..................................................................................................................................... 40 Chapter 6..................................................................................................................................... 50 Chapter 7..................................................................................................................................... 60 Chapter 8..................................................................................................................................... 69 Chapter 9..................................................................................................................................... 84 Chapter 10................................................................................................................................... 95 Answers ...................................................................................................................................... 99

Group Activities (G) Chapter 1....................................................................................................................................... 1 Chapter 2....................................................................................................................................... 3 Chapter 3....................................................................................................................................... 5 Chapter 4....................................................................................................................................... 7 Chapter 5....................................................................................................................................... 9 Chapter 6..................................................................................................................................... 11 Chapter 7..................................................................................................................................... 13 Chapter 8..................................................................................................................................... 15 Chapter 9..................................................................................................................................... 17 Chapter 10................................................................................................................................... 19 Answers ...................................................................................................................................... 21

Tests (T) Chapter 1....................................................................................................................................... 1 Chapter 2..................................................................................................................................... 19 Chapter 3..................................................................................................................................... 41 Chapter 4..................................................................................................................................... 60 Chapter 5..................................................................................................................................... 92 Chapter 6................................................................................................................................... 110 Chapter 7................................................................................................................................... 142 Chapter 8................................................................................................................................... 160 Chapter 9................................................................................................................................... 205 Chapter 10................................................................................................................................. 231 Final Exam................................................................................................................................ 249 Tests Answers ........................................................................................................................... 281 Final Exam Answers ................................................................................................................. 325

Mini-Lecture 1.1 Study Skill Tips for Success in Mathematics Learning Objectives: 1. 2. 3. 4. 5. 6. 7.

Get ready for this course. Understand some general tips for success. Know how to use this text. Know how to use text resources. Get help as soon as you need it. Learn how to prepare for and take an exam. Develop good time management.

Examples: 1. Get ready for this course. a) Positive attitude b) Understand how course is presented 2. Understand some general tips for success. a) Organize class materials b) Attend all classes d) Learn from mistakes

c) Avoid schedule conflicts

c) Do your homework and get help if needed e) Turn in all assignments on time

3. Know how to use the text. a) Each chapter is divided into sections, each of which is divided into numbered objectives. b) Each example in every section has a Practice exercise associated with it. c) Each section contains an exercise set with exercises keyed to the examples and objectives, as well as mixed exercises with no objective references. d) Review the meaning of icons used in text. e) Each chapter includes an Integrated Review midway through the chapter. f) Each chapter ends with Vocabulary Checks, Chapter Highlights, Chapter Reviews, Getting Ready for the Tests, Chapter Tests, and Cumulative Reviews. g) Student Resources section at the back of the text book contents Study Skill Builders, Study Guide Outline, a Practice Final, and Answers to Selected Exercises. 4. Know how to use text resources. a) Refer to the Interactive Video Lecture Series and Chapter Test Prep Videos in MyLab Math. b) Use the Video Organizer that accompanies this text. 5. Get help as soon as you need it. a) Try your instructor, a tutoring center, or a math lab, or you may want to form as study group with fellow classmates. 6. Learn how to prepare for and take an exam. a) Review previous homework assignments, class notes, quizzes, etc. b) Read Chapter Highlights to review concepts and definitions. c) Practice exercises in the Chapter Review, Getting Ready for the Test, and Chapter Test. d) When taking a test, read directions and problems carefully. e) Pace yourself. Use all available time. Check your work and answers. 7. Develop good time management. a) Make a list of all weekly commitments with estimated time needed. b) Be sure to schedule study time. Don’t forget eating, sleeping, and relaxing! Teaching Notes: • •

M-1

Mini-Lecture 1.2 Place Value, Names for Numbers, and Reading Tables Learning Objectives: 1. 2. 3. 4.

Find the place value of a digit in a whole number. Write a whole number in words and in standard form. Write a whole number in expanded form. Read tables.

Examples: 1. Find the place value of the digit 7 in each whole number. a) 7,352

b) 607

c) 702,433

d) 17,009,321

c) 17,403

d) 1,067,599

2. Write each whole number in words. a) 62

b) 698

Write each number in standard form. e) nine hundred fifty-two f) three hundred sixty-two thousand, five hundred eighty-six g) three million, four hundred thousand, one hundred two 3. Write each number in expanded form. a) 398

b) 2,907

c) 4,089,347

4. Use the following table of Number of Students Enrolled to answer the questions. Subject Basic Mathematics Statistics

Section 1 23 20

Section 2 27 25

Section 3 19 22

Total 69 67

a) How many total students are enrolled in Basic Mathematics? b) How many students are enrolled in Section 3 of Statistics? Teaching Notes: •

•

Students who do not have English as their first language will need additional assistance learning place value vocabulary. Students who do not have English as their first language may use periods instead of commas in writing numbers.

Answers: 1a) thousands, b) ones, c) hundred thousands, d) million; 2a) sixty-two, b) six hundred ninety-eight, c) seventeen thousand, four hundred three, d) one million, sixty-seven thousand, five hundred ninety-nine, e) 952, f) 362,586, g) 3,400,102; 3a)300+90+8, b) 2000+900+7, c)4,000,000+80,000+9,000+300+40+7; 4a) 69, b) 22.

M-2

Mini-Lecture 1.3 Adding and Subtracting Whole Numbers, and Perimeter Learning Objectives: 1. 2. 3. 4.

Add whole numbers. Subtract whole numbers. Find the perimeter of a polygon. Solve applications by adding or subtracting whole numbers.

Examples: 1. Add. a) 3 + 9

b) 40 + 70

c) 1900 + 17

d) 5703 + 0

g) 93 + 145 + 69

51 +27

7329 + 683

6,403 793 17,187

2. Subtract. Check by adding. a) 11 – 7

b) 15 – 8

c) 22 – 22

d) 31 – 0

27 - 13

198 - 94

3004 -2965

20,003 -16,867

3. Find the perimeter of each figure a)

17 feet

b) 8 meters 10 meters

19 feet

6 meters

11 feet 4. Solve the following word problems. a) What is the sum of 8,932 and 14,799? b) Subtract 376 from 803. c) The Library Renovation Project has set a goal of $75,000 to fundraise. To date, $47,908 has been fundraised. How much more money does the Library Renovation Project need to fundraise? d) On Monday, Ichika drove 57 miles; on Tuesday, she drove 39 miles; and on Wednesday, she drove 92 miles. How many total miles did Ichika drive on these three days? Teaching Notes: • Some students need additional practice with basic addition and subtraction facts. • Remind students that it is acceptable to write the carry digit in order to obtain the correct answer. • Most students will find this section easy but may need assistance with word problems. • Many students need to write the borrowing/regrouping step to maintain accuracy. • Many students are challenged when borrowing with zeros. Answers: 1a) 12, b) 110, c) 1917, d) 5703, e) 78, f) 8012, g) 307, h) 24,383; 2a)4, b) 7, c) 0, d) 31, e) 14, f) 104, g) 39, h) 3,136; 3a) 66 ft., b) 24 m.; 4a) 23,731, b) 427, c) $27,092, d) 188 miles. Copyright © 2025 Pearson Education, Inc.

M-3

Mini-Lecture 1.4 Rounding and Estimating Learning Objectives: 1. Round whole numbers. 2. Use rounding to estimate sums and differences. 3. Solve applicationss by estimating.

Examples: 1. Round to the nearest ten. a) 31

b) 57

c) 346

d) 2,795

g) 8,672

h) 1,899

Round to the nearest hundred. e) 312

f) 6,658

2. Round to the nearest thousand to find the estimated sum or difference. a)

4892 – 2305

2731 + 3020

17,032 – 12,513

24,803 + 14,587

3. Solve. a) At the last 3 dances, attendance was 657 students, 403 students, and 559 students. Estimate the total attendance by rounding each to the nearest hundred. b) Enrollment figures at the Town of Johnson’s School Department increased from 6,721 students to 7,653 students. Round each number to the nearest hundred to estimate the increase. c) The Hernandez family needs to buy a refrigerator for $999, a stove for $459, and a dishwasher for $449. Round each cost to the nearest hundred to estimate the total cost. Teaching Notes: • •

•

Some students need to be repeatedly reminded to look at the digit to the right of the rounding position. Have students draw a line after the digit in the rounding position. A common error students make is to leave the digits to the right of the rounding position the same instead of changing them to zeros after rounding. Stress the importance of rounding and estimating with applications.

Answers: 1a) 30, b) 60, c) 350, d) 2,800, e) 300, f) 6,700, g) 8,700, h) 1,900; 2a) 3000, b) 6000, c) 4000, d) 40,000; 3a) 1700, b) 1,000, c) $1,900

M-4

Mini-Lecture 1.5 Multiplying Whole Numbers and Area Learning Objectives: 1. 2. 3. 4.

Use the properties of multiplication. Multiply whole numbers. Find the area of a rectangle. Solve applications by multiplying whole numbers.

Examples: 1. Multiply. a) 37 · 1

b) 1 · 22

c) 0 · 183

d) 9 · 5 · 0

Use the distributive property to rewrite each expression. e) 2( 5 + 4)

f) 5(1 + 9)

g) 10(9 + 6)

h) 15(0 + 14)

2. Multiply. a)

37 × 6

643 × 27

412 × 4

309 × 800

1708 × 9 825 × 1,000

337 ×_ 25 2,477 × 963

3. Find the area of a rectangle with length 14 feet and width 8 feet. 4. At a recent football game, 413 adult tickets were sold at a price of $5 each. There were 127 child tickets sold at a price of $3 each. How much total amount of money in ticket sales for the game? Teaching Notes: • • •

•

Some students need additional practice with basic multiplication facts. Some students do not know the different types of symbols used for multiplication. When using distributive property, many students forget to distribute over both terms. When multiplying, remind students to carefully line up the ones, tens, hundreds, etc.

Answers: 1a) 37, b) 22, c) 0, d) 0, e) 18, f) 50, g) 150, h) 210; 2a) 222, b) 1648, c) 15,372, d) 8,425, e) 17,361, f) 247,200, g) 825,000, h) 2,385,351; 3) 112 sq .ft.; 4) $2,446

M-5

Mini-Lecture 1.6 Dividing Whole Numbers Learning Objectives: 1. 2. 3. 4.

Divide whole numbers Perform long division. Solve applications that require dividing by whole numbers. Find the average of a list of numbers

Examples: 1. Find each quotient. Check by multiplying. c)

5 5

a) 228 ÷ 4

572 7

c) 1570 ÷ 3

d) 14 7070

97 41,270

f) 603 604,911

a) 3 12

b) 13 ÷ 1

d) 15 ÷ 15

e) 0 5

2. Divide. Check by multiplying.

a) Find the quotient of 94 and 5. b) Recently, Hasina earned $1,722 selling calendars. If each calendar cost $14, how many calendars did Hasina sell?

During the semester, Kashif’s test scores were: 87, 93, 62, 83 and 100. What was Kashif’s average for the semester?

Teaching Notes: • •

•

Some students need additional practice with basic division facts. Many students confuse division by zero (undefined) and zero divided by any non-zero number ( = 0). Many students need to be cautious with placement of digits in quotient and dividend. Be sure appropriate place values are lined up. Stress organization!

Answers: 1a) 4, b) 13, c) 1, d) 1, e) undefined; 2a) 57, b) 81r5, c) 523 r1; d) 505, e) 425 r45, f) 1003 r102; 3a) 18 r4, b) 123; 4) 85

M-6

Mini-Lecture 1.7 Exponents and Order of Operations Learning Objectives: 1. 2. 3. 4.

Write repeated factors using exponential notation. Evaluate expressions containing exponents. Use the order of operations. Find the area of a square.

Examples: 1. Write using exponential notation. a) 2 · 2 · 2 · 2 · 2 · 2

b) (7)(7)(7)

c) 4 · 4 · 3 · 3 · 3

d) 5 · 5 · 8 · 8 · 5 · 5

b) 73

c) 36

d) 104

2. Evaluate. a) 52

3. Using order of operations, simplify. a) 3 · 4 – 10 ÷ 2

b) 62 ÷ 3 · 2

c) 8 · 4 + {27 ÷ [8 – (3 + 2)]}

4. a) Find the area of a square whose side measures 6 feet. b) Find the area of a square whose side measures 23 miles. Teaching Notes: • •

• • • •

Students may confuse exponent and base. Many students have trouble with order of operations. Avoid “PEMDAS” as many students will multiply before dividing and add before subtracting. Stress to students that all multiplication/division must be preformed in order from left to right. Stress to students that addition/subtraction is preformed in order from left to right. Stress to students you can only add/subtract after all multiplication/division is complete.

Answers: 1a)26, b) 73, c) 33 · 42, d) 54 · 82; 2a) 25, b) 343, c) 729, d) 10,000; 3a) 7, b) 24, c)41; 4a)36 sq ft, b) 529 sq ft.

M-7

Mini-Lecture 1.8 Introduction to Variables, Algebraic Expressions, and Equations Learning Objectives: 1. Evaluate algebraic expressions given replacement values. 2. Identify solutions of equations. 3. Translate phrases into variable expressions. Addition (+) Sum, plus, added to, more than, increased by, total

Subtraction (–) Difference, minus, subtract, less than, decreased by, less

Multiplication (•) Product, times, multiply, multiply by, of, double, triple

Division (÷) Quotient, divide, shared equally among, per, divided by, divided into

Examples: 1. Evaluate each expression for x = 12 , y = 4 , and z = 3 a) x − y + z

e) x − 4 y

b) x − ( y + z )

f) y − 2 x

c) 5 ( 3x + 7 )

d) 2 xy − 3 z

3x yz − g) 4 3

 2y z − x  h)   2  

2. Determine whether the given number is a solution of the given equation. a) Is 10 a solution of n – 3 = 7? b) Is 3 a solution of 2n = 12? Determine which numbers in each set are solutions to the corresponding equations. c) n – 3 = 12; {11, 12, 15} d) 4n = 24; {3, 6, 20} 3. Write each phrase as a variable expression. Use x to represent “a number.” a) The sum of a number and eleven b) Fifteen added to a number c) The difference between a number and three hundred d) A number subtracted from forty-two e) The product of sixteen and a number f) A number times thirteen g) The quotient of thirty and a number h) Seven divided by a number i) The quotient of eighteen and a number, decreased by two Teaching Notes:

• • • •

Remind students that order of operations apply with variables. Stress to students that an equation ha an equal sign and an expression does not. Many students will have difficulty translating a phrase into an algebraic expression. Refer students to textbook for Translating Phrases into Variable Expressions Chart.

Answers: 1a) 11, b) 5, c) 215, d) 87, e) 128, f) 40, g) 5, h)216; 2a) yes, b) no, c) 15, d) 6; 3a) x + 11, b) 15 + x, c) x – 300, d) 42 – x, e) 16x, f) 13x, g) 30/x, h) 7/x, i) M-8

18 −2 x

Mini-Lecture 2.1 Introduction to Integers Learning Objectives: 1. 2. 3. 4. 5. 6.

Represent real-life situations with integers. Graph integers on a number line. Compare integers. Find the absolute value of a number. Find the opposite of a number. Read bar graphs containing integers.

Examples: 1. Represent each quantity using an integer. a) A scuba diver is swimming 25 feet below sea level. b) The record high temperature for the town is 113°F. c) The number of televisions sold reflected a 35 percent loss from the previous year. 2. Graph each integer in the list on the same number line. a) 1, 3, 5, 6

b) 2, -2, 3, -3

c) 4, 0, -2, -5

d) 0, –1, –2, –5

3. Insert < or > between each pair of integers to make a true statement. c) −42 ____ − 38

b) 0 ____ −3

a) 5 ____ 10

d) −22 ____ 22

4. Simplify.

−12

a) 2

e) − 45

f) − −103

−3

d) − 14

g) − x if x = -25

c) 0

d) −16

x if x = −8

5. Find the opposite of each integer. b) −15

a) 9

6. The bar graph shows the January temperatures for four days in Boston..

Which day was the coldest?

Which day was the warmest?

10 5 0 -5 -10

Mon.

Tues.

Wed.

Thurs.

Teaching Notes: • •

Many students will confuse absolute value and opposite. Encourage students to list everyday situation where negative numbers are used.

Answers: 1a) –25; b) +113; c) –35; 2a) d)

; 3a) <; b) >; c) <; d) <; 4a) 2, b) 12; c) 3; d) -14; e) -45; f) -103;

M-9

Mini-Lecture 2.2 Adding Integers Learning Objectives: 1. Add integers. 2. Evaluate an algebraic expression by adding. 3. Solve applications by adding integers. Examples: 1. Add. a) 23 + 12

b) −23 + (−17)

c) −11 + (−2)

d) −21 + (−13)

e) 6 + ( −8)

f) −3 + 5

g) −74 + 27

h) −51 + (24)

i) −8 + ( −13)

j) −79 + 97

k) 46 + ( −54)

l) −4 + (−24)

m) 23 + (−19) + (−8)

n) 14 + 25 + (−16)

o) −25 + (−4) + (−2) + (−6)

2. Evaluate x + y for the given replacement values.

a) x = −5 and y = 14

b) x = −33 and y = −27

c) x = −43 and y = 38

3. Solve. a) Find the sum of −7 and 25.

b) Find the sum of −52, 13, and − 82

c) During a storm in Anchorage Alaska, the temperature was 10°F at Noon. At 1 p.m., the temperature had dropped 7°. At 2 p.m., the temperature dropped another 5°; and finally, at 3 p.m., the temperature had dropped an additional 9°. Use positive and negative numbers to represent his situation. Then find the present temperature. Teaching Notes: • • •

Some students need to see adding integers done on a number line first. Many students have a better understanding if they think of depositing and withdrawing money from a bank account. Refer students to the rules for adding signed numbers in the textbook.

Answers: 1a) 35; b) –40; c) –13; d) –34; e) –2; f) 2; g) –47; h) –27; i) –21; j)18; k) –8; l) –28; m) –4; n) 23; o) –37; 2a) 9; b) –60; c) –5; 3a) 18; b) –121; c) –11°.

M-10

Mini-Lecture 2.3 Subtracting Integers Learning Objectives:

1. Subtract integers. 2. Add and subtract integers. 3. Evaluate an algebraic expression by subtracting. 4. Solve applications by subtracting integers. Examples:

1. Subtract. a) −9 − (−2)

b) −14 − (−2)

c) 4 − ( −3)

d) 20 − 20

e) 2 − 5

f) −2 − 12

g) −150 − 410

h) −147 − (−85)

b) −1 − 11 − 12

c) −1 − 20 + 10

d) −16 + 11 − 18 + ( −4)

2. Simplify. a) 6 + 20 − 15

3. Evaluate x − y for the given replacement values. a) x = − 2 and y = − 8 c) x = −9 and y = − 9

b) x = 8 and y = − 32 d) x = 3 and y = − 15

4. Solve. a) A student has $545 in her checking account. She writes a check for $257, makes a deposit of $75, and then writes another check for $409. Find the balance in her account. (Write the amount as an integer.) b) The city of Manchester has an elevation of 13,005 feet above sea level while the city of Catherine has an elevation of 17,532 feet below sea level. Find the difference in elevation between those two cities. c) The temperature on a January morning in Worcester is −5° F at 2 a.m. If the temperature drops 4° by 3 a.m., rise 6° by 4 a.m., and then drops 8° by 5 a.m., find the temperature by 8 a.m. Teaching Notes: • • •

Many students find subtracting signed numbers difficult at first. Some students like to see subtracting signed numbers on a number line. Many students make errors when evaluating x – y when y is a negative number. Encourage students to make a direct substitution first so they do not forget to write the subtraction symbol.

Answers: 1a) –7; b) –12; c) 7; d) 0; e) –3; f) –14; g) –560; h) –62; 2a) 11; b) –24; c) –11; d) –27; 3a) 6; b) 40; c) 0; d) 18; 4a) –$46; b) 4527 ft.; c) –11°F.

M-11

Mini-Lecture 2.4 Multiplying and Dividing Integers Learning Objectives:

1. 2. 3. 4.

Multiply integers. Divide integers. Evaluate an algebraic expression by multiplying or dividing. Solve applications by multiplying or dividing integers.

Examples:

1. Multiply. a) 7(−6)

b) −4(10)

c) −20(13)

d) −10(−19)

e) ( −4)(−3)(6)

f) (−50)(0)(−5)(8)

g) (−4)( −5)( −4)(−3)

h) (−2)(3)(−1)( −4)(2)

i) −42

j) ( −3)

k) −33

l) ( −8)

2. Divide, if possible. a) 21 ÷ 7

b) 36 ÷ (−6)

−48 6

x for the given replacement values. y b) x = − 30 and y = − 10 a) x = 8 and y = −4

−17 0

3. Evaluate xy and also

c) x = 0 and y = −16

4. Solve. a) Find the product of −13 and −5 . b) Find the quotient of 63 and −9 . c) Better Electric Co. marked $15 off the price of each microwave in stock. If there are 57 microwaves in stock, write the total reduction in price of all microwaves as an integer. d) During a cold front in Canada the temperature dropped 4°F each hour for 7 hours. Express the total drop in temperature as an integer. Teaching Notes: • • •

Some students need a review of basic multiplication and division facts before they begin working with integers. Some students mix up the rules for addition of integers and the rules for multiplication/division of integers. Many students have a hard time understanding the difference between −32 and ( −3)

Answers:1a) –42; b) –40; c) –260; d)190; e) 72; f) 0; g) 240; h) –48; i) –16; j) –27; k) –27; l) 64; 2a) 3; b) –6; c) –8; d) undefined; 3a) –2; b) 3; c) 0;4a) 65; b) –7; c) –$855; d) –28°F.

M-12

Mini-Lecture 2.5 Order of Operations Learning Objectives:

1. Simplify expressions by using the order of operations. 2. Evaluate an algebraic expression. 3. Find the average of a list of numbers. Examples:

1. Simplify. a) −2 + 5  6

b) −2 − 5(5 − 8)

c) 2( −5)(7 − 3) − 7

d) 80 ÷ (−8) − 15

e) 33 − 8(2)

f) 8 − 2 7 − 22 + 3

g) 82 − 2(6) + 45 ÷ 5

h) 3( −2) + (8 − 10) 2

i) 21 ÷  7( −15 ÷ ( −5 ) ) 

8(−2) − 4 + 3 −85 ÷ 5

(

[ −36 ÷ (−4) − 1] [ 2 − (−2)]

)

20(−1) − (−5)(−2) 3.[−12 ÷ (−3 − 3)]

2. Evaluate each expression for x = −3, y = 6, and z = −1 . a) x + y + z

b) 2 y − 3z + x

c) x 2 − y + z

8x 2y

e) 5 y − x 2

f) x 3 + yz

3. Find the average of each list of numbers. a) –20, –9, –1, 0, 4, 6 , 6

b) –50, –30, –15, –5

Teaching Notes: • • •

Many students confuse the addition/subtraction rules with the multiplication/division rules when working with many operations in one expression. Encourage students to perform one operation at a time. Refer students to Order of Operations in the textbook.

M-13

Mini-Lecture 2.6 Solving Equations: The Addition and Multiplication Properties Learning Objectives: 1. Identify solutions of equations. 2. Use the addition property of equality to solve equations. 3. Use the multiplication property of equality to solve equations. Examples: 1. Determine whether the given number is a solution of the given equation. a) Is 12 a solution of x + 3 = 15?

b) Is 8 a solution of z – 15 = 23?

c) Is –2 a solution of 4k = k – 6?

d) Is 5 a solution of 6(x – 2) = 3x + 1?

e) Is

1 a solution of –3x = 5x + 1? 2

f) Is –2 a solution of –2x + 5 = 6x – 5x + 7?

2. Solve. Check each solution. a) a + 7 = 25

b) d – 4 = –19

c) 10z = 9z – 13

d) –14 = 15 + x

3. Solve. Check each solution. x =5 −5

a) 3x = 18

c) –5y = 0

d) –20x = –20

Teaching Notes: • • • •

Encourage students to write down all steps in a neat, organized manner. This habit will help students as equations increase in difficulty. Encourage students to use the addition property in such a way that the variable ends up with a positive coefficient. Mention to students that it does not matter on which side of the equation you isolate the variable. Remind students to always check their final answer by substituting it back into the original equation.

Answers: 1a) yes, b) no, c) yes, d) no, e) no, f) no; 2a) 18, b) –15, c) –13, d) –29; 3a) 6, b) –25, c) 0, d) 1

M-14

Mini-Lecture 3.1 Simplifying Algebraic Expressions Learning Objectives: 1. 2. 3. 4.

Use properties of numbers to combine like terms. Use properties of numbers to multiply expressions. Simplify expressions by multiplying and then combining like terms. Find the perimeter and area of figures.

Examples: 1. Simplify each expression by combining like terms. a) 3x + 5x

b) 11y – 9y

c) 5a – 19a

d) 6z + 15z – 5z + 7

e) 4.2 + 8.7x – 1.9 – 3.3x

4 2 1 1 x− + x− 5 3 3 5

2. Multiply. a) 7(4x)

b) –12(6a)

2 ( −15x ) 5

d) –5(3y –2)

3. Simplify each expression. Use the distributive property to remove parentheses. a) 5(y + 3) – 6

b) 2(7 – 3a) + a

c) –3(x + 1) + 4(8 – x) – 18

4. Find the perimeter of the figures.

a) 5 in

5 in

6 in

b) 6 ft 12 ft

Teaching Notes: • • • •

Some students need to practice identifying “like terms”. Some students do not know that a variable without a numerical coefficient actually has a coefficient of 1. Many students tend to make careless errors associated with the distributive property. Remind students that perimeter is the distance around an object.

Answers: 1a)8x, b) 2y, c) –14a, d) 16z + 7, e)5.4x + 2.3, f)17/15 x – 13/15; 2a) 28x, b) –72a, c) –6x, d) –15y +10; 3a) 5y + 9, b) 14 – 5a, c) –7x + 11; 4a) 16in, b) 36 ft

M-15

Mini-Lecture 3.2 Solving Equations: Review of the Addition and Multiplication Properties Learning Objectives: 1. Use the addition property or the multiplication property to solve equations. 2. Use both properties to solve equations. 3. Translate word phrases to mathematical expressions. Examples: 1. Solve. Check each solution. a) –14 = 15 + x c)

b) 3y – 7y = 12

x = 11 − 5 4

d) 5x + 2 – 4x = 7 – 19

e) 3(3x – 5) = 10x

f) 13x = 4(3x – 1)

2. Solve each equation. Check each solution. a) 5y + 2 = 17

b) 3x – 5 = 10

x −12 3. Write each phrase as a variable expression. Use x to represent “a number.” a) Eight subtracted from a number b) The product of a number and 5 c) The quotient of a number and negative 7 d) The total of twice a number and 3 c) –4(x + 2) – 60 = –8

d) 9 − 14 =

Teaching Notes: • • • •

Answers: 1a) –29, b)–3, c) 24, d) –14, e) –15, f) –4; 2a) 3, b) 152, c) –15, d) 60; 3a) x – 8, b) 5x, c) x/–7, d) 2x + 3

M-16

Mini-Lecture 3.3 Solving Linear Equations in One Variable Learning Objectives: 1. Solve linear equations using the addition and multiplication properties. 2. Solve linear equations containing parentheses. 3. Write numerical sentences as equations. Examples: 1. Solve each equation. Remember to check your answer by substitution. a) 2x – 20 = 0

b) 3p + 5 = 4p + 11

d) 10z = 7z + 10 + 2z e) –2a + 24 = –8a – 6a

c) 6y + 21 = 5y + 9 f) 40 – 5y + 5 = –2y – 10 – 4y

2. Solve each equation. Remember to check your answer by substitution. a) 5(y + 5) = 6(y – 8)

b) 3(y + 5) = 4(y – 4)

c) 4(2x – 4) = 7(x + 5)

d) 6(2a – 3) = 9(a + 4)

e) –7y + 6(–3y – 7) = –64 – 3y

f) 6b + 5(–3b – 2) = –12 – 7b

g) –2(8y – 6) – 2(–7y – 3) = –8

h) 5(2z – 2) = 9(z + 5)

3. Write each sentence as an equation. a) The sum of –57 and 49 is –8. b) The difference of negative 31 and 15 is negative 46. c) The quotient of –10 and 2 amounts to –5. Teaching Notes: • Encourage students to write out each step rather than doing it in their head. • Remind students that it does not matter which side you isolate the variable. • Caution students to take their time using the distributive property. • Refer students to the textbook for Steps for Solving an Equation. • Refer students to the textbook for Key Words or Phrases chart. • Remind students to always check their final answer by substituting it back into the original equation. Answers: 1a) 10, b) –6, c) –12, d) 10, e) –2, f) -55; 2a) 73, b) 31, c) 51, d) 18, e) 1, f) 1, g) 13, h) 55; 3a) –57 + 49 = -8, b) –31 – 15 = –46, c) –10/2 = –5.

M-17

Mini-Lecture 3.4 Linear Equations in One Variable and Problem Solving Learning Objectives: 1. Write sentences as equations. 2. Use problem-solving steps to solve applications.

Examples: 1. Write each sentence as an equation. Use x to represent “a number”. Do not solve. a) A number added to –12 equals 15. b) Two subtracted from a number amounts to 55. c) Ten subtracted from ten times a number is equal to 150. d) The product of a number and –4 is twice the sum of the number and 2. e) The quotient of 10 and a number is 130. 2. Translate each to an equation. Then solve the equation. a) Six times a number yields 36. Find the number. b) A number subtracted from 16 amounts to the quotient of 42 and 6. Find the number. c) The difference of –6 times some number and 12 gives –8 times the sum of the number and –8. Find the number. d) A Ford Taurus is traveling three times as fast as a Honda CRV. If their combined speed is 96 miles per hour, find the speed of each car. Teaching Notes: • • • •

Refer students to Key Words and Phrases table. Refer students to Problem-Solving Steps box. Remind students that a phrase is translated into an expression; a sentence is translated into an equation. Many students have difficulty translating words into mathematical symbols. This section will be a challenge to most students.

Answers: 1a) x + (–12) = 15, b) x – 2 = 55, c) 10x – 10 = 150, d) –4x=2(x+2), e) 10x=130; 2a) 6x + 36, x = 6, b) 16 – x = 42/6, x = 9, c) –6x – 12 = –8[x + (–8)], x = 38, d) x + 3x = 96; Ford = 72 mph, Honda = 24 mph.

M-18

Mini-Lecture 4.1 Introduction to Fractions and Mixed Numbers Learning Objectives: 1. Identify the numerator and the denominator of a fraction. 2. Write a fraction to represent parts of figures or real-life data. 3. Graph fractions on a number line. 4. Review division properties of 0 and 1. 5. Write mixed numbers as improper fractions. 6. Write improper fractions as mixed numbers or whole numbers. Examples: 1. Identify the numerator and the denominator of a fraction.

3 12 10 13 b) c) d) 7 13 7 13 2. Write a fraction to represent the shaded part and unshaded part of each figure as a fraction. a)

Draw and shade a part of a diagram to represent each fraction. 1 5 c) of a diagram d) of a diagram. 6 9 Write a fraction to represent the following information. e) Of the 207 students taking Basic Mathematics, 143 are freshman. What fraction of the class is freshman? 3. Graph each fraction on a number line. a)

1 4

4. Simplify. 4 a) 4

8 3

9 7

3 5

−7 1

0 2

−12 0

5. Write each mixed number as an improper fraction. a) 3 6.

1 2

b) 2

8 9

c) 13

2 9

d) 103

3 11

Write each improper fraction as a mixed number or a whole number. a)

16 3

38 5

156 12

159 143

Teaching Notes: • Students need to have a firm grasp of fraction vocabulary before continuing. 0 x • Many students confuse with . Be sure to stress the difference. 0 x • Many students can write a fraction to represent a real-life situation, but they do not truly understand the meaning. Answers: 1a) n = 3, d = 7; b) n = 12, d = 13; c) n = 10, d = 7; d) n = 13, d = 13; 2a) 2/5, 3/5; b) 4/8, 4/8; 2c – 2d) diagrams will vary; e) 143/207; 3a) – 3d) number lines; 4a) 1; b) –7; c) 0; d) undefined ; 5a) 7/2; b) 26/9; c) 119/9; d) 1136/11; 6a) 5 1/3; b) 7 3/5; c) 13; d) 1 16/143 Copyright © 2025 Pearson Education, Inc.

M-19

Mini-Lecture 4.2 Factors and Simplest Form Learning Objectives: 1. 2. 3. 4.

Write a number as a product of prime numbers. Write a fraction in simplest form. Determine whether two fractions are equivalent. Solve problems by writing fractions in simplest form.

Examples: 1. Write the prime factorization of each number. a) 30

b) 75

2. Write each fraction in simplest form. 10 36 a) b) 16 63 e)

11 34

−27 36

c) 170

d) 360

77 88

d) −

12 42

30 80

6 −105

6 0 and 0 6

3. Determine whether each pair of fractions is equivalent. a)

5 11 and 10 22

7 8 and 21 24

2 8 and 7 15

4. Solve. Write each fraction in simplest form. a) Monae was scheduled to work 6 hours at the tanning salon. What fraction of Monae’s shift is represented by 4 hours? b) There are 36 inches in a yard. What fraction of a yard is represent by 9 inches? c) There are 140 students in a freshman lecture class. If 16 students are absent, what fraction of the students are absent? Teaching Notes: • • •

Many students will understand equivalent fractions if they are shown drawings. Some students will confuse cross products and simplifying. Stress that cross products is only a check to determine equality of fractions. Some students prefer to reduce fractions by factoring the numerator and denominator as products of prime numbers, then canceling all common factors. Others prefer to repeatedly divide the numerator and the denominator by a common factor.

Answers: 1a) 235 , b) 352 ,c) 2517 , d) 23 32 5 ; 2a) 5/8, b) 4/7, c) 7/8, d) –2/7, e) cannot be simplified, f) –3/4, g) 3/8, d) -2/35; 3a) yes, b) yes, c) no, d) no; 4a) 2/3; b) ¼; c) 4/35

M-20

Mini-Lecture 4.3 Multiplying and Dividing Fractions Learning Objectives: 1. 2. 3. 4. 5.

Multiply fractions. Evaluate exponential expressions with fractional bases. Divide fractions. Multiply and divide given fractional replacement values. Solve applications that require multiplication of fractions.

Examples: 1. Multiply. Write the product in simplest form. 1 1 2 1 5 2  b) c) −  − a)  9 7 3 4 6 3 e)

5 18  2 15

2. Evaluate. 2 1 a)   2

7 0 8

 1 b)  −   3

3. Divide. Write all quotients in simplest form. 3 4 1 1 ÷ a) ÷ b) 5 7 4 4 e) −

2 3 ÷− 17 17

1 ÷0 14

d) −

1 3 1  −  2 5 5

 2 c)  −   5

1 9 c) − ÷ 5 19 g)

27 4 ÷ −7 7

7 6  2 3 h)

−

12 3 2 − − 14 9 10 2

2 1 d)    7 4

8 12 ÷ 17 15

h) 0 ÷

−3 11

4. Given the following replacement values, evaluate (a) xy and (b) x ÷ y . 1 4 5 5 a) x = − and y = b) x = and y = − 3 9 7 9

5. Solve. Write each answer in simplest form. a) Find

1 of 48. 3

b) Find

c) A bike trail is 27 miles long. Marshawn bikes

3 of −63 7

2 of the trail. How many miles did 3

Marshawn bike? Teaching Notes: • •

Encourage students to divide out common factors in the numerator and denominator before multiplying. When dividing, encourage students take the time and rewrite the problem by changing the division symbol to multiplication and multiply by the reciprocal. Many students begin “simplifying” and forget to multiply by the reciprocal.

Answers: 1a) 1/63, b) 1/6, c) 5/9, d) –7, e) 3, f) 0, g) -3/50, h) –2/35; 2a) ¼, b) 1/81, c) –8/125, d) 1/49, 3a) 21/20, b) 1, c) –19/45, d) 10/17, e) 2/3, f) undefined, g) –27/4, h) 0; 4a) –4/27, –3/4; b) –25/63, –9/7; 5a) 16, b) –27, c) 18. Copyright © 2025 Pearson Education, Inc.

M-21

Mini-Lecture 4.4 Adding and Subtracting Like Fractions, Least Common Denominator, and Equivalent Fractions Learning Objectives: 1. 2. 3. 4. 5.

Add or subtract like fractions. Add or subtract given fractional replacement values. Solve applications by adding or subtracting like fractions. Find the least common denominator of a list of fractions. Write equivalent fractions.

Examples: 1. Add and simplify. 1 4 + a) 9 9

1 9 + 10 10

d) −

3  5 +−  14  14 

25  7  − − 42  42 

h) −

28  5  − − 13  13 

1 5 + 8 8

c) −

6 5 − 21 21

Subtract and simplify. e)

6 3 − 8 8

f) −

2. Evaluate each expression if x = a) x + y 3.

3 1 and y = − . 5 5

b) x  y

c) x − y

d) x ÷ y

Solve.

6 9 5 inch, inch, and inch. 25 25 25 2 3 4 of her book on Friday, of her book on Saturday, and of her book on b) A student read 11 11 11 Sunday. What part of her book has she read?

a) Find the perimeter of a triangle with sides:

4. Find the LCD of each list of fractions. a)

7 5 , 10 12

3 4 , 12 14

2 8 , 30 35

1 6 15 , , 30 20 50

2 = 3 24

5. Write each fraction as an equivalent fraction with the given denominator. a)

4 = 9 18

7 = 11 55

5 = 2 6

Teaching Notes:

• •

•

Many students add or subtract both numerator and denominator. When subtracting, some students may need to take the intermediate step of writing out the operations 1  3 1  3  −1 + 3 2 = . performed on the numerators. For example: − −  −  = − +  +  = 7  7 7  7 7 7 Some students forget to multiply the numerator when building equivalent fractions.

Answers: 1a) 5/9, b) ¾, c) 4/5, d) –4/7, e) 3/8, f) –11/21, g) 16/21, h) –23/13; 2a) 2/5, b) –3/25, c) 4/5, d) –3; 3a) 4/5 inch, b) 9/11, 4a) 60, b) 84, c) 210, d) 300; 5a) 8, b) 35, c) 15, d) 16. M-22

Mini-Lecture 4.5 Adding and Subtracting Unlike Fractions Learning Objectives: 1. 2. 3. 4.

Add or subtract unlike fractions. Write fractions in order. Evaluate expressions given fractional replacement values. Solve applications by adding or subtracting unlike fractions.

Examples: 1. Add or subtract as indicated. 1 2 + a) 10 5

1  2  b) − +  −  5  25 

1  9 +−  7  10  7  1 −−  9  12 

3 1 + 5 20

4 3 − 5 20

5  1 g) − −  −  7  2

1 8 − 20 15

7 8 4 i) − + + 5 16 20

5 1 _____ 12 2

5 4 c) − _____ − 6 5

2. Insert < or > to form a true sentence. a)

2 1 ____ 3 9

3. Evaluate each expression if x = a) x + y

1 3 and y = − . 4 5

b) x  y

c) x − y

d) x ÷ y

4. Solve. a) Find the perimeter of a rectangle with width

3 3 feet and length feet. 4 14

b) Mahnoor is making matching holiday outfits for her three children. Each outfit required

7 yards. 8

How many yards of material will be needed to make the three outfits? Teaching Notes:

• • • •

Refer students back to Section 4.4: Method 1: Finding the LCD of a List of Fractions Using Multiples of the Largest Number and Method 2: Finding the LCD of a List of Denominators Using Prime Factorization. Some students try to cross-cancel when adding or subtracting. Some students add and subtract both the numerator and denominator. Some students forget to multiply the numerator when building equivalent fractions.

Answers: 1a) ½, b) –7/25, c) –53/70, d) 13/20, e) 13/20, f) 31/36, g) –3/14, h) –29/60, i) –7/10; 2a) >, b) <, c) <; 3a) –7/20, b) –3/20, c) 17/20, d) –5/12; 4a) 27/14 feet, b) 21/8 yd

M-23

Mini-Lecture 4.6 Complex Fractions and Review of Order of Operations Learning Objectives: 1. Simplify complex fractions. 2. Review the order of operations. 3. Evaluate expressions given replacement values. Examples: 1.

Simplify each complex fraction. 1 6 a) b) 2 3

16 7 8 7

1 1 + 6 2 1 3 + 3 4

Use the order of operations to simplify each expression. a)

1 1 1 +  4 4 3

3 7 7  ÷ + 2  8 16 

1 2 1 + − 2  3  3

2 1 d)    5 2

3. Evaluate each expression if x = − a) 3x − z

 2 3  2 3  c)  +  −   7 14  7 14 

y z

2 1 1  + 6 3  4 2 

1 3 7 , y = , and z = . 2 5 10 c) x 2 + 2 y

x+ y z

Teaching Notes:

• • •

Many students make careless errors when using Method 2 for simplifying complex fractions. If this is the case, encourage students to use Method 1 (rewrite as a division problem). Remind students that when dividing fractions, you must change division to multiplication and multiply by the reciprocal. Some students will try to apply procedures for simplifying complex fractions to adding and subtracting fractions.

Answers: 1a) ¼, b) 2, c) 8/13; 2a) 1/3, b) 8/7, c) 1/28, d) 2/25, e) 11/18, f) 3; 3a) –11/5, b) 6/7, c) 29/20, d) 1/7

M-24

Mini-Lecture 4.7 Operations on Mixed Numbers Learning Objectives:

1. 2. 3. 4. 5.

Graph positive and negative fractions and mixed numbers. Multiply or divide mixed or whole numbers. Add or subtract mixed numbers. Solve applications containing mixed numbers. Perform operations on negative mixed numbers.

Examples:

1. Graph each list of numbers on a number line. 1 3 a) −3, − 3 , − 1, , 2 2 4 2. Multiply or divide.

3 1 1 4, − 3 , 0, 1 , − 4 5 2

2 1 a) 2  3 2

3 3 b) 3 1 4 5

1 1 c) 4 ÷ 5 5

1 3 d) 3 ÷ 2 3 5

3. Add or subtract. 1 1 a) 10 + 7 2 9

1 4 b) 6 + 13 3 9

2 7 c) 4 + 9 3 9

1 2 2 d) 8 + 2 + 3 3 3 9

1 13 g) 17 − 5 6 24

h) 13 − 6

8 2 e) 15 − 6 9 9

f) 19

1 1 −7 25 5

5 9

4. Solve. Write answer in simplest form. 4 a) Alba rode her bicycle 9 miles on each of 9 days. What is the total distance Alba rode? 15 3 b) Qian cuts a board 13 feet long from one 20 feet long. How long is the remaining piece? 7 5. Perform the indicated operation. 7 1 7  2 1  1 5 a) −5 ÷ 5 b) −7 +  −4  c) −8   −1  d) 5 ÷ ( −9 ) 9  9 3  5 8 4 8

3  18  e) 19 −  −5  5  20 

3 1  f)  −15  + 14 7 5 

1  5 g) 10 +  −5  9  9

 7 h) −9   5   12 

Teaching Notes:

• • •

Most students forget that mixed numbers must be changed to improper fractions before multiplying. Some try to multiply the whole number parts together, and then multiply the fractional parts together. Many students confuse the rules for multiplication with adding/subtracting rules. Many students are challenged by word problems. Students have trouble deciding which operation to use for the word problems. Answers: 1a) 1b) 2a) 4/3, b) 6, c) 21, d) 1 11/3; 3a) 17 11/18; b) 19 7/9; c) 14 4/9; d) 14 2/9; e) 9 2/3; f) 11 21/25; g) 11 5/8; h) 6 4/9; 4a) 83 2/5; b) 6 4/7; 5a) –47/42; b) –12; c) 10; d) –5/8; e) 25 ½; f) –1 8/35; g) 4 5/9; h) –50 ¼

M-25

Mini-Lecture 4.8 Solving Equations Containing Fractions Learning Objectives:

1. Solve equations containing fractions. 2. Solve equations by multiplying by the LCD. 3. Review adding and subtracting fractions. Examples:

1. Solve each equation. Check your proposed solution. 1 1 4 3 3 5 a) x + = − b) z − = c) − = x − 2 2 15 15 8 6

3 11 d) 8 x − − 7 x = 5 15 2. Solve each equation. 1 x=5 a) 3

3 7 y=− 5 25

3 4 b) − x = − 8 5

x − x = −5 7

c) −8a =

16 25

b b 7 = + 3 5 3

3. Add or subtract as indicated. n 4 5c c + − a) b) 2 7 8 4

Teaching Notes:

• • • •

Emphasis checking proposed solutions. When adding or subtracting fractions, the denominators need to be the same. Review properties: Addition Property of Equality Multiplication Property of Equality Make sure students understand the difference between solving an equation containing fractions (multiply both sides of the equation by the LCD of the fractions) and adding or subtracting two fractions (create equivalent fractions).

Answers: 1a) –1, b) b)

7 11 4 32 2 7 35 35 7n + 8 , c) , d) ; 2a) 15, b) ,c ) − ; d) − , e) , f) ; 3a) , 15 24 3 15 25 15 6 2 14

3c 8

M-26

Mini-Lecture 5.1 Introduction to Decimals Learning Objectives: 1. 2. 3. 4. 5.

Know the meaning of place value for a decimal number and write decimals in words. Write decimals in standard form. Write decimals as fractions. Compare decimals. Round decimals to given place values.

Examples: 1. Determine the place value for the digit 9 in each number. a) 90 b) 900 c) 0.9 Write each decimal number in words. e) 8.54 f) – 0.382 2. Write each decimal number in standard form. a) two and seven tenths c) seven hundred three and two hundred fifty–five thousandths

d) 0.09 g) 7002.09

b) negative eleven and five hundredths d) negative ninety–five ten thousandths

3. Write each decimal as a fraction or a mixed number. Write your answer in simplest form. a) 0.7 b) – 0.35 c) 0.094 d) – 2.4005 4. Insert <, >, or = to form a true statement. a) 0.2 ____ 0.5 c) 0.6401 ___ 0.6410

b) 0.14 ___ – 0.14000 d) – 15.0037 ___ 15.00037

5. Round each decimal to the given place value. a) 0.39 to the nearest tenth c) 1.4782 to the nearest thousandth

b) – 0.174 to the nearest hundredth d) – 22.099 to the nearest hundredth

Round each monetary amount to the nearest cent or dollar as indicated. e) $0.058 to the nearest cent f) $17.88 to the nearest dollar Teaching Notes: • • •

Most students find problems 1 and 2 easy. Some students have difficulty with example 3 when a whole number is involved. Some students become confused when rounding monetary values. When rounding to the nearest cent, it is important to remind them that this is the hundredths position (one–hundredths–of–a–dollar position).

Answers: 1a) tens, b) hundreds, c) tenths, d) hundredths, e) eight and fifty–four hundredths, f) negative three hundred eighty–two thousandths, g) seven thousand two and nine hundredths; 2a) 2.7, b) –11.05, c) 703.255, d) –0.0095; 7 7 47 801 3a) , b) − , c) , d) −2 ; 4a) <, b) >, c) <, d) <; 5a) 0.4, b) –0.17, c) 1.478, d) –22.10, e) $0.06, f) $18 10 20 500 2000

M-27

Mini-Lecture 5.2 Adding and Subtracting Decimals Learning Objectives: 1. 2. 3. 4. 5.

Add or subtract decimals. Estimate when adding or subtracting decimals. Evaluate expressions with decimal replacement values. Simplify expressions containing decimals. Solve applications that involve adding and subtracting decimals.

Examples: 1. Add. Be sure to estimate to see if the answer is reasonable. a) 0.5 + 0.1

b) – 2.7 + – 3.2

c) 7.2 + 3.27

d) – 372 + 9.302

e) 43.097 + 289.3887

f) 5.03 + 16.988 + 0.006

2. Subtract. Be sure to estimate to see if the answer is reasonable. a) 0.8 – 0.2

b) – 7.5 – 2.3

c) 187.5 – 8.39

d) 8.2 – 5.006

e) – 632.021 – (–295.9)

f) 1000 – 3.0947

3. Evaluate each expression for x = 2.4, y = 3, and z = 0.51. a) x + z

b) y – x

c) x + y – z

4. Solve. a) Recently, Lakeith went shopping and spent $18.92 at the bookstore, $68.03 at the grocery store, and $129.76 at a department store. What is the total amount of money Lakeith spent? b) Find the perimeter of a rectangular lawn that measures 40.93 feet by 27.09 feet. Teaching Notes: • • • •

Remind students to work in a vertical format and line–up the decimal point and corresponding place values. Some students need to be shown how to add extra zeros to the ends of the decimal part of the numbers and where to place the decimal point with whole numbers. Some students must be reminded of how to borrow across zeros when subtracting. Refer students to the Adding or Subtracting decimals box in the textbook.

Answers: 1a) 0.6, b) –5.9, c) 10.47, d) –362.698, e) 332.4857, f) 22.024; 2a) 0.6, b) –9.8, c) 179.11, d) 3.194, e) –336.121, f) 996.9053; 3a) 2.91, b) 0.6, c) 4.89; 4a) $216.71, b) 136.04 ft

M-28

Mini-Lecture 5.3 Multiplying Decimals and Circumference of a Circle Learning Objectives: 1. Multiply decimals. 2. Estimate when multiplying decimals. 3. Multiply decimals by powers of 10. 4. Evaluate expressions with decimal replacement values. 5. Find the circumference of circles. 6. Solve applications by multiplying decimals. Examples: 1. Multiply. a) 0.7 × 0.2 d) ×

0.856 3.1

b) 1.33 × –0.5

c) 7.2 × 5.8

e) ×

–2.00033 –6.9

2. Multiply. Check by estimating. a) (6.8)(3.2) b) (8.4)(1.8)

0.0896 × 0.345

c) (5.8)(0.7)

3. Multiply. . a)

4.3 × 10

b) 17.693 × 100

c) –0.0027 × 1000

–0.07 × –0.1

f) 2.908 × 0.001

9.07 × 0.01

4. Evaluate each expressions for x = 2, y = –0.3, and z = 7.3. a) xy

b) xz – y

c) –3y + z

5. Find the circumference of a circle with the given information. Use π = 3.14. a) 6.

radius = 7 feet

b) diameter = 16 inches

c) radius = 10.3 meters

a) Write 57.6 million in standard form. b) A 1-ounce serving of hot cocoa contains 0.375 grams of fat. How many grams of fat are in an 8 oz. mug of hot cocoa?

Teaching Notes: • Some students do not see the pattern that develops when multiplying by powers of 10, they must be shown. • Many students prefer to multiply numbers by a power of ten the long way. • Some students will attempt to line up the decimal point (like adding) when multiplying. • Remind students that C = 2лr. Answers: 1a) 0.14, b) –0.665, c) 41.76, d) 2.6536, e) 13.802277, f) 0.030912; 2a) 21.76; b) 15.12; c) 4.06; 3a) 43, b) 1769.3, c) –2.7, d) 0.007, e) 0.0907, f) 0.002908; 4a) –0.6, b) –6.7, c) 8.2; 5a) 43.96 ft. b) 50.24 in., c) 64.684 m; 6a) 57,600,000; b) 3g

M-29

Mini-Lecture 5.4 Dividing Decimals Learning Objectives: 1. Divide decimals. 2. Estimate when dividing decimals. 3. Divide decimals by powers of 10. 4. Evaluate expressions with decimal replacement values. 5. Solve applications by dividing decimals. Examples: 1. Divide. a)

1.5 ÷ 5

b) 26 7.826

−518 0.9324

8.9 22.25

0.02 0.8

–1411.51 ÷ –36.1

Divide. Then estimate to see if your answer is reasonable. a) 2.5 18.5 b) 2.4 35.4

Divide decimals by powers of 10. a)

7.74 10

1000 −887.73

c) 1.047 ÷ 100

4. Evaluate each expression for x = 3.02, y = –0.3, and z = 1.51. a)

z÷y

b) z ÷ x

c) x ÷ y

5. Solve. a) Divide 0.894 by –0.041 and round the quotient to the nearest hundredth. b) Preparing for a picnic, Amitra went to the deli and purchased: 0.52 pounds of salami at $3.29/pound; 0.48 pounds of sliced turkey at $8.99/pound; 1.04 pounds of ham at $3.99/pound; and 0.98 pounds of cheese at $4.29/pound. What was the total amount Amitra spent at the deli? (Round your answer to the nearest cent.) Teaching Notes: • Most students have forgotten the mechanics of long division with decimals. • Stress the importance of neatness so that the decimal ends up in the correct position. Some students find it helpful to do the division on graph paper or on lined paper that is rotated so the lines are vertical. • Remind students when rounding the quotient to a specific place, you need to carry your division one more place than the rounding place. Answers: 1a) 0.3, b) 0.301, c) –0.0018, d) 2.5, e) 39.1, f) 40; 2a)7.4; b)14.75; 3a) 0.774, b) –0.88773, c) 0.01047; 4a) –5.0 3 , b) 0.5, c) –10.0 6 ; 5a) –21.81, b) $14.38

M-30

Mini-Lecture 5.5 Fractions, Decimals, and Order of Operations Learning Objectives:

1. 2. 3. 4. 5.

Write fractions as decimals. Compare decimals and fractions. Simplify expressions containing decimals and fractions using order of operations. Solve area applications containing fractions and decimals. Evaluate expressions given decimal replacement values.

Examples:

1. Write each fraction or mixed number as a decimal. 3 3 a) b) − 5 20 d)

5 16

e) −

13 11

1 3

f) −1

7 8

2. Insert <, >, or = to form a true statement. a) –0.0832 ____ –0.0823

b) 0.501 ____

1 2

c) 0.428 ____

Write the numbers in order from smallest to largest. 1 15 d) 0.331, , 0.330 e) 2.15 , 2.142 , 3 7

1.5833, 1

3 7

21 38 , 36 36

3. Simplify each expression. a) (0.3)2 – 0.4

1 − 3(6.5) 4

4. a) 6 ft

b) (7.3)(100) – (7.2)(10)

3 − (6.4)(−3) 5

4 + 0.42 −2

1 (−9.1 − 3.3) 5

5. Evaluate for x = –2, y = 0.5, and z = 3.6: x a) + 2 z y

4.8 ft

Teaching Notes:

• • •

Most students need a review of order of operations. Most students, once taught how to convert from fraction to decimal, have little problem doing so. Some students have difficulty ordering numbers when they are mixed. Suggest that after converting to decimal, line the decimal points up vertically and compare corresponding place value.

Answers: 1a) 0.6, b) –0.15, c) 0. 3 , d) 0.3125, e) –1. 18 , f) –1.875; 2a) <, b) > c) <, d) 0.330, 0.331, 1/3, e) 2.142, 15/7, 2.15, f) 38/36, 1.5833, 1 21/36; 3a) –0.31, b) 658, c) –2.21, d) –19.25, e) 19.8, f) –2.48; 4a)14.4ft; 5a) 3.2

M-31

Mini-Lecture 5.6 Solving Equations Containing Decimals Learning Objectives:

1. Solve equations containing decimals. Examples:

1. Solve each equation. a. z + 0.8 = 2.5 =

b. 0.27 x = −0.81 =

c. 3.9 = 1.5 + 0.6x

d. 5 x + 3.6 = 8 x + 12.9

e. 6.8 − 4 x = 5 ( x − 13.7 )

Teaching Notes: • Review steps for solving equations involving a variable. • Stress neatness in lining up decimals when adding or subtracting, especially when an integer is present. • Remind students the advantage of multiplying by powers of 10 to eliminate decimals when solving equations.

Answers: 1a) 1.7, b) –0.3, c) 4, d) –3.1, e) 3.7

M-32

Mini-Lecture 5.7 Decimal Applications: Mean, Median, and Mode Learning Objectives:

1. Find the mean (or average) of a list of numbers. 2. Find the median of a list of numbers. 3. Find the mode of a list of numbers. Examples:

1. Find the mean of the list of numbers. If necessary, round to the nearest tenth. a) 71, 47, 71, 99, 47

b) 143, 83, 225, 16

c) 4.1, 1.4, 8.7, 1.9, 13.1, 7.6

d) Find the GPA (grade point average) for a particular student. Use the following for point values of the grades: A-4, B-3, C-2, D-1, F-0. If necessary, round the grade point average to the nearest hundredth. Grade A B A C

Credit Hours 3 3 4 3

2. Find the median of the following list of numbers. Round to the nearest hundredth, if necessary. a) 3, 3, 14, 27, 31, 37, 50

b) 10, 2, 1, 28, 45, 48, 36

c) 8, 2, 21, 19, 23, 49, 39, 38

0.1, 3.3, 2.5, 0.3, 4, 2.7

3. Find the mode of the following list of numbers. a) 20, 43, 46, 43, 49, 43, 50

b) 90, 57, 32, 57, 29, 90

c) 5, 9, 46, 3, 2, 8, 18, 1, 6, 19

d) 7.08, 7.41, 7.56, 7.08, 7.88, 7.99, 7.62

Teaching Notes:

• • •

Most students are familiar with mean (average) but have not worked with median or mode. Encourage students to estimate when finding the mean. Stress the importance of ordering the data when finding median and mode.

Answers: 1a) 67, b) 116.8, c) 6.1, d) 3.31; 2a) 27, b) 28, c) 22, d) 2.6; 3a) 43, b) 57,90, c) no mode, d) 7.08

M-33

Mini-Lecture 6.1 Ratio and Rates Learning Objectives: 1. 2. 3. 4.

Write ratios as fractions. Write rates as fractions. Find unit rates. Find unit prices.

Examples: 1. Write each ratio as a fraction in simplest form. Write the fraction in simplest form. a)

20 to 30

b) 16 to 10

1.17 days to 9.9 days

e) 3

1 hours to 6 hours 4

c) 1.5 to 10 f) 7

1 1 to 7 5 3

2. Write each rate as a fraction in simplest form. a) 81 miles in 42 minutes c) 6 tests for 30 students

b) 1036 miles in 63 hours d) 246 miles on 54 gallons

3. Write each rate as a unit rate. a) 84 miles in 4 hours c) 1161 cars in 387 households

b) 468 miles on 18 gallons of gas d) 380 people in 10 buses

4. Find each unit price. Round to the nearest cent, if necessary. a) $0.90 for 10 ounces c) $11.92 for 8 pounds of apples

b) $1.39 for 17 ounces d) $118.58 for 121 pens

e) Which is the better buy (lowest cost per ounce)? Round to the thousandth, if necessary. Shampoo: $0.70 for 11 ounces or 17 ounces for $1.39. f) There were 3 men and 6 women on a volleyball team. Find the ratio of men to women. Teaching Notes: • • •

Some students need a quick review on converting mixed numbers to improper fractions and dividing fractions, decimals, etc. Remind students that order is important when solving ratio and rate word problems. Remind students that unit rate has a denominator of one.

Answers: 1a) 2/3, b) 8/5, c) 3/20, d) 13/110, e) 13/24, f) 54/55; 2a) 27miles/14minutes, b) 148 miles/9hours, c) 1 test for 5 students, d) 41 miles per 9 gallons; 3a) 21 miles/hr, b) 26 miles/gallons, c) 3 cars/household, d) 38 people/bus; 4a) $0.09/oz, b) $0.08/oz, c) $1.49/pound, d) $0.98/pen; e) 11 ounces for $0.07, f) 1/2

M-34

Mini-Lecture 6.2 Proportions Learning Objectives: 1. Write sentences as proportions. 2. Determine whether proportions are true. 3. Find an unknown number in a proportion.

Examples: 1. Write each sentence as a proportion. a) 70 pencils is to 28 students as 5 pencils is to 2 students b) 96 guests is to 12 tables as 8 guests is to 1 student. 1 6 c) 2 ounces of pasta is to 3 grams of protein as 12 ounces of pasta is to 18 grams 11 11 of protein. 2. Determine whether each proportion is true or false.

3 4 = a) 6 8

1 b)

10 7 3 13 = 13 3 6

0.6 3.0 = c) 0.5 2.6

2 8 8 9= 9 7 30

2 d)

3. For each proportion, find the unknown number n. a)

n 1 = 54 18

1 n = −2 15

1 n = 1 19 9 2

−4.5 −2.5 = n 5.5

Teaching Notes:

• • •

It is important to stress the easy problems 1a, 1b, and 2a to show that they are equivalent fractions. Many students have difficulty when complex fractions are involved in the proportion. Many students need to actually circle the cross products so they can be clear of the procedure.

96 8 70 5 = , c) Answers: 1a) = , b) 28 2 12 1 9.9

1 6 12 11 = 11 ; 2a) yes, b) yes, c) false, d) false; 3a) 3, b) –7.5 or 7 1/2, c) 2, d) 3 18

M-35

Mini-Lecture 6.3 Proportions and Problem Solving Learning Objectives:

1. Solve applications by writing proportions. Examples:

1. On a map of the city of Worcester, 1-inch corresponds to 12 miles. How far away is a town if the distance on the map measures 3 inches? 2. The ratio of students to faculty is 25 to 2. How many faculty members will be needed for a student population of 623? Round to the nearest whole number. 3. A particular piece of machinery made 1200 revolutions in 20 minutes. How many revolutions will occur in 23 minutes? 4. Four cups of water is needed to make 9.2 cups of rice. How many cups of rice can be made with 21 cups of water?

5. If 2 servings of a recipe calls for

3 teaspoon of butter, how many servings can be made from 9 4

teaspoons of butter? 6. One bag of lawn & garden fertilizer covers 1000 square feet of lawn. How many bags of fertilizer must you purchase to cover a lawn 440 feet by 220 feet? Remember, you cannot purchase a fractional part of a bag. Teaching Notes:

• • •

Many students have trouble correctly setting up the proportion. Students tend to write the numbers as they appear rather than “lining up” the units. Remind students to estimate to see if their answer is reasonable. Refer students to the four steps for problem solving: Understand, Translate, Solve, and Interpret.

Answers: 1) 36 miles, 2) 50 faculty members; 3) 1380 revolutions; 4) 48.3 cups; 5) 24 servings; 6) 97 bags.

M-36

Mini-Lecture 6.4 Square Roots and the Pythagorean Theorem Learning Objectives:

1. Find the square root of a number. 2. Approximate square roots. 3. Use the Pythagorean Theorem. Examples:

1. Find each square root. a)

1 64

16 25

25 49

2. Use a calculator or the appropriate Appendix to approximate each square root to the thousandths position. a)

3. Sketch each right triangle and find the length of the side not given. If necessary, round the answer to the nearest thousandth. a) leg = 3, leg = 4

b) leg = 12, hypotenuse = 15

c) hypotenuse = 6.4, leg = 3

Solve. If necessary, round to the nearest thousandth. d) A section of land is a square with each side measuring 2 miles. Find the length of the diagonal of the section of land. e) A garden is in the shape of a rectangle. The diagonal length of the garden is 25 feet, and the length of one of the sides is 15 feet. Find the length of the other side. Teaching Notes:

• • •

Some students have never done square roots on a calculator and will need guidance. When approximating square roots, encourage students to mentally estimate the answer. That way if they use the calculator incorrectly they might be able to notice the incorrect result. Most students do not have trouble using the Pythagorean Theorem for finding a hypotenuse, but some have trouble using it for finding a missing leg.

Answers: 1a) 2, b) 5, c) 7, d) 9, e) 1/8, f) 4/5, g) 5/7; 2a) 1.414, b) 2.449, c) 3.317, d) 6.245; 3a) 5, b) 9, c) 5.653; d) 2.828 mi, e) 20 ft

M-37

Mini-Lecture 6.5 Congruent and Similar Triangles Learning Objectives:

1. Decide whether two triangles are congruent. 2. Find the ratio of corresponding sides in similar triangles. 3. Find unknown lengths of sides in similar triangles. Examples:

1. Determine whether each pair of triangles is congruent. 80º

7 in

5 in

7 in

63º

12 in 8 in

8 in

135º

5 in

12 in. 63º

80º

c) 12 m 12 m 135º 4m

2. Find each ratio of the corresponding sides of the given similar triangles. 1.6 m

b) 50 cm

1.2m

25 cm 1.8 m 24 cm

2.4 m

48 cm

3. Solve. Round to the nearest tenth. a) A tree casts a shadow of 26 ft. Nearby, a 5-ft pole casts a shadow of 3 ft. What is the height of the tree? b) A rock climber is 6 feet tall and her shadow measures 9 feet long. The rock she wants to climb casts a shadow of 580 feet. How tall is the rock? Teaching Notes:

• •

Some students need a review on solving proportions. Encourage students to mentally visualize applied problems and draw and label a diagram before solving.

Answers: 1a) yes, b) yes, c) yes; 2a) ½, b) 4/3; 3a) 43 1/3 ft., b) 386 2/3 ft

M-38

Mini-Lecture 7.1 Percents, Decimals, and Fractions Learning Objectives: 1. Understand percent. 2. Write percents as decimals or fractions. 3. Write decimals or fractions as percents. 4. Solve applications with percents, decimals, and fractions. Examples: 1. Write the percent described by the sentence. In a recent survey of 100 students, 37 of the students indicated that they eat lunch at the dining hall. What percent of the students surveyed do not eat lunch in the dining hall? 2. Write each percent as a decimal. a) 32% b) 5.7%

c) 120%

d) 0.02%

Write each percent as a fraction or mixed number in simplest form.

1 2

f) 8.1%

g) 275%

h) 5 %

3. Write each decimal as a percent. a) 0.77 b) 2.6

c) 0.081

d) 3.00

e) 15%

Write each fraction or mixed number as a percent. 23 3 7 e) f) g) 5 12 50 Complete the table. Round to the nearest thousandth, if necessary Percent Fraction Decimal

2 5

i) j) k)

h) 4

27% 0.55 2

1 6

4. Write each percent as a decimal and a fraction. a) A family decides to spend no more than 27.5% on its monthly income on rent. Write 27.5% as a decimal and a fraction. 3 b) Provincetown’s budget for waste disposal increased by 1 times over the budget from last year. 4 What percent increase is this? Teaching Notes: • Some students have trouble remembering which way to move the decimal point. • Some students become very confused with converting between percents, fractions, and decimals, and need to see many examples. Answers: 1) 37%, 63%; 2a) 0.32, b) 0.057, c) 1.2, d) 0.0002, e) 3/20, f) 81/1000, g) 2 ¾ , h) 11/200; 3a) 77%, b) 260%, 1 c) 8.1%, d) 300%, e) 46%, f) 60%, g) 58. 3 % or 58 %, h) 440%, i) 15%, 0.15, j) 27/100, 0.27, k) 55%, 11/20, l) 3 216.7%, 2.167; 4a) 0.275,11/40, b) 175% Copyright © 2025 Pearson Education, Inc.

M-39

Mini-Lecture 7.2 Solving Percent Problems with Equations Learning Objectives: 1. Write percent problems as equations. 2. Solve percent problems. Examples: 1. Translate each to an equation. Do not solve. a) 12% of 80 is what number?

b) What percent of 60 is 20?

c) 40% of what number is 20?

d) 1.8 is what percent of 9?

2. Solve the following equations for the amount. a) What number is 8% of 50?

e) 70% of what number is 35?

b) 20% of 65 is what number? 3 d) What number is 15 % of 50? 4 f) 18 is 4% of what number?

g) 8.1 is 36% of what number?

h) 22.5% of what number is 2.7?

i) What percent of 200 is 16?

j) 0.06 is what percent of 100?

k) 210 is what percent of 60?

l) What percent of 1041 is 333.12?

c) 125% of 16 is what number?

Teaching Notes: • •

Many students get confused between “amount” and “base”. Some students have better success at solving these types of equations by working in Section 7.3, Solving Percent Problems Using Proportions.

Answers: 1a) (0.12)(80) =x, b) x• 60=20, c) 0.40 x = 20, d) 1.8 = 9x; 2a) 4, b) 13, c) 20, d) 7.875; e) 50, f)450, g) 22.5, h) 12; i) 8%, j) 0.06%, k) 350%, l) 32%

M-40

Mini-Lecture 7.3 Solving Percent Problems with Proportions Learning Objectives: 1. Write percent problems as proportions. 2. Solve percent problems. Examples: 1. Translate each to a proportion. Use n to indicate the unknown. Do not solve. a) 50% of 24 is what number?

b) What percent of 40 is 14?

c) 60% of what number is 2?

d) What percent of 50 is 50?

2. Solve. a) 30% of 350 is what number?

b) What number is 8% of 625?

c) What number is 250% of 60?

d) 20% of 8.5 is what number?

e) 8% of what number is 4?

f) 84 is 70% of what number?

1 1 g) 5 % of what number is 2 ? 2 5

h) 1716 is 143% of what number?

i) What percent of 120 is 90?

j) 616 is what percent of 560?

k) 2.1 is what percent of 70?

l) What percent of 800 is 6?

Teaching Notes: • • •

amount percent = base 100 Many students find it difficult to identify the different parts of the proportion. Remind students that the “base” is after the word “of”.

Remind students that the percent proportion is:

50 n 14 n 2 60 50 n , b) , c) , d) ; 2a) 105, b) 50, c) 150, d) 1.7; e) 50, f) 120, g) 40, = = = = 100 24 40 100 n 100 50 100 h) 1200 i) 75%, j) 110%, k) 3%, l) 0.75%

Answers: 1a)

M-41

Mini-Lecture 7.4 Applications of Percent Learning Objectives: 1. Solve applications involving percent. 2. Find percent of increase and percent of decrease. Examples: 1. Solve. a) 15% of an employee’s paycheck of $1200 is paid towards health care. How much money is paid towards health care? b) During a recent inspection, the fire department found 112 faulty smoke alarms. If this is 0.08% of the total inspected, how many smoke alarms were inspected? c) One day, 18 students were out sick with the flu. What percent of the students were absent if there should be a total of 80 students in the class? 2. Solve. a) Find the percent of increase if the original amount was 80 and the new amount is 100. b) Find the percent of increase if the original amount was 20 and the new amount is 65. c) Recently, a bookstore announced that all their books would increase in price 5%. How much will a book cost if the original price was $4.50. Round to the nearest cent. d) Find the percent of decrease if the original amount was 16 and the new amount is 10. e) Find the percent of decrease if the original amount was 140 and the new amount is 91. f) On a recent shopping trip, a sign read: “Jeans! Originally $40, Now $34!”. Find the percent of decrease in price. Teaching Notes: •

Most students find this section difficult. Refer students to the following formulas: percent of increase =

amount of increase original amount

percent of decrease =

amount of decrease original amount

Answers: 1a) $180, b) 140,000, c) 22.5%; 2a) 25%, b) 225%, c) $4.73; d) 37.5%, e) 35%, f) 15%

M-42

Mini-Lecture 7.5 Percent and Problem Solving: Sales Tax, Commission, and Discount Learning Objectives: 1. Calculate sales tax and total price. 2. Calculate commissions. 3. Calculate discount and sale price. Examples: 1. Solve. a) Find the sales tax and the total price on the purchase of a $230 mobile phone where the sales tax rate is 6.5%. b) The sales tax on a $1050 computer system is $63. What is the sales tax rate? c) A mobile phone costs $185 and the phone case costs $30. What is the total paid to the cashier if the sales tax rate is 7%? 2. Solve. a) A book salesperson is paid a commission of 3.1% of her monthly sales. For the month of August, she sold $180,000 worth of books. What was the amount of her commission for the month? b) A salesperson earned a commission of $3,842.50 for selling $53,000 worth of merchandise. Find the salesperson’s commission rate. 3. Solve. a) Find the amount of discount when the original price is $72 and the discount rate is 20%. b) Find the sale price when the original price is $58 and the discount rate is 9%. c) A $3300 diamond bracelet is part of a “25% off” sale. Find the discount and the sale price for the bracelet. Teaching Notes: • •

Most students find these types of problems difficult and frustrating. Refer students to the following formulas: Sales Tax = tax rate · purchase price Total Price = purchase price + sales tax Commission = commission rate · sales Amount of Discount = discount rate · original price Sale Price = original price – amount of discount

Answers: 1a) $14.95, $244.95, b) 6%, c) $230.05; 2a) $5580, b) 7.25% or 7 ¼%; 3a) $14.40, b) $52.78, c) $825, $2475

M-43

Mini-Lecture 7.6 Percent and Problem Solving: Interest Learning Objectives: 1. Calculate simple interest. 2. Calculate compound interest. Examples: 1. Solve. a) Upon graduation, Ayla is given money that totals $4,700. If this money is invested at 9.5% simple interest for 7 years, find the total amount. b) Akiko borrows $10,500 for 4

1 years at a rate of 9.5% simple interest. Find the total amount. 2

c) $250,000 is borrowed to buy a house. If the simple interest rate on the 30-year loan is 6.75%, find the total amount paid on the loan.

2. Solve. Round answers to the nearest cent. a) A physical therapist deposited $2,850 in a compound interest account for 5 years. If the account earns 10% interest compounded quarterly, find the total amount. (Compound interest factor is 1.63862) b) $4,690 is compounded semi-annually at a rate of 8%. Find the total amount of compound interest earned at the end of 15 years. (Compound interest factor is 3.2434). c) $2500 is compounded daily at a rate of 8% for 10 years. Find the total amount.

Teaching Notes: •

Refer students to the following formulas in the textbook: Simple Interest = principal · rate · time Total Amount = principal + interest Compound Interest Tot. Amt. =original principal · compound interest factor principal + interst Monthly Payment = total number of payments

Answers: 1a) $7825.50, b) $14,988.75, c) $756,250; 2a) $4670.07, b) $15,211.54, c) $5563.36.

M-44

Mini-Lecture 8.1 Reading Pictographs, Bar Graphs, Histograms, Line Graphs and Introduction to Statistics Learning Objectives: 1. Read pictographs. 2. Read and construct bar graphs. 3. Read and construct histograms (or frequency distribution graphs). 4. Read line graphs. 5. Calculate range, mean, median, and mode from a frequency distribution table or graph. Examples:

1. The following bar graph shows the points scored per quarter in a basketball game. 40 35 30 25 20 15 10 5 0

Lakers Celtics

2. The following pictograph shows the number of satisfied customers at a restaurant. Use the information to answer the following questions. Month Jan. Feb. Mar.

Number of Satisfied Customers

= 100 Satisfied Customers 1st Qtr 2nd Qtr

3rd 4th Qtr Qtr

a) How many points did the Celtics score in the 4th quarter? b) How many points did the Lakers score in the 1st quarter? c) What was the total score for the game? 3. The following line graph shows the total sales per month.

a) Which month had the most satisfied customers? b) What was the total number of satisfied customers during the 3month period? 4. The following is a list of scores on a recent math exam. Use this list to complete the frequency distribution table: 100, 85, 89, 75, 60, 55, 92, 85, 85, 76, and 77

Class Intervals 90 - 100

b c d e

80 - 89 70 - 79 60 - 69 Below 60

30000 25000 20000 15000 10000

Tally Class Frequency

5000 0 Jan

Feb

Mar

Apr

May

Jun

5. Find the (a) range, (b) mean, (c) median, and (d) mode from the frequency distribution table completed in Example 4.

a) What month had the least sales? b) What is the difference between the highest and lowest month? Teaching Notes: • Encourage students to look around their environment and find different graphs.

M-45

Mini-Lecture 8.2 Circle Graphs Learning Objectives: 1. Read circle graphs. 2. Draw circle graphs. Examples: 1. The freshman class held elections for class president. The following circle graph shows the results of how the 200 students voted. Use the circle graph to answer the accompanying questions. Votes For Class President

a) Who won the election? b) Who received the fewest votes?

Ray 26%

Amy 33%

c) How many votes did the winner receive? d) Write a ratio of students voting for Haiwin to total number of students.

Haiwin 15% Chris 26%

2. A recent survey of college students asked how many cans of soda did they consume per day. Complete the table and draw a circle graph to represent the information given in the table.

a b c d e

Number of Cans of Soda Consumed in One Day 0 1-2 3-4 5-6 7 or more

Percent of College Students 5% 42% 20% 15% 18%

Degrees of a Circle

Teaching Notes: • • •

Remind students that the order of a ratio is important. Some students will need instruction on the use of a protractor. Encourage students, when drawing a circle graph, to estimate first.

Answers: 1a) Amy, b) Haiwin, c) 66, d) 3/20; 2a) 18°, b) 151.2°, c) 72°, d) 54°, e) 64.8°

M-46

Mini-Lecture 8.3 The Rectangular Coordinate System and Paired Data Learning Objectives: 1. Plot points on a rectangular coordinate system. 2. Determine whether ordered pairs are solutions of equations. 3. Complete ordered pair solutions of equations. Examples: 1. Plot each order pair. State in which quadrant or on which axis each point lies.  2 1 a) (–2, –5) b) (0, -4) c)  2 , 4  d) (–1, 4)  3 2

2. Determine whether each ordered pair is a solution of the given linear equation. a) x – y = 4; (1, 2)

b) x + 3y = 7; (4,1)

c) y = 5x + 7; (–2, –3)

3. Complete each ordered pair so that it is a solution of the given linear equation. 1 1  a) x + 2 y = 6 ; (2, ), ( , -3) b) y = x − 2; ( 6, ) ,  , −  3 3 

Teaching Notes: • Many students have trouble putting meaning to an ordered pair. • Remind students that an ordered pair is (x, y), which is in alphabetical order.

Answers: 1a) III, b) y-axis, c) I, d) II; 2a) no, b) yes, c) yes; 3a) (2, 2), (12, -3), b) (6, 0), (5, −

1 ) 3

M-47

Mini-Lecture 8.4 Graphing Linear Equations in Two Variables Learning Objectives:

1. Graph linear equations by plotting points. Examples:

1. For each equation, find three ordered pair solutions by completing the table. Then use the ordered pairs to graph the equation. 1 x y a) x − y = 2 b) y = − x − 2 x y 6 3 3 –4 –2 0 –1

2 x 3

–6

y=3

0 10 3

2 –1 0

Graph the following linear equations. e) x + y = 0

y = −2 x − 1

g) x – 2 = 0

Teaching Notes:

• • •

Examples 1a) – d) tend to pose the least amount of challenge for students. Some students become very confused when they can choose any value for x or y as a starting point for finding an ordered-pair solution. Many students do not understand Examples 1d) and 1g) and must memorize the form for an equation for a horizontal or a vertical line.

Answers (for all graphs, see Mini-Lecture graphing answers at end of section): 1a) (3,1), (0, –2), (–1, –3); 1b) (6, –4), (6, –4), (–6, 0); 1c) (–6, –4), (0,0), (5,

10 ); 1d) (–2,–3), (–1, –3), (0, –3); 1e ) – 1g) see graphing 3

answers

M-48

Mini-Lecture 8.5 Counting and Introduction to Probability Learning Objectives:

1. Use a tree diagram to count outcomes. 2. Find the probability of an event. Examples:

1. Draw a tree diagram to find the number of possible outcomes. a) Choose a number: 2, 4, or 6, and then a letter: a, b, or c. b) Choose a shirt: red or blue, and then pants: white, black, or brown. c) Toss a coin and then toss a single six-sided dice. d) Toss four coins. 2. Find the probability of each event occurring. a) If a single die is tossed, find the probability of a 2. b) If a single die is tossed, find the probability of a number greater than 2. c) If a single choice is made from a bag of marbles containing 2 red, 2 blue, 2 green, find the probability of choosing a red marble; find the probability of choosing a red or green marble. d) Recently, a weight loss drug was tested on 300 people. The results were as follows: 220 people lost weight; 60 people stayed the same weight, 20 people gained weight. What is the probability that a person taking this drug would lose weight? What is the probability that a person would not lose weight (either stay the same or gain)? Teaching Notes:

• • •

Refer students to the textbook for sample tree diagrams and the formula for The Probability of an Event. Most students, once shown tree diagrams, are able to satisfactory draw a tree diagram. Some students need to list or draw the possible outcomes to find or visualize the probability.

Answers: 1a) 9, b) 6, c) 12, d) 16; 2a) 1/6, b) 2/3, c) 1/3, 2/3, d) 11/15, 4/15

M-49

Mini-Lecture 9.1 Lines and Angles Learning Objectives: 1. 2. 3. 4.

Identify lines, line segments, rays, and angles. Classify angles as acute, right, obtuse, or straight. Identify complementary and supplementary angles. Find measures of angles.

Examples: 1. Draw an example of each term. a) line

b) ray

c) segment

d) angle

c) obtuse

d) straight

2. Draw an example of each angle. a) acute

b) right

3. Find each complementary or supplementary angle as indicated. a) Find the complement of a 35º angle. c) Find the supplement of a 152º angle.

b) Find the complement of a 71º angle. d) Find the supplement of a 83º angle.

4. Find the measures of angles x, y, and z in each figure. a)

b) m  n

122º x

z y

139º x z

m n

Teaching Notes: • •

Some students are unfamiliar with the vocabulary and need repetition. Refer students to the textbook, Chapter Highlights, for a condensed listing of definitions and concepts.

Answers: 1a) – d) Answers may vary; 2a – d) Answers may vary; 3a) 55°, b) 19°c) 28°, d) 97°; 4a) x = 58°, y = 122°, z = 58°, b) x = 139°, y = 139°, z = 41°

M-50

Mini-Lecture 9.2 Perimeter Learning Objectives: 1. Use formulas to find perimeters. 2. Use formulas to find circumferences.

Examples: 1. Find the perimeter of the following figures. a)

15 m

25 ft 23 ft

10 ft

23 ft

8 ft 2 ft

7 ft

18 ft d) A rectangular field measures 441 feet by 108 feet. Find the cost of constructing a fence if fencing costs $29.50 per yd. 2. Find the circumference of each circle. Give the exact and then an approximation by using π = 3.14. Round to the nearest hundredth. a) radius = 7 ft.

b) diameter = 84 m.

c) radius = 34.9 ft

d) A circular room has a radius of 10.3 feet. Find the distance around the room. e) A circular statue has a base with a diameter of 11 ft. Find the distance around the base of the statue. f) Find the distance around a circular Jacuzzi with a diameter of 8 ft. For this problem, use 22 π= 7

Teaching Notes: • • •

Refer students to textbook for formulas of perimeter and circumference. Many students have difficulty understanding the difference between approximation and exact value when working with π . Remind students to read carefully and take note of radius vs. diameter.

Answers: 1a) 42 m, b) 89 ft., c) 50 ft., d) $10,797; 2a) 14π, 43.96 ft, b) 84π, 263.76 m, c) 69.8π, 219.17 ft, d) 20.6π, 64.69 ft, e) 11π, 34.54 ft., f) 8π, 27 1/7 ft.

M-51

Mini-Lecture 9.3 Area, Volume, and Surface Area Learning Objectives: 1. Find the area of plane regions. 2. Find the volume and surface area of solids. Examples: 1. Find the area of the following. Use π = 3.14. Round to the nearest hundredth. a) 3 yd

7 yd

c) 8 ft

d = 24 miles

3.8 ft

d) A small rug is in the shape of a trapezoid. The bases measure 19.8 in and 22.4 in. and the height is 8 in. Find the area of the rug. e) A side of a square towel measures 7.5 in. Find how many square inches of material is needed to make 8 towels. 2. Find the volume and surface area of each rectangular solid or cube. Round to the nearest tenth. a) Rectangular solid: length = 8.3 in.; width = 3 in.; height = 9 in. b) Cube: side = 7.5 cm Find the volume and surface area of each sphere or circular cylinder. Give the exact answers and an approximate. Use π = 3.14 . Round to the nearest hundredth. c) Sphere: diameter = 12 in. d) Circular cylinder: radius = 8 feet; height = 3 feet Find the volume and surface area of each cone or square-based pyramid. If necessary, give the exact answer and an approximate. Use π = 3.14 . Round to the nearest hundredth. e) Cone: height = 4 m; radius = 3 m f) Square-based pyramid: height = 11 cm; edge of base = 10.5 cm; slant height = 12.19 cm Teaching Notes: • Refer students to textbook for Area Formulas of Common Geometric Figures and Area Formula of a Circle. • Many students have difficulty when the height lies outside of a triangle, parallelogram, or trapezoid. • Refer students to the textbook for Volume and Surface Area Formulas of Common Solids. • Some students need to be able to visualize the solids. Encourage students to seek out actual items in their environment for these common solids. • Remind students that volume is measured in cubic units, and surface area in square units. Answers: 1a) 21 yd 2, b) 15.2 ft2, c) 452.16 mi2, d) 168.8 in2, e) 450 in2, : 2a) V = 224.1 in3, SA = 253.2 in2, b) V = 421.9 cm3, SA = 337.5 cm2; c) V = 288 π in3 ≈ 904.32 in3, SA = 144 π in2 ≈ 452.16 in2, d) V = 192π ft3 ≈ 602.88 ft3, SA = 176 π ft2 ≈ 552.64 ft2; e) V = 12π m3 ≈ 37.70 m3, SA = 24 π m2 ≈ 75.40 m2, f) V = 404.25 cm3, SA = 366.24 cm2

M-52

Mini-Lecture 9.4 Linear Measurement Learning Objectives: 1. 2. 3. 4. 5.

Define U.S. units of length and convert from one unit to another. Use mixed U.S. units of length. Perform arithmetic operations on U.S. units of length. Define metric units of length and convert from one unit to another. Perform arithmetic operations on metric units of length.

Examples: 1. Convert each measurement as indicated. a) 72 in to feet b) 8 yd to feet

c) 3 mi to feet

d) 120 in to yd.

2. Convert each measurement as indicated. a) 89 in = __ft __in

b) 4 yd 2 ft = __ in

c) 11,213 ft = ___mi___ft

3. Perform each indicated operation. Simplify the result if possible. a) 11 ft 3 in. + 5 ft 10 in.

42 yd 1 ft – 38 yd 2 ft

c) 18 ft 8 in. ÷ 2

d) A garden is 7 ft. 11 in. long by 8 ft. 7 in. wide. What is the total length of fencing needed to completely enclose the garden? 4. Convert as indicated. a) 60 mm to cm

b) 500 m to km

c) 8.2 m to cm

d) 34,000 mm to m

e) 3.7 km to m

f) 50.6 mm to dm

5. Perform each indicated operation. Simplify the result if possible. a) 26 cm + 11.9 m

b) 33 mm – 1.443 cm c) 4.3 mm ÷ 5

d) A 2.8-m board has 1.3 cm trimmed from each end. How long is the remaining board? Teaching Notes: • •

Refer students to U.S. Units of Length chart and Metric Units of Length chart. Remind students to always put the unit they are converting to in the numerator of the unit fraction.

Answers: 1a) 6 ft, b) 24 ft., c) 15,840 ft, d) 3 1/3 yd; 2a) 7 ft 5 in, b) 168 in., c) 2 mi 653 ft; 3a) 17 ft 1 in, b) 3 yd 2 ft, c) 9 ft 4 in, d) 33 ft; 4a) 6 cm, b) 0.5 km, c) 820 cm, d) 34 m, e) 3700 m, f) 0.506 dm; 5a) 12.16 m, b) 18.57 mm, c) 0.86 mm, d) 277.4 cm or 2.774 m

M-53

Mini-Lecture 9.5 Weight and Mass Learning Objectives: 1. 2. 3. 4.

Define U.S. units of weight and convert from one unit to another. Perform arithmetic operations on U.S. units of weight. Define metric units of mass and convert from one unit to another. Perform arithmetic operations on units of mass.

Examples: 1. Convert as indicated.

2 oz = ? lb 5

a) 192 oz = ? lb

b) 8,800 lbs = ? ton

d) 6.7 lb = ? oz

e) 16 lb 3 oz = ? oz

f) 3.1 ton = ? oz

b) 59 lb 2 oz – 18 lb 15 oz

c) 16 tons 1400 lb ÷ 5

2. Perform each indicated operation. a) 37 lb 12 oz + 22 lb 7 oz

d) A company wishes to ship 15 boxes of books. If each box weighs 3 lb 10 oz, what is the total weight of 15 boxes? 3. Convert as indicated. a) 27 kg = ? g

b) 310 g = ? kg

c) 22 g = ? mg

d) 1,035 mg = ? g

e) 8,360 cg = ? kg

f) 16 hg = ? mg

b) 5 g – 1301 mg

c) 9 kg ÷ 4

4. Perform each indicated operation. a) 11.7 mg + 3.2 mg

d) A bottle weighs 125 grams. Find the weight in kilograms of 2 dozen bottles. Teaching Notes: • • •

Refer students to the U.S. Units of Weight chart and the Metric Units of Mass chart in the textbook. Review the use of unit fractions. Remind students that the prefixes are the same in the metric system for mass and length.

Answers: 1a) 12 lb, b) 4.4 ton, c) 1/40 or 0.025 lb, d) 107.2 oz, e) 259 oz, f) 99.2 oz; 2a) 60 lb 3 oz, b) 40 lb 3 oz, c) 6680 lb, d) 54 lb 6 oz; 3a) 27,000g, b) 0.31 kg, c) 22,000 mg, d) 1.035 g, e) 0.0836 kg, f) 1,600,000 mg; 4a) 14.9 mg, b) 3.699g, c) 2.25 kg, d) 3 kg

M-54

Mini-Lecture 9.6 Capacity Learning Objectives: 1. 2. 3. 4.

Define U.S. units of capacity and convert from one unit to another. Perform arithmetic operations on U.S. units of capacity. Define metric units of capacity and convert from one unit to another. Perform arithmetic operations on metric units of capacity.

Examples: 1. Convert each measurement as indicated. a) 72 fl oz = ? c

b) 10 pt = ? c

d) 26 qt = ? gal

e) 3

1 qt = ? fl oz 8

c) 6

1 qt = ? pt 2

f) 5 gal = ? fl oz

2. Perform each indicated operation. a) 8 gal 5 qt + 6 gal 2 qt

b) 5 pt – 2 pt 1 c

c) 3 gal 8 cups × 3

d) A recipe calls for 45 fluid ounces of water. How many cups is this? 3. Convert each measurement as indicated. a) 6 L = ? ml

3.2 L = ? cl

c) 1800 ml = ? L

d) 0.135 L = ? kl

e) 0.072 dl = ? ml

f) 43,000 L = ? hl

b) 12,520 ml – 0.6 L

c) 13.5 L ÷ 1.8

4. Perform each indicated operation. a) 17.5 L + 16.8 L

d) A chemistry student accidentally added 69 cl of a chemical to a mixture instead of 38 ml. How much extra of the chemical did the student add? Teaching Notes: • • •

Refer students to U.S. Units of Capacity chart and Metric Unit of Capacity chart in textbook. Remind students to use unit fractions whenever possible. With metrics, many students will memorize prefixes and move the decimal point accordingly.

Answers: 1a) 9c, b) 20c, c) 13 pt, d) 6.5 gal, e) 100 fl oz, f) 640 fl oz; 2a) 15 gal 3 qt, b) 2 pt 1 c, d) 10 gal 8 c; 3a) 6000 ml, b) 320 cl, c) 1.8 L, d) 0.000135 kl, e) 7.2 ml, f) 430 hl; 4a) 34.3 L, b) 11.92L, c) 7.5L, d) 652 ml

M-55

Mini-Lecture 9.7 Temperature and Conversions Between the U.S. and Metric Systems Learning Objectives: 1. Convert between the U.S. and metric systems. 2. Convert temperatures from degrees Celsius to degrees Fahrenheit. 3. Convert temperatures from degrees Fahrenheit to degrees Celsius. Examples: 1. Convert as indicated. If necessary, round answers to two decimal places. a) 12 km = ? mi

b) 6 in = ? cm

c) 5.5 ft = ? m

d) 20 L = ? gal

e) 34 qt = ? L

f) 27 L = ? qt

g) 3 kg = ? lb

h) 16 oz = ? g

i) 14.3 lb = ? kg

2. Covert from degrees Celsius to degrees Fahrenheit. When necessary, round to the nearest tenth of a degree. a) 60°C to degrees Fahrenheit ° b) 32°C to degrees Fahrenheit c) 50°C to degrees Fahrenheit 3. Convert from degrees Fahrenheit to degrees Celsius. When necessary, round to the nearest Tenth of a degree. a) 68°F to degrees Celsius b) 113°F to degrees Celsius c) 20°F to degrees Celsius Teaching Notes: • • •

Remind students to use their unit fractions. Refer students to the textbook for conversions involving length, capacity, and weight. Some students may need to find objects that correspond to the conversions so they can visualize these equivalences.

Answers: 1a) 7.44 mi, b) 15.24 cm, c) 1.65 m, d) 5.2 gal, e) 32.3 L, f) 28.62 qt, g) 6.6 lb, h) 453.6 g, i) 6.44 k; 2a) 140◦ F, b) 89.6◦F, c) 122◦ F; 3a) 20◦ C b) 45◦C, c) –6.7◦ C

M-56

Mini-Lecture 10.1 Adding and Subtracting Polynomials Learning Objectives: 1. Add polynomials. 2. Subtract polynomials. 3. Evaluate polynomials at given replacement values. Examples: 1. Add the polynomials. a) ( 3 y + 7 ) + ( −9 y − 14 )

( x − 4 x − 3) + ( 5 x − 6 x ) c) ( − z − 4.2 z + 11) and ( 9 z − 1.9 z + 6.3) 2

2. Subtract the polynomials. a) ( 9b + 8 ) − (11b − 20 )

b) ( −4a + 20 ) − ( −14a + 36 )

(

) (

c) 11x 2 + 7 x + 2 − 15 x 2 + 4 x

)

( ) ( ) e) Subtract ( 3 x − 12 x ) from ( −4 x + 20 x + 17 )

d) Subtract −7 y 2 + y − 4 from −3 y 2 + 5 y 2

3. Find the value of each polynomial when y = 3. a) 2 y 3 + y 2 − 6 4 y2 − 14 2 c) 4 y 2 − 5 y + 10

Teaching Notes: • Stress the definition of like terms. • Stress the concept of distributing the negative in front of parentheses. • Caution students to think about order when asked to subtract one polynomial from another polynomial. Which polynomial is the minuend, and which polynomials is the subtrahend?

Answers: 1a) –6y – 7, b) 6x2 – 10x – 3, c) 8z2 – 6.1z + 17.3; 2a) –2b + 28, b) 10a – 16, c) –4x2 + 3x + 2 , d) 4y2 + 4y + 4, e) –7x2 + 32x + 17; 3a) 57, b) 4, c) 31

M-57

Mini-Lecture 10.2 Multiplication Properties of Exponents Learning Objectives:

1. Use the product property for exponents. 2. Use the power property for exponents. 3. Use the power of a product property for exponents. Examples:

1. Use the product property to simplify each expression. Write the results using exponents. a) x5 · x3

b) (4z3)(9z5)

c) (–3x3y2)(–5x4y6)

d) (9ab2c4)(–11a3b)(–2b2c5)

2. Use the power property to simplify each expression. a) (x7)3

b) (y3)11

c) (xy)5

d) (m7)5 ∙(m2)3

3. Use the power of a product property to simplify each expression. a) (–3x)3 (2x3)4

b) (–3x)(2x2)4

c) (2x)(–2x3y2)2

d) (2x2y4)4(3x6y9)2

Teaching Notes: • Most students need a lot of practice to master these properties.

Answers 1a) x8; b) 36z8; c) 15x7y8; d) 198a4b5c9; 2a) x21; b) y33; c) x5y5; d) m41; 3a) –432x15; b) –48x9; c) 8x7y4; d)144x20y34

M-58

Mini-Lecture 10.3 Multiplying Polynomials Learning Objectives:

1. 2. 3. 4. 5.

Multiply a monomial and any polynomial. Multiply two binomials. Square a binomial. Use the FOIL order to multiply binomials. Multiply any two polynomials.

Examples: 4. Multiply. a) 4 y 8 y 2 + 5

(

)

(

b) −2a 2a 2 − a + 12

5. Multiply, a) ( x + 4 )( x + 5 )

)

(

)

c) 3r 8r 2 − r + 11

b) ( b + 3)( b + 5 )

c) ( 7 x − 1)( 5 x + 4 )

b) ( y − 3)

c) ( 6 y − 1)

6. Multiply. a) ( b + 3)

7. Use FOIL order to multiply. b) ( 3x + 2 )

a) (10 x − 7 )( 2 x + 3) 8. Multiply. a) ( x + 3) x 2 + 2 x − 2

(

)

(

)

b) ( 2 x + 5 ) x 2 + 4 x − 1

Teaching Notes: • Remind students: FOIL order can only be used to multiply two binomials. • Show several examples on squaring a binomial to reinforce using the definition of an exponent.

Answers: 1a) 32 y + 20 y , b) − 4a + 2a − 24 a , c) 24 r − 3r + 33r ; 2a) x + 9 x + 20 , b) b + 8b + 15 3

, c) 35 x + 23 x − 4 ; 3a) b + 6b + 9 , b) y − 6 y + 9 , c) 36 y − 12 y + 1 ; 4a) 20 x + 16 x − 21 , b) 2

9 x 2 + 12 x + 4 ; 5a) x 3 + 5 x 2 + 4 x − 6 , b) 2 x 3 + 13 x 2 + 18 x − 5

M-59

Mini-Lecture 10.4 Introduction to Factoring Polynomials Learning Objectives:

1. Find the greatest common factor of a list of integers. 2. Find the greatest common factor of a list of terms. 3. Factor the greatest common factor from the terms of a polynomial. Examples:

1. Find the greatest common factor for each list. a) 16, 6

b) 18, 24

c) 15, 21

d) 12, 28, 40

c) –28x4, 56x5

d) 21m2n5, 35mn4

2. Find the GCF for each list. a) 15m2, 25m5

b) 40x2 , 20x7

3. Factor out the GCF from each polynomial. a) 5a + 15

b) 56z + 8

c) y3 + 2y

d) 5x3 + 10x4

e) 16z5 + 8z3 – 12z

f) x(y2 – 2) + 3(y2 – 2)

g) 6a8b9 – 8a3b4 + 2a2b3 + 4a5b3

Teaching Notes: • • •

Many students remove common factors, but not necessarily the greatest common factor. Encourage students to factor in a step-by-step manner: first factor out the GCF for the coefficients, then the GCF for each variable. Remind students that they can check their work by multiplying.

Answers: 1a) 2; b) 6; c) 3; d) 4; 2a) 5m2; b) 20x2; c) 28x4; d) 7mn4; 3a) 5(a+3); b) 8(7z+1); c) y(y2+2); d) 5x3(1+2x); e) 4z(4z4+2z2-3); 3f) (y2-2)(x+3); g)2a2b3(3a6b6-4ab+1+2a3

M-60

Mini-Lecture Graphing Answers Section 8.4 continued

Section 4.1 3a) 3b) 3c)

•

3d) 0

•3

1d)

• •

1e)

Section 8.4 1a)

1b)

1f)

1g)

1c) 2a) x axis: 1 mark = 1 year y axis: 1 mark = $10,000

M-61

Name: Instructor:

Date: Section:

Additional Exercises 1.2 Determine the place value of the digit 7 in each whole number. 1. 2784

1. __________________________

2. 29,706,142

2. __________________________

3. 397

3. __________________________

4. 127,851

4. __________________________

Write each whole number in words. 5. 3965

5. __________________________

6. 50,918

6. __________________________

7. 84,921

7. __________________________

8. 190,025

8. __________________________

9. 2,000,175

9. __________________________

10. 30,000,108

10. __________________________

Write each number in the sentence in words. 11. The population of Texas was recently reported as 24,326,974.

11. __________________________

12. The film opening of “The Dark Knight” earned a record breaking $18,489,000.

12. __________________________

Write each whole number in standard form. 13. Thirty-one thousand, eight hundred twenty-four.

13. __________________________

14. Seven thousand, six hundred four.

14. __________________________

15. One hundred twenty-five thousand, three hundred sixty-four.

15. __________________________

16. Seventeen thousand, eight hundred forty-one.

16. __________________________

17. Three million, ten thousand, eight.

17. __________________________

Write each whole number in expanded form. 18. 295

18. __________________________

19. 71,028

19. __________________________

20. 45,012,260

20. __________________________

E-1

Name: Instructor:

Date: Section:

Additional Exercises 1.3 Add or subtract as indicated. 1. 63 + 97

1. __________________________

2. 2698 + 350

2. __________________________

52 – 28

3. __________________________

4. 2651 –348

4. __________________________

5. 172 + 36 + 97

5. __________________________

6. __________________________ 6. 7000 – 3296

7. 4326 – 2152

7. __________________________

8. 4021 + 3607 + 892 + 6389

8. __________________________

9. 4210 – 1865

9. __________________________

10. Find the sum of 395 and 276.

10. __________________________

11. Find the difference of 70 and 28.

11. __________________________

E-2

Name: Instructor:

Date: Section:

Additional Exercises 1.3 (cont’d) 12. Subtract 97 from 101.

12. __________________________

13. Find the total of 82, 37, 76, and 98.

13. __________________________

14. What is 508 decreased by 149?

14. __________________________

15. Find 726 increased by 85.

15. __________________________

Find the perimeter of each figure. 16.

7 ft.

16. __________________________

9 ft. 12 ft.

17.

17. __________________________

60 ft. 28 ft. 44 ft.

26 ft. 33 ft. 63 ft.

Solve 18. Abraham Lincoln was born in 1809 and became President of the United States when he was 52 years old. What year did he become President?

18. __________________________

19. When the Silva family started on a trip, the odometer of their car read 30,246; when the trip was over, it read 33,205. How many miles did they drive on their trip?

19. __________________________

20. The Nile River is 4160 miles long. The Mississippi River is 2340 miles long. How much longer is the Nile than the Mississippi?

20. __________________________

E-3

Name: Instructor:

Date: Section:

Additional Exercises 1.4 Round each number to the given place 1. 705 to the nearest ten.

1. __________________________

2. 2634 to the nearest ten.

2. __________________________

3. 127,849 to the nearest thousand.

3. __________________________

4. 11,249 to the nearest hundred.

4. __________________________

5. 3942 to the nearest ten-thousand.

5. __________________________

6. 68,321,482 to the nearest thousand.

6. __________________________

7. 78,264,105 to the nearest million.

7. __________________________

Complete the table by estimating the given number to the given place value. Tens Hundreds Thousands 8. 9214 9. 7778 10. 94,073 11. A new baseball stadium has a seating capacity of 47,895 people. Round this number to the nearest thousand.

11. __________________________

12. According to a recent census, the U.S. population was 334,914,895. Round this number to the nearest million.

12. __________________________

Estimate each sum or difference by rounding each number to the nearest hundred. 13.

812 914 + 785

13. __________________________

14.

903 − 794

14. __________________________

Estimate each sum or difference by rounding each number to the nearest thousand. 15.

2840 4742 + 1690

15. __________________________

16. 42,910 − 16,245

16. __________________________

E-4

Name: Instructor:

Date: Section:

Additional Exercises 1.4 (cont.) Estimate each sum to determine whether the given answer could be correct. 17.

6430 +1218; 7648

17. __________________________

18. 51,800 + 32,560 + 13,189; 96,549

18. __________________________

Solve each application by estimating. 19. A homeowner needs a new washer and dryer. The washer she likes costs $399 and the dryer costs $321. Round each cost to the nearest hundred to estimate the total cost.

19. __________________________

20. Hank Aaron batted in 2,297 runs during his major league baseball career and Babe Ruth batted in 2,213 runs during his. Round each number to the nearest hundred to estimate how many more runs Hank Aaron hit than Babe Ruth?

20. __________________________

E-5

Name: Instructor:

Date: Section:

Additional Exercises 1.5 Use the distributive property to rewrite each expression. 1. 3(2 + 6)

1. __________________________

2. 4(2 + 5)

2. __________________________

3. 6(12 + 5)

3. __________________________

Multiply 4.

78 × 65

4. __________________________

298 × 3

5. __________________________

6. 1036 × 2

6. __________________________

379 × 13

7. __________________________

907 × 12

8. __________________________

328 × 94

9. __________________________

10. (700)(20)

10. __________________________

11. (23)(2)(4)

11. __________________________

12. (13)(626)(0)

12. __________________________

13. (470)(10)(1)

13. __________________________

E-6

Name: Instructor:

Date: Section:

Additional Exercises 1.5 (cont.) 14.

6083 × 419

14. __________________________

15.

3158 × 4672

15. __________________________

Estimate the product by rounding each factor to the nearest hundred. 16. 895 × 368

16. __________________________

17. 218 × 752

17. __________________________

Solve. 14 m 18. Find the area of the rectangle.

18. __________________________ 6m

19. One bagel has 230 calories. How many calories are in a package of 12 bagels?

19. __________________________

20. A marching band has 9 rows with 17 people in each row. How many people are in the band?

20. __________________________

E-7

Name: Instructor:

Date: Section:

Additional Exercises 1.6 Divide and then check by multiplying. 1.

6 288

1205 ÷ 5

3. 423 ÷ 9 4.

1032 ÷ 4

9 639

1. __________________________

2. __________________________ 3. __________________________ 4. __________________________

5. __________________________

6. 83 ÷ 0

6. __________________________

7. 179 ÷ 6

7. __________________________

8. 3624 ÷ 16

8. __________________________

25 973

9. __________________________

10. 4000 ÷ 47

10. __________________________

11. 9870 ÷ 42

11. __________________________

12. 364 ÷ 13

12. __________________________

13. 0 ÷ 418

13. __________________________

E-8

Name: Instructor: Additional Exercises 1.6 (cont.)

Date: Section:

14. A truck has 5600 pounds of apples to deliver to a distribution center. The truck can haul 350 pounds each trip. How many trips does it take?

14. __________________________

15. A movie theater collected $784 on one showing of a movie. If tickets cost $8 per person, how many people attended the showing?

15. __________________________

16. Find the number of yards in a mile. (A mile is 5280 feet; a yard is 3 feet.)

16. __________________________

17. A pharmaceutical sales representative in Oklahoma travels four days a week. She drove 330 miles on Monday, 285 miles on Tuesday, 251 miles on Wednesday, and 118 miles on Thursday. What was her average mileage per day?

17. __________________________

18. Rashid had scores of 68, 85, 76, 77, and 89 on his math quizzes. What was his quiz grade average?

18. __________________________

Romanian Cities Major City Bucharest Brasov Constanta Cluj – Napoca Sibiu

Population 2,300,000 353,000 316,000 318,000 169,000

Find the average population for the five cities

19. __________________________

20. How many more people live in Cluj – Napoca than in Constanta?

20. __________________________

19.

E-9

Name: Instructor:

Date: Section:

Additional Exercises 1.7 Write each using exponential notation. 1. 4 ⋅ 4 ⋅ 4 ⋅ 3 ⋅ 3 ⋅ 3

1. __________________________

2. 21 ⋅ 21 ⋅ 21⋅ 21

2. __________________________

3. 8 ⋅ 8 ⋅ 6 ⋅ 6 ⋅ 6 ⋅ 3 ⋅3 ⋅3 ⋅3

3. __________________________

Evaluate. 4. 811

4. __________________________

5. 34

5. __________________________

6. 110

6. __________________________

7. 5 ⋅ 62

7. __________________________

8. 103

8. __________________________

9. 4 2 ⋅ 2 3

9. __________________________

Simplify. 10. 32 – 6 ⋅ 4

10. __________________________

11. 48 ÷ 8 – 2

11. __________________________

12. 12 ⋅5 + 2 2 ⋅ 6

12. __________________________

13. (15 + 10) ÷ 5

13. __________________________

14. 16 ÷ (6 – 2)

14. __________________________

5(6 − 2) + 7 3

15. __________________________

15.

16. 40 − 2 ⎡⎣32 + (18 ÷ 3 + 2) ⎤⎦

16. __________________________

17. 50 – [52 + (12 – 8) ⋅ 2] + 6 ⋅ 2

17. __________________________

Find the area of each square. 18.

18. __________________________ 12 m

19.

19. __________________________ 15 ft.

20. A square lawn is 72 feet on each side. If four bags of fertilizer are available and each covers 1200 square feet, is there enough fertilizer to cover the lawn? E-10

20. __________________________

Name: Instructor:

Date: Section:

Additional Exercises 1.8 Evaluate each following expression for x = 2, y = 3, and z = 4. 1. 3x + 4

1. __________________________

2. 4x – 2z

2. __________________________

3. 3x + yz

3. __________________________

4. x2 – y

4. __________________________

5. 5 x − 2 y + z 2

5. __________________________

2 xyz 3x

6. __________________________

7. (2z – y)2

7. __________________________

8. 2 y 2 (3 x − z )

8. __________________________

Write each phrase as a variable expression. Use x to represent “a number.” 9. The sum of a number and twice the number.

9. __________________________

10. Seven decreased by a number.

10. __________________________

11. The square of a number less five.

11. __________________________

12. Eighteen less than the product of four and a number.

12. __________________________

13. The quotient of a number and eight.

13. __________________________

14. A number subtracted from twelve.

14. __________________________

Decide whether the given number is a solution to the given equation. 15. Is 3 a solution to x + 8 = 11?

15. __________________________

16. Is 4 a solution to3 x − 8 = 8?

16. __________________________

17. Is 8 a solution to 4x − 16 = 2x?

17. __________________________

18. Is 5 a solution to 7x + 40 = 5?

18. __________________________

Determine which numbers in each set are solutions to the corresponding equation. 19. n + 5 = 10 {2, 5, 7} 20. 2x + 3 = 11 {3, 4, 8}

19. __________________________ 20. __________________________

E-11

Name: Instructor:

Date: Section:

Additional Exercises 2.1 Represent each quantity by an integer. 1. A deep-sea diver is 350 feet below the surface of the ocean.

1. __________________________

2. A company reports a loss of $2,000,000 for last year.

2. __________________________

Graph each integer in the list on the same number line. 3. 4, −2, 3

4. −3, 0, −1, 4

Insert < , > or = between each pair of integers to make a true statement. 5. –7

5. __________________________

6. 5

6. __________________________

7. −|3|

|−3|

7. __________________________

8. |15|

|−15|

8. __________________________

Find each absolute value. 9. |0|

9. __________________________

10. −|6|

10. __________________________

11. |−20|

11. __________________________

E-12

Name: Instructor: Additional Exercises 2.1 (cont.)

Date: Section:

Find the opposite of each integer. 13. −6

13. __________________________

14. 8

14. __________________________

15. −3

15. __________________________

Simplify. 16. −(−10)

16. __________________________

17. −|−5|

17. __________________________

18. −|12|

18. __________________________

Evaluate. 19. |−x| if x = 4.

19. __________________________

20. |x| if x = −8.

20. __________________________

E-13

Name: Instructor:

Date: Section:

Additional Exercises 2.2 Add. 1. 7 + (−3)

1. __________________________

2. (−10) + 5

2. __________________________

3. 3 + (−5)

3. __________________________

4. −18 + (−21)

4. __________________________

5. −3 + (−15)

5. __________________________

6. −18 + 12

6. __________________________

7. −5 + 5

7. __________________________

8. 17 + (−2)

8. __________________________

9. −45 + 15

9. __________________________

10. 27 + (−83)

10. __________________________

11. −19 + 12

11. __________________________

12. 110 + (−24)

12. __________________________

13. 8 + (−2)

13. __________________________

14. −21 + (−11)

14. __________________________

15. −5 + 7 + (−8)

15. __________________________

16. 20 + (−11) + (−3) + 6

16. __________________________

17. Evaluate x + y for x = 3 and y = −8.

17. __________________________

18. Evaluate 2x + y for x = −2 and y = 6.

18. __________________________

19. A student has $290 in her checking account. She writes checks for $102 and $75 and then makes a deposit of $170. Find the amount left in her account.

19. __________________________

20. Suppose a deep-sea diver dives from the surface to 125 feet below the surface. She then dives 12 more feet. What is her present depth as a signed number?

20. __________________________

E-14

Name: Instructor:

Date: Section:

Additional Exercises 2.3 Subtract. 1. −7 − (−3)

1. __________________________

2. 7 – 4

2. __________________________

3. –14 – (–14)

3. __________________________

4. −6 − 8

4. __________________________

5. 8 – (−8)

5. __________________________

6. 14 – 28

6. __________________________

7. −41 – (–7)

7. __________________________

8. 6 + (−6)

8. __________________________

9. 18 – 53

9. __________________________

10. 9 – (−4)

10. __________________________

11. Subtract −3 from −12.

11. __________________________

12. Find the difference of −15 and −3.

12. __________________________

Simplify. 13. 15 – 5 – 4

13. __________________________

14. −8 – 3 – (−7)

14. __________________________

15. −12 + (−5) −3

15. __________________________

16. 21 − (−15) + (−8)

16. __________________________

17. Evaluate x − y for x = 5 and y = −8.

17. __________________________

18. Evaluate 3x − y for x = −4 and y = −2.

18. __________________________

19. On December 21, the average high temperature was −5° F. On June 21, the average high temperature was 62° F. How many degrees warmer was the temperature in June than in December?

19. __________________________

20. A student has $87 in their checking account. They make a deposit of $130 and write two checks for $19 and $100. Find the amount left in their account.

20. __________________________

E-15

Name: Instructor:

Date: Section:

Additional Exercises 2.4 Multiply. 1. −3(−8)

1. __________________________

2. −2(5)

2. __________________________

3. 0(−8)

3. __________________________

4. 4(−2)

4. __________________________

5. −8(−4)

5. __________________________

6. 6(−5)

6. __________________________

7. −7(9)

7. __________________________

Evaluate. 8. (−3) 4

8. __________________________

9. −2 4

9. __________________________

Multiply. 10. −1 ⋅ (−1) ⋅ (−1) ⋅ (1)

10. __________________________

11. −2(−7)(3)

11. __________________________

12. 2(–3)(6)

12. __________________________

Divide. 13. −28 ÷ 7

13. __________________________

14.

−27 0

14. __________________________

15.

−25 −5

15. __________________________

16.

−185 5

16. __________________________

17. Evaluate ab for a = −3 and b = −2. a 18. Evaluate for a = −12 and b = 4. b

17. __________________________

19. A gambler lost $125 on each of three consecutive days at the horse races. Find his total loss as a signed number.

19. __________________________

20. A company declared a loss of $10,500 for each of five consecutive months. Find the company’s total loss as a signed number.

20. __________________________

E-16

18. __________________________

Name: Instructor:

Date: Section:

Additional Exercises 2.5 Simplify. 1. 3 + (−12) ÷ 3

1. __________________________

2. 5 + 4 ⋅ 2 – 7

2. __________________________

3. (−12) + 16 ÷ 4

3. __________________________

4. 6 ⋅ 2 3

4. __________________________

80 −2 − 3

5. __________________________

6. [2 + (−3)]2

6. __________________________

7. 23 – 27

7. __________________________

8. (8 – 16) ÷ 4

8. __________________________

9. (−17) + 8 ÷ 2

9. __________________________

10. |3 – 9| ⋅ (−2) ÷ 6

10. __________________________

11. 4 – 3 ⋅ 5 – 6

11. __________________________

12. 4(7 – 4) + (−2)3

12. __________________________

13. (−5 ÷ 5) – (5 ÷ 5)

13. __________________________

14. (8 – 2)(5 – 12)

14. __________________________

15. 92 – (5)2

15. __________________________

16. 10 – [8 – (2 – 7)]

16. __________________________

(−5)(−2) − (7)(3) 11

17. __________________________

17.

18. 42 − (2)2

18. __________________________

19. Evaluate 2x − y − z for x = 2 and y = 5 and z = −3.

19. __________________________

20. Evaluate − x 2 for x = −4.

20. __________________________

E-17

Name: Instructor:

Date: Section:

Additional Exercises 2.6 Determine whether the given number is a solution of the given equation. 1. Is 8 a solution of x − 4 = 4?

1. __________________________

2. Is 3 a solution of x + 6 = 6?

2. __________________________

3. Is −4 a solution of −2x = 4 − x?

3. __________________________

4. Is 1 a solution of 2(x − 4) = 6?

4. __________________________

Solve. Check each solution. 5. x + 5 = 12

5. __________________________

6. x + 7 = −8

6. __________________________

7. x − 5 = −3

7. __________________________

8. 14x = −14

8. __________________________

9. 12x = 11x − 5

9. __________________________

10. 3 = x + 5

10. __________________________

11. 5 = x − 4

11. __________________________

12. 2x = 18

12. __________________________

13. 3x = −12

13. __________________________

14. −2x = −4

14. __________________________

15.

x =2 −2

15. __________________________

16.

x = −1 12

16. __________________________

17. 40 = 8 x

17. __________________________

Translate each phrase to an algebraic expression. Use x to represent “a number.” 18. Subtract a number from –7 18. __________________________ 19. The quotient of –6 and a number. 19. __________________________ 20. four times a number 20. __________________________

E-18

Name: Instructor:

Date: Section:

Additional Exercises 3.1 Simplify each expression by combining like terms. 1. 5x – 4x

1. __________________________

2. 7y + 8y

2. __________________________

3. 6b + 10b – b – 7b

3. __________________________

4. −5a + 2a – 9a + 16 – (−2a)

4. __________________________

Multiply. 5. 8(12x)

5. __________________________

6. 4(y + 5)

6. __________________________

7. 2(6x – 8)

7. __________________________

8. −5(3b – 7)

8. __________________________

9. −9(6 + 3x)

9. __________________________

Simplify each expression. First use the distributive property to multiply and remove parentheses. 10. 4(x – 6) + 3x

10. __________________________

11. −5z + 7(2 – 3z)

11. __________________________

12. 2(5n – 6) + 3n

12. __________________________

13. −8 + 2(n – 4) + 5(n – 1)

13. __________________________

14. −3(5w + 1) − 6w

14. __________________________

15. 5(2x – 3) – 7x

15. __________________________

16. −3(n – 1) – 6n

16. __________________________

17. 8z – (5z − 2)

17. __________________________

18. Find the perimeter.

x feet

18. __________________________

2x – 3 feet 19. Find the perimeter of the square. (4y) cm

19. __________________________

20. Find the perimeter.

20. __________________________ 2x yds.

2x yds. x yds.

E-19

Name: Instructor:

Date: Section:

Additional Exercises 3.2 Solve each equation. First, combine any like terms on each side of the equation. 1. 6x + 2x = 32

1. __________________________

2. 24 = 10x – 6x

2. __________________________

3. 2x – 7x = 40

3. __________________________

4. 5 – 17 = −y

4. __________________________

5. 8x – 17x = 13 + (−13)

5. __________________________

6. 6x + 12x = 18

6. __________________________

7. −64 = −4y + 12y

7. __________________________

x = −20 + 16 4

8. __________________________

Solve each equation. 9. −5 x = −35

9. __________________________

10. 16a = 3 ( 2a + 10)

10. __________________________

11. 4 ( 3 y − 6 ) = 36

11. __________________________

12. 8 x − 11 − 5 x = 10 − 6

12. __________________________

13. 12 y − 4 y = 6 ( y + 2 )

13. __________________________

14. −12 x − 18 = −9 x

14. __________________________

x = −7 − 5 −3

15. __________________________

15.

Write each phrase as a variable expression. Use x to represent “a number.” 16. Twice a number, increased by twenty.

16. __________________________

17. Twice the sum of a number and 6.

17. __________________________

18. The difference of a number and twice the number.

18. __________________________

19. The quotient of three times a number and seven.

19. __________________________

20. The product of 13 and a number added to 33.

20. __________________________

E-20

Name: Instructor:

Date: Section:

Additional Exercises 3.3 Solve each equation. 1. 3x – 21 = 0

1. __________________________

2. −7x + 9 = 37

2. __________________________

3. −8 = 3y + 1

3. __________________________

4. 3n – n = 21 − 5

4. __________________________

5. 8 − 9 x = −4 x − 12

5. __________________________

6. 16 = 2y + 8

6. __________________________

7. 9x – 3 = 6x + 15

7. __________________________

8. 7x – 8 = 5x + 10

8. __________________________

9. 12y – 11y = 18 – (−4)

9. __________________________

10. 18 + 3 y = −5 + 4 y

10. __________________________

11. 3(x + 2) = 2x – 7

11. __________________________

12. 7 – 3 = x + 4

12. __________________________

13. 8 ( 4 + 7 x ) = 12 + 55 x

13. __________________________

14. −2(x − 3) = −8

14. __________________________

15. 7 (2a − 5) + 12 = 8a + 1

15. __________________________

16. 8(3 + x) = 7(x + 3)

16. __________________________

Write each sentence as an equation and solve. 17. The sum of −38 and a number is −21.

17. __________________________

18. Twice the difference of 7 and a number yields 14.

18. __________________________

19. The product of −3 and 4 plus a number is −33.

19. __________________________

20. The quotient of 120 and ten is twice a number.

20. __________________________

E-21

Name: Instructor:

Date: Section:

Additional Exercises 3.4 Write each sentence as an equation. Use x to represent “a number.” 1. The sum of a number and 6 is 9.

1. __________________________

2. The difference of a number and 7 is 13.

2. __________________________

3. The product of −3 and a number gives 21.

3. __________________________

4. 20 decreased by twice a number equals 10.

4. __________________________

Translate each sentence into an equation. Use x to represent “a number.” Then solve the equation. 5. Twelve increased by a number is negative seven.

5. __________________________

6. The difference of nine and a number equals six times the number minus five. Find the number.

6. __________________________

7. Twenty-five less a number is equal to four times the number. Find the number.

7. __________________________

8. The sum of ten, eighteen and a number is 50. Find the number.

8. __________________________

9. A number added to 10 is –30. Find the number.

9. __________________________

10. Four subtracted from a number equals 12. Find the number.

10. __________________________

11. The quotient of a number and 11 is −33. Find the number.

11. __________________________

12. Six added to twice a number gives −14. Find the number.

12. __________________________

E-22

Name: Instructor: Additional Exercises 3.4 (cont.)

Date: Section:

Solve. 13. A number less 7 is 31. Find the number.

13. __________________________

14. Six times a number added to −10 is −16. Find the number.

14. __________________________

15. The sum of 5, 2 and a number amounts to 4. Find the number.

15. __________________________

16. Twice the difference of a number and 3 is equal to 48. Find the number.

16. __________________________

17. At a school fundraiser the girls’ soccer team sold twice as many t-shirts as the boys’ soccer team. Together the teams sold 201 shirts. How many shirts did each team sell?

17. __________________________

18. A bicyclist is traveling twice as fast as someone walking. If their combined speed is 21 miles per hour, find the speed of the person walking.

18. __________________________

19. During the Student Government Association elections, one candidate received 38 more votes for Senator than the other candidate. If the total number of votes cast for the two candidates was 164, how many votes did each candidate get?

19. __________________________

20. A washing machine with installation costs $420. If the washing machine costs 5 times as much as the installation, find the cost of installation.

20. __________________________

E-23

Name: Instructor:

Date: Section:

Additional Exercises 4.1 Write a fraction to represent the shaded part of each figure. 1.

1. _________________________

2. __________________________

Simplify by dividing. 3.

7 7

3. __________________________

−6 1

4. __________________________

−4 −4

5. __________________________

6 1

6. __________________________

0 −4

7. __________________________

8 0

8. __________________________

Write each mixed number as an improper fraction. 1 9. 3 4 10. 3

E-24

4 5

9. __________________________

10. __________________________

Name: Instructor: Additional Exercises 4.1 (cont.)

11. 2

Date: Section:

4 7

12. 17

11. __________________________

2 3

12. __________________________

Write each improper fraction as a mixed number or a whole number. 13.

21 8

14.

32 9

14. __________________________

15.

78 7

15. __________________________

16.

96 12

16. __________________________

17.

191 172

17. __________________________

18.

112 11

18. __________________________

13. __________________________

19. In a bag of 8 cookies, there are 3 sugar cookies and 5 chocolate cookies. What fraction of the cookies are chocolate?

19. __________________________

20. In a bag class of 27 marbles, 5 are black and the rest are red. What fraction of the marbles are black?

20. __________________________

E-25

Name: Instructor:

Date: Section:

Additional Exercises 4.2 Write the prime factorization of each number. Write any repeated factors using exponents. 1. 21

1. __________________________

2. 72

2. __________________________

3. 210

3. __________________________

4. 186

4. __________________________

5. 115

5. __________________________

6. 238

6. __________________________

7. 625

7. __________________________

8. 936

8. __________________________

Write each fraction in simplest form. 9.

40 85

9. __________________________

10.

30 x 45

10. __________________________

11.

27 x 60 xy

11. __________________________

12.

40 xy 15 y

12. __________________________

E-26

Name: Instructor: Additional Exercises 4.2 (cont.)

13.

14.

15.

Date: Section:

18 x 45

13. __________________________

36abc 2

14. __________________________

60ab 2 c

−20 x 40 xy

15. __________________________

Determine whether each pair of fractions is equivalent.

16.

2 3

4 6

17.

3 5

25 45

17. __________________________

18.

8 12

10 15

18. __________________________

16. __________________________

19. There are 5280 feet in a mile. What fraction of a mile is represented by 280 feet?

19. __________________________

20. A work shift is 8 hours. What fraction of a 24-hour day does this shift represent?

20. __________________________

E-27

Name: Instructor:

Date: Section:

Additional Exercises 4.3 Multiply. Write the product in simplest form. 1.

2 5 ⋅ 3 8

1. __________________________

5 1 ⋅ 8 15

2. __________________________

5 8

3. __________________________

5 16 ⋅ 8 15

4. __________________________

x2 y3 ⋅ y4 x

5. __________________________

3. 6a 2 ⋅

3 y

⋅

y 9

6. __________________________

Divide. Write all quotients in simplest form. 7.

1 1 ÷ 4 2

7. __________________________

1 2 ÷ 2 3

8. __________________________

4 1 ÷ 5 5

9. __________________________

10.

2x 3 ÷ 5 4

10. __________________________

11.

2 3 ÷ 5 5

11. __________________________

12.

3 2 ÷ 7 5

E-28

12. __________________________

Name: Instructor: Additional Exercises 4.3 (cont.)

Date: Section:

Evaluate. 1 1 and y = . 13. __________________________ 2 4 2 3 14. x ÷ y if x = and y = . 14. __________________________ 3 5 Determine whether the given replacement value is a solution to the given equation. 2 a solution to 6x = 4? 15. Is 15. __________________________ 3 3 2 16. Is 10 a solution to x = ? 16. __________________________ 5 3

13. xy if x = −

2 of the regular 3 price. If the regular price of a particular trip is $324, what will the attendant pay?

17. A flight attendant can purchase a ticket for

2 of the frame’s length. What is 3 the width if the length is 18 inches?

18. A picture frame’s width is

17. __________________________

18. __________________________

Find the area of each rectangle. 19. __________________________ 19.

3 meter 5 2 meter 5

20.

20. __________________________ 7 inches 8 1 inches 3

E-29

Name: Instructor:

Date: Section:

Additional Exercises 4.4 Add or subtract as indicated and simplify your answer. 1.

1 3 + 8 8

1. __________________________

2 1 + 5 5

2. __________________________

2 x  3x  + − 7  7 

3. __________________________

3. −

7 11 + 30 30

4. __________________________

3 6 − z z

5. __________________________

6 2 − 13 13

6. __________________________

11 7 − 26 26

7. __________________________

1 10 3 + − 15 15 15

8. __________________________

7 4 3 + + 19 19 19

9. __________________________

10.

7 4 − 15 15

10. __________________________

11. Evaluate x + y for x =

2 1 and y = . 4 4

12. Evaluate x − y for x =

9 7 and y = . 10 10

E-30

11. __________________________

12. __________________________

Name: Instructor: Additional Exercises 4.4 (cont.)

Date: Section:

Find the LCD of each list of fractions. 13.

14.

7 15

13. __________________________

6 3 , y 4

14. __________________________

Write each fraction as an equivalent fraction with the given denominator. 15.

16.

3 = 8 24

2 5

15. __________________________

16. __________________________

40a

17. Find the perimeter of the triangle.

17. __________________________

9 meter 17 5 meter 17 11 meter 17

18. Find the perimeter of the square. 18. __________________________ 1 yard 4 1 3 cup milk and cup 4 4 water. How much liquid is needed for this recipe?

19. A recipe for banana bread calls for

19. __________________________

20. A recipe for cookies is doubled to save time. The original 1 recipe calls for 2 cups of sugar. How much sugar is 2 needed for the recipe when it is doubled?

20. __________________________

E-31

Name: Instructor:

Date: Section:

Additional Exercises 4.5 Add or subtract as indicated. 5 1 + 7 3

1. __________________________

3x 1 − 4 2

2. __________________________

1 3 3. − + 3 4

3. __________________________

11 2 x + 35 7

4. __________________________

1. −

3z 8

6. 3 −

5. __________________________

2 5

6. __________________________

3 1 5 + − 16 16 16

7. __________________________

2 1 3 8. − + + 5 5 10

8. __________________________

x x 1 + + 5 4 4

9. __________________________

10.

3 5

Evaluate the expression if x = 11. x + y 12. 3x − y

E-32

10. __________________________

1 1 and y = . 3 4

11. __________________________ 12. __________________________

Name: Instructor: Additional Exercises 4.5 (cont.)

Date: Section: 13. __________________________

13. xy 14. __________________________ 14. x ÷ y 15. __________________________ 15. 2x + y Insert <, >, or = to form a true statement. 16.

3 7

3 5

−

12 16

18 24

17. −

18.

9 21

16. __________________________

3 4

17. __________________________

18. __________________________

19. Find the perimeter of the triangle.

20. Out of 210 students,

1 are juniors. Find the number of 3

19. __________________________

20. __________________________

juniors.

E-33

Name: Instructor:

Date: Section:

Additional Exercises 4.6 Simplify each complex fraction.

1 3 3 4

1. __________________________

7 3

2. __________________________

3 7 10

+ −

4 5 2

3. __________________________

2x 7 4 5

4. __________________________

Use the order of operations to simplify each expression. 1 5. 23 −   3

1 6. 42 −   4

5. __________________________

6. __________________________

9 1 1 ÷ ⋅ 16 2 4

7. __________________________

 1 1  1 1   3 − 15  5 + 3    

8. __________________________

1 4 3 ÷ 9  

9. __________________________

Additional Exercises 4.6 (cont.) E-34

Name: Instructor:

Date: Section: 2

1 1 10.   +   2 4

1 1 11.  +  2 4

10. __________________________

Evaluate each expression if x = −

11. __________________________

1 1 1 , y = , and z = . 2 3 4

12. x + y

12. __________________________

13. 2x − y

13. __________________________

14. x2

14. __________________________

15.

x+ y z

15. __________________________

16.

x z

16. __________________________

17. 2y

17. __________________________

18. (1 + x )(1 − z )

18. __________________________

Answer true or false for each statement. 19. The sum of two negative fractions is always a negative number. 20. The product of two negative fractions is always a negative number.

19. __________________________

20. __________________________

E-35

Name: Instructor:

Date: Section:

Additional Exercises 4.7 Multiply or divide. 1. __________________________ 1. 3 ÷ 1

4 5

2. __________________________ 1 2 2.  2   4   5  3 

3. __________________________ 1 3 3. 3 ⋅1 3 5

4. __________________________ 7 1 4. ÷2 9 3

5. __________________________  1  1  5.  2     3  5 

6. __________________________  2  1  6.  3   1   5  7 

 2  4  7.    2   3  5 

7. __________________________

 2 1 8.  2  ÷    3 5

8. __________________________

5 4 9.   ÷   9 9

9. __________________________

Add or subtract 10. 3

E-36

2 7

10. __________________________

1 1 11. 3 + 4 8 2

11. __________________________

3 1 12. 8 − 2 5 4

12. __________________________

Name: Instructor: Additional Exercises 4.7 (cont.)

Date: Section:

1 2 13. 7 − 4 3 5

13. __________________________

3 5 14. 18 − 17 5 6

14. __________________________

Perform each indicated operation. 1 1 15. 2 + 4 3 2

15. __________________________

2 1 16. 6 + 3 3 9

16. __________________________

1 3 17. 3 + 4 + 9 5 10

17. __________________________

5 7 −7 24 12

18. __________________________

18. 15

1 feet and length 3 2 2 feet 3 1 1 feet 3

19. __________________________

20. Find the perimeter of a regular pentagon for which each side 3 measures inch. 8

20. __________________________

19. Find the area of the rectangle with width 1 2

2 feet. Use the formula A = w. 3

E-37

Name: Instructor:

Date: Section:

Additional Exercises 4.8 Solve each equation. 1. x +

4 5 = 17 17

1. __________________________

3 2 − 4x = − 14 7

2. __________________________

2. 5 x +

1 x=5 3

3. __________________________

5 y = −3 9

4. __________________________

4 2 y= 9 3

5. __________________________

11 1 x= 13 26

6. __________________________

x 7 +3 = 2 3

7. __________________________

1 3 x −1 = 5 5

8. __________________________

2 x = 3 12

9. __________________________

10.

1 1 x − = 2 5 10

10. __________________________

5. −

E-38

Name: Instructor:

Date: Section:

Additional Exercises 4.8 (cont.) Add or subtract. x 1 − 7 3

11. __________________________

4x 3

12. __________________________

13.

5x x + 9 4

13. __________________________

14.

3x x + 5 10

14. __________________________

3x 5

15. __________________________

16.

2 1 x= 3 2

16. __________________________

17.

2 1 y= 5 10

17. __________________________

18.

2 x 4 − = 3 5 15

18. __________________________

19.

x 1 −2 = 5 5

19. __________________________

20.

x − x = −2 2

11.

12. 3 −

15. 4 +

Solve.

20. __________________________

E-39

Name: Instructor:

Date: Section:

Additional Exercises 5.1 Write each decimal in words. 1. 2.95

1. __________________________

2. 3.621

2. __________________________

3. –0.037

3. __________________________

4. 36.401

4. __________________________

Write each decimal number in standard form. 5. Twenty-one and thirty-five hundredths

5. __________________________

6. Negative forty-three and five hundredths

6. __________________________

7. Nine and two hundred two thousandths

7. __________________________

8. Thirty-five ten thousandths

8. __________________________

9. Twelve and twelve ten thousandths

9. __________________________

Write each decimal as a fraction or mixed number. Write your answer in simplest form. 10. 0.08

10. __________________________

11. 0.205

11. __________________________

12. 10.5

12. __________________________

13. 4.25

13. __________________________

14. –50.024

14. __________________________

Insert <, >, or = between each pair of numbers to form a true statement. 15. 0.036

0.094

16. 0.16

0.156

17. −0.23

−0.31

18. −7.01

−7.1

15. __________________________ 16. __________________________ 17. __________________________ 18. __________________________

19. A new dining room set cost $1295.86 and is financed for one year. Find the monthly payments rounded to the nearest cent.

19. __________________________

20. A student attends a college that is 785.89 miles away from home. Round this distance to the nearest mile.

20. __________________________

E-40

Name: Instructor:

Date: Section:

Additional Exercises 5.2 Perform the indicated operations. 1. 3.6 + 2.98

1. __________________________

2. 8.14 + 3.16

2. __________________________

3. 0.0006 + 1.134

3. __________________________

4. 31 + 0.165 + 2.63

4. __________________________

5. 57.4 + 6.95 + 0.084

5. __________________________

6. 9.3 – 4.71

6. __________________________

7. 6 – 4.12

7. __________________________

8. –3.25 + (–61.07)

8. __________________________

9. –17.35 – (–1.901)

9. __________________________

10. Subtract 3.5 from 5.8.

10. __________________________

11. Subtract 7.6 from 12.35

11. __________________________

12. Subtract 12.9 from 15.9

12. __________________________

E-41

Name: Instructor: Additional Exercises 5.2 (cont.)

Date: Section:

13. A student purchased several new items for their first semester at college. Below is a list of the items and the cost of each item. Find the total amount he spent. Calculator: Cell phone: Laptop: Printer:

13. __________________________

$89.99 $259.85 $699.89 $124.59

14. The total cost of a bottle of shampoo is $3.88 including the tax. If the price of the shampoo is $3.59, how much is the tax?

15. How much fencing is needed to enclose the field with the dimensions shown? 12.5 yds. 6.2 yds.

14. __________________________

15. __________________________

10.5 yds. 14.6 yds.

16. A shopper bought $38.09 worth of groceries. If they paid with two $20 bills, what was the change?

16. __________________________

17. Simplify by combining like terms: 1.04x + 1.3x

17. __________________________

18. Evaluate x + y if x = 1.37 and y = 0.32

18. __________________________

19. Evaluate x – z + y if x = 10 and y = 5.7 and z = 3.89

19. __________________________

20. Simplify by combining like terms: 0.5x – 3.2x + 3.57x

20. __________________________

E-42

Name: Instructor:

Date: Section:

Additional Exercises 5.3 Multiply. 1.

0.49 × 7

1. __________________________

3.7 × 0.5

2. __________________________

0.78 × 8.22

3. __________________________

(−3.7)(5.6)

4. __________________________

(−4.2)(0.36)

5. __________________________

3.89 × 10

6. __________________________

1.02 × 100

7. __________________________

5.013 × 100

8. __________________________

3.5 × 0.001

9. __________________________

10.

(3.2)(0.2)(0.5)

10. __________________________

11.

Find the exact circumference of a pizza with a diameter of 8 inches. Then use 3.14 for π to approximate the circumference.

11. __________________________

12.

Find the exact circumference of a mini trampoline with a diameter of 3.5 feet. Then use 3.14 for π to approximate the circumference.

12. __________________________

13.

Find the exact circumference of a circle with diameter of 20 cm. Then use 3.14 to approximate the circumference.

13. __________________________

14.

One cracker has 0.75 grams of fat. How much fat is in 8 crackers?

14. __________________________

15.

A meter is approximately equal to 39.37 inches. Susie is 1.2 meters tall. Find her approximate height in inches.

15. __________________________

16.

Evaluate xy if x = 3.25 and y = 0.7

16. __________________________

17.

Evaluate 3x if x = 1.2

17. __________________________

18.

A server at the Dairy Bar worked 32 hours last week. Calculate their pay before taxes if they make $7.85 an hour.

18. __________________________

19.

Find the perimeter of a square with sides of length 3.2 yards.

19. __________________________

20.

Find the area of a rectangle with width 2.35 ft and length 4.8 ft.

20. __________________________

E-43

Name: Instructor:

Date: Section:

Additional Exercises 5.4 Divide. 1. 0.8 0.48

1. __________________________

2. 5 2.55

2. __________________________

3. 0.07 21

3. __________________________

4. 0.6 24.12

4. __________________________

5. 7 4.97

5. __________________________

6. 6.25 ÷ 3.5 (Round the quotient to the nearest hundredth.)

6. __________________________

7. 12 ÷ 1.02 (Round to the nearest thousandth.)

7. __________________________

8. 54 ÷ 0.09

8. __________________________

9. 7.95 ÷ 0.05

9. __________________________

10. Divide 589.68 by 1.8

10. __________________________

11. Divide 49.6 by 0.3 (Round to the nearest hundredth.)

11. __________________________

12. 36.52 ÷ 100

12. __________________________

13. 8.36 ÷ 1000

13. __________________________

14. 0.73 ÷ 0.001

14. __________________________

15. Determine whether the given value is a solution of the x equation. = 2.46; x = 14.76 6 16. Determine whether the given value is a solution of the x equation. = 1.8; x = 57.6 32

15. __________________________

17. The total cost of a loan is $11,287.64. How many monthly payments of $217.07 would it take to pay off the loan?

17. __________________________

18. A high school basketball player scored 386 points in 16 games. What was her average number of points per game? Round to the nearest tenth.

18. __________________________

19. Evaluate x ÷ y

if x = 2.5 and y = 0.02.

19. __________________________

20. Evaluate 2x ÷ y

if x = 0.025 and y = 2.5.

20. __________________________

E-44

16. __________________________

Name: Instructor:

Date: Section:

Additional Exercises 5.5 Write each fraction as a decimal. 1.

1 8

1. __________________________

3 5

2. __________________________

7 8

3. __________________________

Write each fraction as a decimal. Round to the nearest hundredth. 4.

2 3

4. __________________________

1 7

5. __________________________

3 7

6. __________________________

7. 6

3 20

7. __________________________

8 11

8. __________________________

8. 15

Write each fraction as a decimal rounded to the nearest thousandth. 9. A basketball player made

12 of their shots for the season. 17

10. In a recent election for the U.S. Senate, one candidate got about 65 of the votes. 109

9. __________________________

10. __________________________

E-45

Name: Instructor: Additional Exercises 5.5 (cont.)

Date: Section:

Insert <, >, or = to form a true statement. 11. 0.0438

0.0435

11. __________________________

12.

2 7

4 15

12. __________________________

13.

7 8

0.9167

13. __________________________

Write the numbers in order from smallest to largest. 14. 0.725

0.7252

0.72152

14. __________________________

Simplify each expression. 15. (0.05)2

15. __________________________

16. ( 5.3)(100 ) − (12.2 )(10 )

16. __________________________ 1 17. __________________________

17. 0.4 ( 7.5 − 1.9 ) 18.

2.75 + 1.25 10

18. __________________________

Find the area of each triangle. Use the formula A =

1 bh . 2

19. __________________________

19. 5.3 in.

4.2 in. 20.

20. __________________________ 1.6 m 2.8 m

E-46

Name: Instructor:

Date: Section:

Additional Exercises 5.6 Solve and check. 1. x + 1.6 = 3.4

1. __________________________

2. −3.9 = y + 8.65

2. __________________________

3. 1.1x = 6.6

3. __________________________

x = 3.285 5

4. __________________________

5. 0.50 = 4x

5. __________________________

6. 3(x – 2.1) = 2x – 6.9

6. __________________________

7. 2x + 3x = 3.6

7. __________________________

8. x – 0.5x = −18.5

8. __________________________

9. 3(x + 2) = −12.6

9. __________________________

10. 1.5x – 9.72 = 0.5x + 8.65

10. __________________________

11. 2(x – 1.5) = 8.352

11. __________________________

12. 4x + 7.12 = 2(3x – 5.8)

12. __________________________

13. 0.7x + 18.3 = 2x – 20.7

13. __________________________

14. 45.6 − 300 x = 200 x − 36.9

14. __________________________

15. x – 4.59 = 6.85

15. __________________________

16. 2x + 0.5x = −125

16. __________________________

17. −1.1x = −99

17. __________________________

Solve. 18. Four friends went to lunch. The total bill was $38.64. Find the average cost for each of the four friends.

18. __________________________

19. Five textbooks cost $604.25. What is the average cost of each book?

19. __________________________

20. A local drive-in has “Happy Hour” each afternoon and all soft drinks are half price. If an order of six drinks costs $4.74 during “Happy Hour,” how much does one soft drink cost during regular hours?

20. __________________________

E-47

Name: Instructor:

Date: Section:

Additional Exercises 5.7 True or False 1.

The mean of a set of numbers is the average of the numbers.

1. __________________________

When a set of numbers is in numerical order, the number in the middle is called the median.

2. __________________________

A set of numbers may not have a mode or it may have more than one mode.

3. __________________________

For each set of numbers, find the mean, median, and mode. If necessary, round each mean to one decimal place. 4.

15, 19, 24, 17, 31

4. __________________________

7.5, 8.9, 4.6, 9.5, 8.6, 8.9

5. __________________________

392, 476, 831, 956, 371, 429, 444

6. __________________________

7. 17, 21, 36, 21, 48, 61

7. __________________________

8. 0.3, 0.4, 0.7, 1.2, 1.4, 0.8, 0.7

8. __________________________

9. 98, 97, 84, 79, 88, 88

9. __________________________

10. 6.3, 7.5, 8.2, 8.6, 7.4, 6.3

10. __________________________

11.

11. __________________________

Find the grade point average using 4 points for A, 3 points for B, 2 points for C, 1 point for D, and 0 points for F. Round to two decimal places. Grade B C B A

E-48

Credit Hours 3 3 4 3

Name: Instructor: Additional Exercises 5.7 (cont.)

Date: Section:

Use the following table for problems 12 - 20. Tallest Buildings in Gotham City Building A Building B Building C Building D Building E Building F Building G Building H Building I Building J

Height in Feet 1250 1046 1046 952 927 915 861 850 814 813

Height in Meters 381 319 319 290 283 279 262 259 248 248

Floors 102 77 52 66 70 59 72 69 75 60

12. Find the mean height in feet.

12. __________________________

13. Find the mean height in meters.

13. __________________________

14. Find the mean number of floors.

14. __________________________

15. Find the median for the height in feet.

15. __________________________

16. Find the median for the height in meters.

16. __________________________

17. Find the median for the number of floors.

17. __________________________

18. Find the mode for the height in feet.

18. __________________________

19. Find the mode for the height in meters.

19. __________________________

20. Find the mode for the number of floors.

20. __________________________

E-49

Name: Instructor:

Date: Section:

Additional Exercises 6.1 Write each ratio as a ratio of whole numbers using fractional notation in simplest form. 1. 6 to 17

1. __________________________

2. 8 to 24 1 3 to 3 3. 2 4 8

2. __________________________ 3. __________________________

4. 3.5 to 10

4. __________________________

5. 12 to 18

5. __________________________

6. 2.8 to 5.6

6. __________________________

7. 60 miles to 15 miles

7. __________________________

8. 280 acres to 120 acres

8. __________________________

9. 78 gallons to 86 gallons

9. __________________________

10. $85 to $120

10. __________________________

11. 0.6 meters to 3 meters

11. __________________________

A rectangular storage building is 20 feet long and 15 feet wide. 12. Find the ratio of the width to the length in simplest terms.

12. __________________________

13. Find the ratio of the length to the perimeter of the building.

13. __________________________

14. Find the ratio of the width to the perimeter of the building.

14. __________________________

A book shelf contains 18 world history books and 10 math books. 15. Find the ratio of math books to world history books.

15. __________________________

16. Find the ratio of math books to total books.

16. __________________________

Write each rate as a unit rate. 17. 156 calories in 12 ounces

7. __________________________

18. 434 miles in 7 hours

8. __________________________

Find each unit price and decide which is a better buy. 19. Treated lumber: $3.79 for an 8-foot board or $6.18 for a 12foot board.

19. __________________________

20. 50 aspirin tablets for $3.79 or 100 aspirin tablets for $5.85.

20. __________________________

E-50

Name: Instructor:

Date: Section:

Additional Exercises 6.2 Write each sentence as a proportion. 1. 3 eggs is to 6 cups of flour as 12 eggs is to 24 cups of flour.

1. __________________________

2. 0.5 meters is to 5 kilometers as 3 meters is to 30 kilometers. 1 page is to 20 minutes as 1 page is to 60 minutes. 3. 3

2. __________________________ 3. __________________________

Determine whether each proportion is true or false. 4.

15 5 = 9 3

7 42 = 8 56

5. __________________________

3 12 = 4 16

6. __________________________

7 1 = 32 4

7. __________________________

4 28 = 5 35

8. __________________________

1 2 = 1.5 3 9

4. __________________________

9. __________________________

E-51

Name: Instructor: Additional Exercises 6.2 (cont.)

Date: Section:

Solve each proportion for the given variable. 10.

18 x = 30 15 5 7.5 = x 9

11.

12.

x 15 = 30 25

13.

16 6 = 8 x

14.

1 2 = 20 1 x 5

10. __________________________

11. __________________________

12. __________________________

13. __________________________

14. __________________________

= 8 1 x 6

15. __________________________

16.

x 15 = 20 25

16. __________________________

17.

1.5 x = 2.5 15

17. __________________________

18.

x 6 = 16 4.8

18. __________________________

19.

x 1.2 = 9 1.8

15.

20.

E-52

1 5 = x 8 15

19. __________________________

20. __________________________

Name: Instructor:

Date: Section:

Additional Exercises 6.3 Solve. Amitha’s car averages 588 miles on a 21 gallon tank of gas. 1. How far can she drive on 6 gallons of gas?

1. __________________________

2. How many gallons of gas would she use on a 975 mile trip? Round to the nearest tenth.

2. __________________________

An 80-pound bag of ready to use concrete mix fills 8 cubic feet. 3. Juan needs 280 cubic feet of concrete, how many bags does he need? 4. If Sharma buys 10 bags, how many cubic feet will that fill?

3. __________________________

4. __________________________

A animal shelter allows 150 square feet of yards space per dog. 5. Find the minimum yard space for 5 dogs.

5. __________________________

6. They plan to fence a rectangular area that is 30 by 10. Find the maximum number of dogs the new yard can accommodate.

6. __________________________

On a road map, 1 inch corresponds to 30 miles. 7. Find the distance represented by a line segment 2 14 inches long.

8. If two cities are 150 miles apart, find the measurement on the map.

7. __________________________

8. __________________________

Local sales tax is $6.25 for every $100 purchase. 9. If sales tax on a sofa is $18.75, what was the purchase price of the sofa?

10. Find the sales tax on a refrigerator priced at $550. (Round to the nearest cent.)

9. __________________________

10. _________________________

E-53

Name: Instructor: Additional Exercises 6.3 (cont.)

Date: Section:

11. A survey revealed that 4 out of 5 people prefer vanilla ice cream to chocolate. In a class of 35, how many students are likely to prefer chocolate ice cream?

12. Another survey indicated that 3 out of 5 people prefer water over iced tea. If 12 people in a room prefer water, how many people would most likely choose iced tea?

11. __________________________

12. __________________________

13. If a family drinks 2 gallons of milk every 3 days, how many gallons of milk do they drink in a month (30 days)?

13. __________________________

12. If Calypso, the family dog, eats 2 cups of dried dog food each day, how many cups are eaten each week?

14. __________________________

15. If Hilda can type and spell check 5 pages in 30 minutes, how long will it take her to finish her research paper which is 25 pages long?

15. __________________________

16. If a recipe calls for 2 12 cups of sugar for 2 dozen cookies, how

16. __________________________

much sugar is needed to make 6 dozen cookies? 1 inch represents 60 miles on a road map, what is the 2 distance represented by 3 12 inches?

17. If

17. __________________________

18. A mix uses two eggs to make 12 pancakes. How many eggs are needed to make 36 pancakes?

18. __________________________

19. You burn about 200 calories while jogging for 45 minutes. How long would you have to jog to burn 600 calories?

19. __________________________

20. A student reads 8 pages in 15 minutes. How many pages can they read in an hour?

20. __________________________

E-54

Name: Instructor:

Date: Section:

Additional Exercises 6.4 Find each square root. 1. __________________________

100

49 64

5. __________________________

1 16

6. __________________________

169

144

225

10.

100 121

2. __________________________

3. __________________________

4. __________________________

7. __________________________

8. __________________________

9. __________________________

10. __________________________

E-55

Name: Instructor: Additional Exercises 6.4 (cont.)

11.

25 49

12.

Date: Section:

11. __________________________

12. __________________________

Use Appendix A.4 or a calculator to approximate each square root to the nearest thousandth. 13.

13. __________________________

14.

14. __________________________

15.

15. __________________________

16.

178

16. __________________________

Find the unknown length in each right triangle. Approximate to the nearest thousandth. 17. 5 ft.

17. __________________________

7 ft. 18.

18. __________________________ 16 cm 8 cm

19. Find the height of a building if a 10 foot ladder is placed 4 feet from a wall.

19. __________________________

10 4 20. Find the length of a guy-wire attached to a 12 foot pole if it is tied 5 feet from the pole. 20. __________________________

12 5

E-56

Name: Instructor:

Date: Section:

Additional Exercises 6.5 Find each ratio of the corresponding sides of the similar triangles. 8

1. __________________________ 6

2. __________________________

3 3 9

3. __________________________ 12

5 20 4. __________________________

4. 0.5 2.5 1.5 7.5 5.

5. __________________________ x

x 5x

3 15

6. __________________________

4 4 8

E-57

Name: Instructor: Additional Exercises 6.5 (cont.)

Date: Section:

Given that the pairs of triangles are similar, find the unknown length of the side labeled with a variable. 7.

6 2

7. __________________________

8. __________________________ 5

15 7

x 9. __________________________

9. 1

1 5

10.

10. __________________________

11.

11. __________________________ 7 14

12.

13.

12. __________________________ 2.3

2.3

8.7

13. __________________________ x

22.5

14.

14. __________________________ 3

x 7.5

E-58

12.5

Name: Instructor: Additional Exercises 6.5 (cont.)

Date: Section:

15.

15. __________________________ 2.7

3.8 x

7.2

16. If a 25 foot tree casts a 10 foot shadow, find the length of the shadow cast by a 40 foot tree.

16. __________________________

17. A flagpole 24 feet tall casts a 36 foot shadow. Find the length of the shadow cast by a 32 foot pole

17. __________________________

18. If a 12 foot tree casts a 7 foot shadow, find the length of the shadow cast by an 30 foot tree.

18. __________________________

19. Find the height of a building if the shadow cast by the building is 72 feet. At the same time, a 5 foot person casts an 8 foot shadow.

19. __________________________

20. Given that triangle ABC is similar to triangle DEF, solve for x and y. D A y 3.2 8.5 x

20. __________________________

C 6

F 9

E-59

Name: Instructor:

Date: Section:

Additional Exercises 7.1 1. At an election, 72 out of every 100 eligible voters turned out to vote. What percent of eligible voters turned out?

1. __________________________

2. At Jacksboro Community College, 61 out of every 100 students must take a developmental math or reading class. What percent of the students take a developmental class?

2. __________________________

Write each percent as a decimal or each decimal as a percent. 3. 3.4%

3. __________________________

4. 0.6%

4. __________________________

5. 179%

5. __________________________

6. 45%

6. __________________________

7. 0.49

7. __________________________

8. 0.003

8. __________________________

9. 0.724

9. __________________________

10. 4.6

10. __________________________

Write each percent as a fraction or mixed number in simplest form. 11. 24%

11. __________________________

12. 1.4%

12. __________________________

13. 120%

13. __________________________

14. 0.25%

14. __________________________

Write each fraction or mixed number as a percent. 15.

13 25

15. __________________________

16.

1 5

16. __________________________

17. 3 14

17. __________________________

Write each percent as a decimal and a fraction. 18. 3.8%

18. __________________________

19. 58.4%

19. __________________________

20. 13 out of every 200 dogs in a community do not have a license. What percent of the dogs don’t have a license?

20. __________________________

E-60

Name: Instructor:

Date: Section:

Additional Exercises 7.2 Translate each into an equation. Do not solve. 1. 12% of 42 is what number?

1. __________________________

2. What number is 30% of 50?

2. __________________________

3. 60% of what number is 18?

3. __________________________

4. What percent of 120 is 30?

4. __________________________

Translate each to an equation and solve. 5. What number is 45% of 360?

5. __________________________

6. 30% of 170 is what number?

6. __________________________

7. 72 is 80% of what number?

7. __________________________

8. 10 is 20% of what number?

8. __________________________

9. 50 is what percent of 250?

9. __________________________

10. 60 is what percent of 240?

10. __________________________

11. 4 is 64% of what number?

11. __________________________

12. 1.5 is 25% of what number?

12. __________________________

13. 20 is 40% of what number?

13. __________________________

14. 875 is 8 34 % of what number?

14. __________________________

15. 3.2 is what percent of 128?

15. __________________________

16. What percent of 45 is 90?

16. __________________________

17. What percent of 20 is 12.5?

17. __________________________

18. What number is 32% of 52?

18. __________________________

19. 150% of what number is 60?

19. __________________________

20. What number is 20% of 86?

20. __________________________

E-61

Name: Instructor:

Date: Section:

Additional Exercises 7.3 Translate each to a proportion. Do not solve. 1. 62% of 80 is what number?

1. __________________________

2. What number is 8% of 75?

2. __________________________

3. 25% of what number is 40?

3. __________________________

4. 7.2 is 15% of what number?

4. __________________________

5. 2.5 is what percent of 25?

5. __________________________

6. 15 is what percent of 50?

6. __________________________

Translate each to a proportion and solve. 7. 40% of 80 is what number?

7. __________________________

8. What number is 95% of 800?

8. __________________________

9. 20 is 25% of what number?

9. __________________________

10. 60% of what number is 81?

10. __________________________

11. 18 is what percent of 24?

11. __________________________

12. 20 is what percent of 50?

12. __________________________

13. 15% of 10 is what number?

13. __________________________

14. What is 33% of 150?

14. __________________________

15. 105 is what percent of 2100?

15. __________________________

16. 125 is 50% of what number?

16. __________________________

17. 110% of 20 is what number?

17. __________________________

18. 8.6 is what percent of 86?

18. __________________________

19. What percent of 1.2 is 1.8?

19. __________________________

20. 110% of what number is 33?

20. __________________________

E-62

Name: Instructor:

Date: Section:

Additional Exercises 7.4 Solve. Round percents to the nearest tenth if necessary. 1. An inspector found 40 defective batteries during an inspection. If this is 2.5% of the total number of batteries inspected, how many batteries were inspected?

1. __________________________

2. An engineer paid 5% of the purchase price of a $145,000 home as a down payment. How much was the down payment?

2. __________________________

3. The Brown family total income is $4200 per month. Last month they spent $210 dining out. What percent of their monthly income was spent on dining out?

3. __________________________

4. On average, 0.5% of the cookies baked by a bakery are broken. If 210 cookies were broken during one week, how many were baked?

4. __________________________

5. A teacher’s salary last year was $40,000. Due to a recession, salaries were cut by 8% this year. What is the teacher’s new salary this year?

5. __________________________

6. Last year, a music teacher had 12 piano students. This year the teacher has 16 piano students. Find the percent increase.

6. __________________________

7. The average number of customers per day at Bygone Toy Store decreased from 140 to 110. Find the percent decrease.

7. __________________________

8. The Rocky Valley Motel charges $120.00 a night during the summer, but $90.00 a night during the fall. What is the percent decrease?

8. __________________________

9. The price of a gallon of milk increased from $3.05 to $3.68. Find the percent increase.

9. __________________________

10. With the addition of a new wing, a rural hospital with 150 beds now has 225 beds. Find the percent increase.

10. __________________________

E-63

Name: Instructor: Additional Exercises 7.4 (cont.)

Date: Section:

11. An employee of the college noticed an increase in their health insurance premium from $368 per month the $413.22 per month. What was the percent increase?

11. __________________________

12. The wholesale cost of a lamp is $15. If a store has a standard mark-up of 40%, what will be the retail purchase price of the lamp?

12. __________________________

13. The size of a family’s cattle herd was 450 before selling 240 heifers. What was the size of the herd after the sale and what was the percent decrease?

13. __________________________

14. A salad dressing has 150 mg of sodium per serving. This is 15% of the daily recommended amount of sodium. What is the daily recommended amount of sodium?

14. __________________________

15. A cereal bar has 150 total calories. If there are 30 fat calories in the cereal bar, what percent of the calories are from fat?

15. __________________________

16. Find the percent increase if 400 is increased to 500.

16. __________________________

17. Find the amount of increase and new amount if 45 is increased by 20%

17. __________________________

18. Find the percent decrease if 150 is decreased to 120.

18. __________________________

19. Find the new amount if $84.50 is decreased by 12%.

19. __________________________

20. A local computer company produces 120 computers per month. If production is increased by 5%, how many are produced monthly now?

20. __________________________

E-64

Name: Instructor:

Date: Section:

Additional Exercises 7.5 Solve. 1. If the sales tax rate is 4%, find the sales tax on a $280 fax machine.

1. __________________________

2. What is the sales tax on a $450 television if the sales tax rate is 4%?

2. __________________________

3. A refrigerator sells for $625. With a sales tax rate of 6.5%, find the total price.

3. __________________________

4. A subcompact care sells for $20,500. If the sales tax rate is 5%, find the total price.

4. __________________________

5. A doctor bought shoes for $80 and a shirt for $215. Find the total price paid if the sales tax rate was 5.5%.

5. __________________________

6. The sales tax is $32.50 on a desk priced at $650. Find the sales tax rate.

6. __________________________

7. The sales tax is $0.72 on a $12 purchase. Find the sales tax rate.

7. __________________________

8. Find the amount of sales tax on a $50.49 item if the sales tax rate is 6.25%.

8. __________________________

9. A real estate agent sells a home for $159,900. If the commission on the sale is 3%, how much does agent earn?

9. __________________________

10. A salesperson receives a 3.5% commission on all sales. If the salesperson sold $1200 worth of merchandise last week, what was the commission?

10. __________________________

11. A salesperson earned a commission of $250 for selling $56,000 worth of hardware. What is the commission rate?

11. __________________________

12. How much commission will a real estate agent receive from the sale of an $89,000 home if the commission is 2% of the sale price?

12. __________________________

13. An insurance agent earned $10,000 for selling $2,500,000 worth of insurance. What is the commission rate?

13. __________________________

E-65

Name: Instructor: Additional Exercises 7.5 (cont.)

Date: Section:

Fill in the tables below with the missing amounts. 14.

Original price $65

Discount Rate 15%

15.

$300

20%

16.

$5000

25%

Original Price

Sales Tax Rate

17.

$45

18.

$120

8.25%

19.

$4000

7.5%

20.

E-66

A couple wants to buy a new television and finds one on sale. The original price is $1800 and is now sale-priced at 30% off. What will the purchase price be for the television? If they must also pay sales tax of 6.25%, what will the final total price be?

Sale Price

Total Price

20. __________________________

Name: Instructor:

Date: Section:

Additional Exercises 7.6 Find the simple interest. Principal

Rate

Time

$300

2 years

1. __________________________

$1500

3 years

2. __________________________

$3000

12%

2 years

3. __________________________

$1500

7 years

4. __________________________

$1200

6.5%

6 months

5. __________________________

$800

12%

8 months

6. __________________________

Solve. 7. __________________________ 7. A student borrows $2000 and agrees to pay it back in 2 years. If the simple interest rate is 12%, find the total amount owed after 2 years. 8. An entrepreneur takes out a 9-month short-term loan of $1500 to buy a new computer. If the interest rate is 12%, find the total amount due at the end of the 9 months.

8. __________________________

9. A certificate of deposit pays simple interest at a 2.5% interest rate. Find the value of a $2000 5-year CD.

9. __________________________

10. Find the total amount due if $500 is borrowed at 8% simple interest for 6 months.

10. __________________________

Find the total amount in each compound interest account. 11. $8000 is compounded annually at a rate of 15% for 3 years.

11. __________________________

E-67

Name: Instructor: Additional Exercises 7.6 (cont.)

Date: Section:

12. $10,000 is compounded monthly at a rate of 5% for 1 year.

12. __________________________

13. $500 is compounded quarterly at 10% for 2 years.

13. __________________________

14. $3000 is compounded semiannually at a rate of 10% for 1 year. 14. __________________________

15. $4000 is compounded quarterly at 8% for 2 years.

15. __________________________

Solve. 16. $4200 is borrowed for 3 years. Find the total amount due if the interest rate is 6% and the amount is compounded annually.

16. __________________________

17. $25,000 is borrowed for 20 years. If the interest rate is 8.5%, and the amount is compounded annually, find the total amount due.

17. __________________________

18. $2000 is borrowed for 9 years. Find the total amount due in 9 years if the interest rate is 12% and it is compounded annually.

18. __________________________

Find the compound interest earned. Principal

Rate

Frequency

Time

19.

$12,000

10%

Monthly

1 year

19. __________________________

20.

$1200

Quarterly

7 years

20. __________________________

E-68

Name: Instructor:

Date: Section:

Additional Exercises 8.1 Use the following bar graph for Exercises 1−5.

Deaths (in thousands)

Heart Disease Deaths 800

770

725

740

1985

1990

1995

600 400 200 0

Source: National Center for Health Statistics 1. Find the difference in heart related deaths from 1985 to 1995. 1. __________________________ 2. Does this difference show a decrease or an increase in the number of deaths?

2. __________________________

3. Find the ratio of deaths in 1985 to 1995.

3. __________________________

4. Find the average number of heart disease related deaths for 1985, 1990, and 1995.

4. __________________________

5. What can be said about heart disease related deaths for this 10 year period?

5. __________________________

Use the following bar graph for Exercises 6−8.

Deaths (in thousands)

HIV Infections 50

40 30

20 10 0 1990

1995

Source: National Center for Health Statistics 6. Find the difference between the number of HIV deaths in 1990 and 1995.

6. __________________________

7. Does this amount represent an increase or a decrease in deaths?

7. __________________________

8. Find the ratio of HIV deaths in 1990 to the number in 1995.

8. __________________________

E-69

Name: Instructor: Additional Exercises 8.1 (cont.)

Date: Section:

Use the following line graph for Exercises 9−12.

% of population

Smokers ages 18 and over

30 25 20 15 10 5 0

9. Estimate the percentage of 18 year old and older smokers for 9. __________________________ Arizona. 10. Approximate the percentage of smokers for California.

10. __________________________

11. Approximate the percentage of smokers for Montana.

11. __________________________

12. What can be said from the line graph about Montana, New Mexico, and New York?

12. __________________________

Use the following bar graph for Exercises 13−17.

FLORIDA UNEMPLOYMENT STATISTICS

Source: Bureau of Labor Statistics 13. Find the percent of unemployment for 2005.

13. __________________________

14. Find the percent of unemployment for 2008.

14. __________________________

E-70

Name: Instructor: Additional Exercises 8.1 (cont.)

Date: Section:

15. What year reflects the highest unemployment rate?

15. __________________________

16. What year reflects the lowest unemployment rate?

16. __________________________

17. Which years had the same percent of unemployment?

17. __________________________

Use the following pictograph for Exercises 18 − 20. Each image represents 10,000 deaths.

DEATHS DUE TO FLU AND PNEUMONIA IN USA

Source: National Center for Health Statistics 18. What year reported the lowest number of deaths due to flu and pneumonia?

18. __________________________

19. What can be said about flu deaths over the years?

19. __________________________

20. Estimate the number of deaths from flu for 2004.

20. __________________________

E-71

Name: Instructor:

Date: Section:

Additional Exercises 8.2 Use the following circle graph for Exercises 1−3. Teens (13 – 19) spending 10 or more hours per week.

Radio 25%

T.V. 36%

Magazine and Newspapers 3%

Source: Rand Youth Poll Knowing how teens spend their time helps advertisers make decisions about strategies in marketing.

Less than 10 hrs 36%

1. ___ 1. Find the ratio of time spent watching T.V. to time spent on magazines and newspapers.

1. __________________________

2. Find the ratio of time spent watching T.V. (more than 10 hours) to time less than 10 hours per week.

2. __________________________

3. Find the ratio of hours spent listening to the radio to time spent watching T.V.

3. __________________________

Use the following information for Exercises 4−7. A local high school has 900 students. Seniors 25% Juniors 20% Sophomores 20% Freshmen 35% 4. Use the information to draw a circle graph.

5. From your graph, find the ratio of sophomores to seniors.

5. __________________________

6. Find the ratio of seniors to total students.

6. __________________________

7. Find the ratio of juniors to freshmen.

7. __________________________

E-72

Name: Instructor: Additional Exercises 8.2 (cont.)

Date: Section:

Use the following information for Exercises 8−11. A survey of 50 cookie eaters revealed that 34 people preferred Chocolate Chip 10 people preferred Oatmeal Raisin 2 people preferred Fig cookies 2 people preferred Coconut cookies 2 people preferred Date Nut cookies 8. Use the information to make a circle graph.

9. Find the ratio of those who preferred Fig to Oatmeal Raisin cookies.

9. __________________________

10. Find the ratio of those who preferred Coconut cookies to Chocolate Chip cookies.

10. __________________________

11. Find the ratio of those who preferred Date Nut cookies to the total number of people in the survey.

11. __________________________

12. Those who preferred Chocolate Chip Cookies represented almost half of the people surveyed. Is this statement true or false?

12. __________________________

E-73

Name: Instructor: Additional Exercises 8.2 (cont.)

Date: Section:

The circle graph represents 100 students enrolled in a junior college algebra class. transfer 9% A's 24%

D's 9%

F's 5%

B's 12% drop (withdrew) 5%

C's 36%

13. Estimate the number of students receiving B’s.

13. __________________________

14. Estimate the number of students receiving a C.

14. __________________________

15. Estimate the number of students who transferred.

15. __________________________

16. Find the ratio of students receiving A’s to students receiving F’s.

16. __________________________

17. Find the ratio of students receiving A’s, B’s, and C’s to the total number of students.

17. __________________________

Rock 1/3

Country 1/4 Jazz 1/8

Classical 1/16

None

Other

Oldies 1/8

Of 100 people surveyed about their musical interests, (round to the nearest whole number): 18. Determine the number of people who like Country Music.

18. __________________________

19. Estimate the number of people who like Rock.

19. __________________________

20. Estimate the number of people who like Jazz.

20. __________________________

E-74

Name: Instructor:

Date: Section:

Additional Exercises 8.3 Plot the points corresponding to the ordered pairs on the same set of axes. 1. (3, 4), (0, −1), (−3, 2), (−4, 0)

1. __________________________

2. (3, 2), (2, −1), (−3, −2), (0, 2)

2. __________________________

Find the x- and y -coordinates of each labeled point. 3.

3. __________________________ y

A D

B x

E-75

Name: Instructor:

Date: Section:

Additional Exercises 8.3 (cont.) 4.

4. __________________________ y

B D

Determine whether each ordered pair is a solution of the given equation. 5. (1, 4)

y = 2x + 3

5. __________________________

6. (0, –2)

x − 3y = 6

6. __________________________

7. (0 , –4)

x − 2 y = −8

7. __________________________

8. (4, 2)

1 x +1 4

8. __________________________

9. (4, 1)

y=x−4

9. __________________________

Plot the three ordered pair solutions of the given equations. 10. 2x + 3y = 6

E-76

(0, 2), (3, 0), (−3, 4)

11. y = 3x + 2

(0, 2), (−2, −4), (−1, −1)

Name: Instructor: Additional Exercises 8.3 (cont.)

Date: Section:

Complete each ordered pair solution of the given equations. y=

1 x+2 2

(2, ), (0, ), (−2, )

12. __________________________

13. x = −3y + 2

(–1, ), (2,

), (

, –1)

13. __________________________

14. y = x – 3

(–1, ), (0, ), (

, –1)

14. __________________________

15. 2x – y = 3

(2, ), (1, ), (0, )

15. __________________________

16. y = 3x – 1

(0, ), (1, ), (2,

)

19. __________________________

17. 2x + y = −4

(0, ), ( , 0), (2,

)

20. __________________________

12.

Determine whether the ordered pair is a solution of the given equation. 18. (3, −5)

1 x−4 3

16. __________________________

19. (1, 2)

3x – 5y = −7

17. __________________________

20. (3, 2)

y=−

2 x 3

18. __________________________

E-77

Name: Instructor:

Date: Section:

Additional Exercises 8.4 Complete each table of values, plot the ordered pair solutions, and graph the equation given. 1. y = x

x -1 0 1

2. y = 4

x -1 0 1

3. x + y = 0

x -2 0 2

E-78

4. 3x – y = 6 x

0 2 4

Name: Instructor: Additional Exercises 8.4 (cont.)

Date: Section:

Graph each equation. 5.

1 x−3 3

y = 3x + 2

7. x = −2

8. 2x + 3y = 6

9. y = −x + 3

10. y = 1.5x

E-79

Name: Instructor: Additional Exercises 8.4 (cont.) 11.

y=−

2 x−2 3

Date: Section:

12. x + 2y = 2

13. y − x = 1

14. y = −2

15. y = x + 1

16. 2x − y = 1

E-80

Name: Instructor:

Date: Section:

Additional Exercises 8.4 (cont.) Determine whether the following equations represent vertical lines, horizontal lines, or neither. 17.

x = –2

17. __________________________

18. x + y = 4

18. __________________________

19. y = 3

19. __________________________

20. y =

1 x 2

20. __________________________

E-81

Name: Instructor:

Date: Section:

Additional Exercises 8.5 An instructor’s bag contains 2 red pens, 3 black pens and 2 blue pens. One pen is taken out at random. 1. What is the probability of a red pen being picked?

1. __________________________

2. If one black pen is taken out, what is the probability that the next pen taken out is black?

2. __________________________

3. What is the probability of a blue pen being picked?

3. __________________________ 4. __________________________

4. What is the probability of a black pen being picked? A bag holds 5 red marbles, 2 blue marbles, 3 green marbles, 2 yellow marbles, 2 orange marbles and 3 “clear” marbles. 5. __________________________ 5. Find the probability that an “orange” marble will be selected. 6. Find the probability that a yellow marble will be selected.

6. __________________________

7. Find the probability that a clear marble will be selected.

7. __________________________

If 20 plastic disks, each labeled 1 – 20 are placed in a bag: 8. Find the probability that an even number will be selected.

8. __________________________

9. Find the probability that an odd number will be selected.

9. __________________________

10. Find the probability that a number less 5 is selected.

10. __________________________

Using a standard deck of playing cards: 11. What is the probability that a heart card will be selected?

11. __________________________

12. What is the probability of the “King of Hearts” being selected?

12. __________________________

E-82

Name: Instructor: Additional Exercises 8.5 (cont.)

Date: Section:

Toss a six-sided die: 13. If a die is rolled, what is the probability that the outcome will be a 3?

13. __________________________

14. If a die is tossed, what is the probability that the outcome will be an even number?

14. __________________________

Draw a tree diagram, and find the number of possible outcomes. 15. Flip a coin twice.

15. __________________________

16. Choose a color: red, blue, green and then a number: 1, 2, or 3.

16. __________________________

17. Explain the meaning of a probability of “0” (zero).

17. __________________________

18. Explain the meaning of a probability of “1”.

18. __________________________

19. Is a probability of

4 possible? Explain. 3

20. What is meant by a probability of

1 ? Explain. 2

19. __________________________

20. __________________________

E-83

Name: Instructor:

Date: Section:

Additional Exercises 9.1 Identify each figure as a line, a ray, a line segment, or an angle. 1.

1. __________________________

2. C

2. __________________________ D

E Y

3. __________________________

X 15° Find the measure of each angle in the figure.

X 105° W

Y 30° Z

4. ∠XWY

4. __________________________

5. ∠XWZ

5. __________________________

6. ∠VWY

6. __________________________

Classify each angle as acute, right, obtuse, or straight. 7.

7. __________________________

8. __________________________

9. __________________________

10. Find the supplement of a 24° angle.

10. __________________________

E-84

Name: Instructor: Additional Exercises 9.1 (cont’d)

Date: Section:

11. Find the complement of a 85° angle.

11. __________________________

12. Find the supplement of a 125° angle.

12. __________________________

Find the measure of ∠x in each figure. 13.

13. __________________________ x

105° 80°

14.

14. __________________________ 3° 57° x

15. Identify the pairs of complementary angles. D E 80° 10° 45° F 45° A B C

15. __________________________

16. Identify the pairs of supplementary angles. P O ● ● 125° ● 55° 55° A ● 125° ● E H

16. __________________________

Find the measures of angle x, y, and z in each picture 17. 122° x z y 18.

17. __________________________

18. __________________________ x 24°

Find the measure of ∠x 19.

19. __________________________ 37°

112° x 20. Find the measures of angle a, b, c, d, e, and f. a 80° 50° b c e f 50° d

20. __________________________

E-85

Name: Instructor:

Date: Section:

Additional Exercises 9.2 Find the perimeter of each figure. 1. 4 ft.

1. __________________________

9 ft. 2.

9 ft.

12 ft.

2. __________________________

18 ft. 3. __________________________

3. 4.8 m 4.8 m 4.

4. __________________________ 2 yd. 3.5 yd.

5. Find the perimeter of a rectangle that is 22 inches long and 14 inches wide.

5. __________________________

6. Find the amount of fencing that is needed to enclose a rectangular field 240 feet by 360 feet.

6. __________________________

Find the perimeter of each figure. 7.

4 cm

7. __________________________

1 cm 6 cm 10 cm 4m

8. __________________________ 5m 10 m 15 m

E-86

Name: Instructor: Additional Exercises 9.2 (cont’d) 9.

Date: Section: 9. __________________________

12 mm 12 mm

14 mm

1 mm 8 in. 10. __________________________

10. 5 in. 10 in.

2 in.

3 in. 3 in.

6 in 11. A regular octagon has sides of length 3 inches. Find its perimeter.

11. __________________________

12. A regular pentagon has sides of length 2.6 centimeters. Find its perimeter.

12. __________________________

13. An interior decorator is planning on putting crown moulding around the dining room of a new house. The dining room has a regular hexagonal shape, and the length of each wall is 7.5 feet. How many feet of crown moulding will be needed?

13. __________________________

14. A landscape designer is planning a flower garden in the shape of an octagon 2.5 feet on a side. The border to the garden costs $3.50 a foot. Find the total cost of the border.

14. __________________________

Find the circumference of each circle. Give the exact circumference and then an approximation. Use π ≈ 3.14. 15.

12 ft.

15. __________________________

16. Find the exact circumference and then an approximate circumference of a circle with a radius of 3.5 meters.

16. __________________________

17. Find the circumference of a circle with a radius of 7 cm.

17. __________________________

18. Find the circumference of a circle with a diameter of 16 feet.

18. __________________________

19. A circular track has a radius of 264 feet. About how many laps around the track does it take to make a mile? Round to a whole number.

19. __________________________

20. The circumference of a round trampoline is approximately 37.68 feet. Find the diameter of the trampoline.

20. __________________________

E-87

Name: Instructor:

Date: Section:

Additional Exercises 9.3 Find the area of the geometric figure. If the figure is a circle, give an exact area and then use π ≈ 3.14 to approximate the area. 1.

1. __________________________ 2.2 m 5m

3. __________________________ 7 ft.

9 ft. 11.4 ft.

4. __________________________ 16 miles 7. __________________________

4. 5m 3.5 m 5. Find the area of a circle that has a diameter of 8 meters.

5. __________________________

6. Find the area of a triangle with a base of 4.5 yards and a height of 9 yards.

6. __________________________

7. Find the area of a parallelogram with a base of 4 feet and a height of 6.3 feet.

7. __________________________

E-88

Name: Instructor: Additional Exercises 9.3 (cont’d) 8.

Date: Section:

9 ft.

8. __________________________

6 ft. 12 ft. 9.

9. __________________________

12 in. 3 in. 7 in.

10 in. 5 in. 4 in. 10 in. 4m 10. __________________________

10.

11. A page on a book measures 8.5 in. by 11 in. Find its area.

11. __________________________

12. A round tablecloth has a 48-inch diameter. Approximate its area. Use the approximation π ≈ 3.14.

12. __________________________

Find the volume of each solid. Use 13.

22 for π. Approximate to the nearest hundredth when necessary. 7

6 mm 12 mm

13. __________________________

50 mm

E-89

Name: Instructor: Additional Exercises 9.3 (cont’d)

Date: Section:

14. 14. __________________________ 30 m

15 m 15.

8 cm 15. __________________________ 12 cm

16. Find the volume of a cone with a radius of 4 meters and a height of 6 meters.

16. __________________________

17. Find the volume of a sphere with a radius of 6 centimeters.

17. __________________________

18. Find the volume of a cube with edges 2.8 feet long.

18. __________________________

19. Approximate to the nearest hundredth, the volume of a cylinder 5 centimeters tall if the base has a diameter of 8 22 centimeters. Use for π. 7

19. __________________________

20. Find the capacity (volume in cubic centimeters) of a rectangular ice chest with inside measurements of 46 centimeters by 40 centimeters by 20 centimeters.

20. __________________________

E-90

Name: Instructor:

Date: Section:

Additional Exercises 9.4 Convert each measurement as indicated. 1 1. 2 miles to feet 3

1. __________________________

2. 7920 feet to miles

2. __________________________

3. 15 feet to yards

3. __________________________

4. 132 in. = ___________ yd. ___________ ft.

4. __________________________

5. 39 feet = ___________ yd.

5. __________________________

6. 8 feet 4 in. = ___________ in.

6. __________________________

Perform each indicated operation. Simplify the result if possible. 7. 3 yd. 2 ft. + 3 yd. 2 ft. + 4 yd. 9 ft.

7. __________________________

8. 7 ft. 2 in. – 3 ft. 9 in.

8. __________________________

9. 6 yd. 8 ft. + 2 yd. 10 ft.

9. __________________________

10. Jenna’s mom is making hair bows for the basketball team. It takes 39 inches of ribbon to make each bow. How many yards of ribbon will she need to make 9 bows?

10. __________________________

11. An airplane is cruising at an altitude of 39,600 feet. How many miles is this?

11. __________________________

Convert each measurement as indicated. 12. 100 m to kilometers

12. __________________________

13. 34.2 mm to decimeters

13. __________________________

14. 0.083 m to millimeters

14. __________________________

15. A 68 cm flag pole is mounted on a 17.5 cm pedestal. Find the height of the top of the flagpole from the ground.

15. __________________________

Perform each indicated operation. Remember to insert units in your answers. 16. 70 cm + 4.4 m

16. __________________________

17. 7 km – 5830 m

17. __________________________

18. 4 ⋅ 17.2 m

18. __________________________

19. Bettye has 15.9 meters of material. She can make 3 bridesmaids’ dresses with the material she has. How much material will be used for each dress?

19. __________________________

20. Clara needs 50 centimeters of ribbon to make a bow. How many bows can she make out of 8 meters of ribbon?

20. __________________________

E-91

Name: Instructor:

Date: Section:

Additional Exercises 9.5 Convert as indicated. 1. 48 oz. to pounds

1. __________________________

2. 3.5 tons to pounds

2. __________________________

3. 3 34 pounds to ounces

3. __________________________

4. 1400 pounds to tons

4. __________________________

5. 65 oz. =

lb.

oz.

5. __________________________

Perform each indicated operation. 6. 18 lb. 10 oz. + 15 lb. 10 oz.

6. __________________________

7. 8 tons 750 lb. – 3 tons 850 lb.

7. __________________________

8. 5 tons 400 lb. ÷ 5

8. __________________________

9. A bottle of cranberry juice contains 12.5 ounces of juice. A recipe for punch requires 52 ounces of cranberry juice. Will a 4-pack of juice be enough to make the punch?

9. __________________________

10. A coal mine produced 1 ton 1500 pounds of coal each of 2 days. How much coal did the mine produce?

10. __________________________

Convert as indicated. 11. 480 g to kilograms

11. __________________________

12. 5.7 g to milligrams

12. __________________________

13. 170 mg to grams

13. __________________________

14. 6.5 g to kilograms

14. __________________________

15. 9.1 kg to grams

15. __________________________

Perform each indicated operation. 16. 5.1 g + 170 mg.

16. __________________________

17. 8 kg – 200 g

17. __________________________

18. A can of spaghetti sauce weighs 751 grams. How many kilograms would a case of 6 cans of sauce weigh?

18. __________________________

19. 1.25 mg ÷ 5

19. __________________________

20. A box of oatmeal weighs 517 grams and has 8 servings. How much does each serving weigh to the nearest tenth of a gram?

20. __________________________

E-92

Name: Instructor:

Date: Section:

Additional Exercises 9.6 Convert as indicated. 1. 6 cups to pints

1. __________________________

2. 4 quarts to pints

2. __________________________

3. 12 pt to gallons 1 4. 3 gal to quarts 4

3. __________________________

5. 82 pt =

4. __________________________ gal

5. __________________________

Perform each indicated operation. 6. 1 gal 2 qt + 2 gal 3 qt

6. __________________________

7. 1 gal 2 qt – 3 qt

7. __________________________

8. 4 ⋅ 1c 3 oz

8. __________________________

9. 6 gal 2qt ÷ 2

9. __________________________

10. An oil drum holds 24 gallons of oil. If it takes 4 quarts to service a car, how many cars can be serviced with 1 oil drum?

10. __________________________

Convert as indicated. 11. 6.2 L to milliliters

11. __________________________

12. 0.0073 Kiloliters to liters

12. __________________________

13. 1800 ml to liters

13. __________________________

14. 0.5 kl to liters

14. __________________________

15. 25 liters to kl

15. __________________________

Perform each indicated operation. Remember to insert units in your answers. 16. 2000 ml + 1.6 l

16. __________________________

17. 8.1 kl – 2900 l

17. __________________________

18. 2 ⋅ 420 ml

18. __________________________

19. 6.8 l ÷ 2

19. __________________________

20. Carol Shumacher drank 180 ml of cola from a 2-liter bottle. How much cola remains in the bottle.

20. __________________________

E-93

Name: Instructor:

Date: Section:

Additional Exercises 9.7 Convert as indicated. When necessary, round to 2 decimal places. 1. 600 milliliters to fluid ounces

1. __________________________

2. 600 grams to ounces

2. __________________________

3. 88 pounds to kilograms

3. __________________________

4. 3 meters to feet

4. __________________________

5. 5 inches to centimeters

5. __________________________

6. 5 fluid ounces to milliliters

6. __________________________

7. 15 centimeters to inches

7. __________________________

8. 160 pounds to kilograms

8. __________________________

9. The speed limit on a country road is 55 mph. Convert to km per hour.

9. __________________________

10. A three year old weighs 66 pounds. Convert this weight to kilograms.

10. __________________________

11. A Kingaire private airplane flies at an altitude of 27,000 feet. How many kilometers is this?

11. __________________________

12. The football team’s star running back is only 5 feet, 9 inches tall. How many centimeters tall is he?

12. __________________________

Convert as indicated. When necessary, round to the nearest tenth of a degree. 13. 35°F to degrees Celsius

13. __________________________

14. 52°F to degrees Celsius

14. __________________________

15. 95°F to degrees Celsius

15. __________________________

16. 25°C to degrees Fahrenheit

16. __________________________

17. 30°C to degrees Fahrenheit

17. __________________________

18. 100°C to degrees Fahrenheit

18. __________________________

19. A cake recipe says to heat the oven to 190°C. What would be the temperature setting on an oven with Fahrenheit controls?

19. __________________________

20. The temperature recorded one summer day was 105°F. Find the corresponding temperature in degrees Celsius. (If needed, round to the nearest tenth.)

20. __________________________

E-94

Name: Instructor:

Date: Section:

Additional Exercises 10.1 Add the polynomials. 1. (2x − 3) + (−5x – 6)

1. __________________________

2. (5x – 4) + (−7x + 8)

2. __________________________

3. (2x2 – 3x + 6) + (−3x2 + 5x – 4)

3. __________________________

4. (3.7x – 4.2) + (0.3x + 6.1)

4. __________________________

Subtract the polynomials. 5. (4x − 3) – (7x + 5)

5. __________________________

6. (4x2 − 2x – 5) – (x2 − 6)

6. __________________________

7. (11x3 + 2x2 – 7) – (2x3 + 5x + 3)

7. __________________________

8. Subtract (2y − 6) from (3y + 4).

8. __________________________

9. Subtract (x2 + 16x – 1) from (6x2 − 4x + 8)

9. __________________________

Perform each indicated operation. 10. (3x– 5) – (3x2 – 4x)

10. __________________________

11. (2x3 + 6x − 3) + (−3x3 + 5x2 – 2x + 4)

11. __________________________

12. (8y2 – 0.5y) – (−3y2 + 7.2y – 3.5)

12. __________________________

13. (8.2x + 4.3) – (8x2 + 3.5x – 2)

13. __________________________

Find the value of each polynomial when x = −4. 14. 3x2 − 2x

14. __________________________

15. −x2 + 2

15. __________________________

5x +x 2

16. __________________________

17. −x2 − 3x + 5

17. __________________________

16.

The distance traveled by a free-falling object in t seconds is given by the polynomial 16t2. 18. Find the distance an object has fallen after 3 seconds.

18. __________________________

19. If an object takes 10 seconds to fall from the top of a building, find the height of the building.

19. __________________________

20. Find the distance an object has fallen in 6 seconds.

20. __________________________

E-95

Name: Instructor:

Date: Section:

Additional Exercises 10.2 Multiply. 1. x7 ⋅ x5

1. __________________________

2. a ⋅ a7

2. __________________________

3. −2x2(3.4x)

3. __________________________

4. (−6x4y2)(3xy)

4. __________________________

5. 3y3 ⋅ y ⋅ y5

5. __________________________

6. b(2b2)(3b3)

6. __________________________

7. x5 ⋅ x7 ⋅ x

7. __________________________

Simplify. 8. (x3)2

8. __________________________

9. (y8)3

9. __________________________

10. (3x2)3

10. __________________________

11. (2a5b)4

11. __________________________

12. (a5b3)3

12. __________________________

13. (xy2)2(x3y)3

13. __________________________

14. (4xy2)3(2x2y2)3

14. __________________________

15. (3x2y)(2x2y3)2

15. __________________________

16. (x5)3(x2)

16. __________________________

17. (–2x)5

17. __________________________

18. (x2)3(y)3

18. __________________________

19. Find the area of

19. __________________________ x 2x – 5

20. Find the area of

20. __________________________ 2x x

E-96

Name: Instructor:

Date: Section:

Additional Exercises 10.3 Multiply. 1. 3(x – 5)

1. __________________________

2. −2(−4y + 8)

2. __________________________

3. 3x(2x2 + 5x – 7)

3. __________________________

4. 5y2(y2 − 3y + 2)

4. __________________________

5. 6x2(3x2 – – 4x + 8)

5. __________________________

Multiply. 6. (x – 5)(x – 6)

6. __________________________

7. (y − 4)(y + 7)

7. __________________________

8. (3y + 5)(y + 3)

8. __________________________

9. (3x + 1)(2x + 5)

9. __________________________

10. (5x + 1)(3x − 7)

10. __________________________

Multiply. 11. (3x − 5)2

11. __________________________

12. (y − 3)(y − 3)

12. __________________________

13. (4x − 5)2

13. __________________________

14. (x − 4)(x + 4)

14. __________________________

15. (y + 5)2

15. __________________________

16. (3y – 5)2

16. __________________________

Multiply. 17. (a + 3)(a2 – 2a + 1)

17. __________________________

18. (x + 4)(x2 − 5x + 2)

18. __________________________

19. (2a + 1)(a2 − 3a – 6)

19. __________________________

20. (x2 + x – 2)(3x – 2)

20. __________________________

E-97

Name: Instructor:

Date: Section:

Additional Exercises 10.4 Find the greatest common factor of each list of numbers. 1. 24 and 12

1. __________________________

2. 10, 25, 55

2. __________________________

3. 24, 30, 54

3. __________________________

4. 54, 36, 81

4. __________________________

Find the greatest common factor of each list of terms. 5. y, y6, y4

5. __________________________

6. 2x5, 6x6, 8x8

6. __________________________

7. 5x5, 7x3, 9x4

7. __________________________

8. 3x2, 21x4, 15x3

8. __________________________

9. 4x2, 12x, 8x5

9. __________________________

10. xy2, x2y2, 2xy2

10. __________________________

11. ab3, a2b2, a3b4

11. __________________________

12. 3xy, 21y2, 36x3y

12. __________________________

Factor. Check by multiplying. 13. 6x3 + 2x2

13. __________________________

14. 15x3 – 10x2 + 5x

14. __________________________

15. 4x2 + 12x – 16

15. __________________________

16. 7y4 − 28y5

16. __________________________

17. 2x2 – 4x + 2

17. __________________________

18. 16x4 + 8x

18. __________________________

19. 18b4 – 9b6

19. __________________________

20. 6 − 15x2 – 12x

20. __________________________

E-98

Chapter 1

Additional Exercises

Answers

1.2 1. hundreds 2. hundred thousands 3. ones 4. thousands 5. three thousand, nine hundred sixty-five 6. fifty thousand, nine hundred eighteen 7. eighty-four thousand, nine hundred twenty-one 8. one hundred ninety thousand, twenty-five 9. two million, one hundred seventy-five 10. thirty million, one hundred eight 11. twenty-four million, three hundred twenty-six thousand, nine hundred seventy-four 12. eighteen million, four hundred eighty-nine thousand 13. 31,824 14. 7604 15. 125,364 16. 17,841 17. 3,010,008 18. 200 + 90 + 5 19. 70,000 + 1000 + 20 + 8 20. 40,000,000 + 5,000 000 + 10,000 + 2000 + 200 + 60

1.3 1. 160 2. 3048 3. 24 4. 2303 5. 305 6. 3704 7. 2174 8. 14,909 9. 2345 10. 671 11. 42 12. 4 13. 293 14. 359 15. 811 16. 28 feet 17. 254 feet 18. 1861 19. 2959 miles 20. 1820 miles

1.4 1. 710 2. 2630 3. 128,000 4. 11,200 5. 0 6. 68,321,000 7. 78,000,000 8. 9210; 9200; 9000 9. 7780; 7800; 8000 10. 94,070; 94,100; 94,000 11. 48,000 12. 335,000,000 13. 2500 14. 100 15. 10,000 16. 27,000 17. correct 18. no 19. $700 20. 100 hits

E-99

Chapter 1

Additional Exercises

Answers (cont’d)

1.5 1. 3 ⋅ 2 + 3 ⋅ 6 2. 4 ⋅ 2 + 4 ⋅ 5 3. 6 ⋅12 + 6 ⋅5 4. 5070 5. 894 6. 2072 7. 4927 8. 10,884 9. 30,832 10. 14,000 11. 184 12. 0 13. 4700 14. 2,548,777 15. 14,754,176 16. 360,000 17. 160,000 18. 84 square m 19. 2760 calories 20. 153

1.6 1. 48 2. 241 3. 47 4. 258 5. 71 6. undefined 7. 29 R 5 8. 226 R 8 9. 38 R 23 10. 85 R 5 11. 235 12. 28 13. 0 14. 16 trips 15. 98 people 16. 1760 yards 17. 246 miles 18. 79 19. 691,200 20. 2000

1.7 1. 43 ⋅ 33 2. 214 3. 82 ⋅ 63 ⋅ 34 4. 81 5. 81 6. 1 7. 180 8. 1000 9. 128 10. 8 11. 4 12. 84 13. 5 14. 4 15. 9 16. 6 17. 11 18. 144 square meters 19. 225 square feet 20. no, need 4.32 or 5 bags

E-100

Chapter 1

Additional Exercises

Answers (cont’d)

1.8 1. 10 2. 0 3. 18 4. 1 5. 20 6. 8 7. 25 8. 180 9. x + 2x 10. 7 − x 11. x 2 − 5 12. 4x − 18 x 13. 8 14. 12 – x 15. yes 16. no 17. yes 18. no 19. 5 20. 4

E-101

Chapter 2

Additional Exercises

Answers

2.1 1. −350 2. −2,000,000 3.

2.2 1. 4 2. −5 3. −2 4. −39 5. −18 6. −6 7. 0 8. 15 9. –30 10. −56 11. −7 12. 86 13. 6 14. −32 15. −6 16. 12 17. −5 18. 2 19. $283 20. −137 ft

2.3 1. −4 2. 3 3. 0 4. −14 5. 16 6. −14 7. −34 8. 0 9. –35 10. 13 11. −9 12. −12 13. 6 14. −4 15. −20 16. 28 17. 13 18. −10 19. 67° 20. $98

5. < 6. < 7. < 8. = 9. 9 10. −6 11. 20 12. −5 13. 6 14. −8 15. 3 16. 10 17. −5 18. −12 19. 4 20. 8

E-102

Chapter 2

Additional Exercises

Answers (cont’d)

2.4 1. 24 2. −10 3. 0 4. −8 5. 32 6. −30 7. −63 8. 81 9. –16 10. −1 11. 42 12. −36 13. −4 14. undefined 15. 5 16. −37 17. 6 18. −3 19. −375 20. −52,500

2.5 1. –1 2. 6 3. –8 4. 48 5. −16 6. 1 7. −19 8. −2 9. −13 10. −2 11. −17 12. 4 13. −2 14. −42 15. 56 16. −3 17. −1 18. 12 19. 2 20. −16

2.6 1. yes 2. no 3. yes 4. no 5. 7 6. −15 7. 2 8. –1 9. −5 10. −2 11. 9 12. 9 13. −4 14. 2 15. −4 16. −12 17. −5 18. –7 – x −6 19. x

E-103

Chapter 3

Additional Exercises

Answers

3.1 1. x 2. 15y 3. 8b 4. −10a + 16 5. 96x 6. 4y + 20 7. 12x − 16 8. −15b + 35 9. −27x − 54 10. 7x − 24 11. −26x + 14 12. 13n − 12 13. 7n − 21 14. −21w − 3 15. 3x − 15 16. −9n + 3 17. 3z + 2 18. (6x − 6) ft 19. 16y cm 20. 5x yds

3.2 1. 4 2. 6 3. −8 4. 12 5. 0 6. 1 7. −8 8. −16 9. 7 10. 3 11. 5 12. 5 13. 6 14. −6 15. 36 16. 2x + 20 17. x − 2x 3x 18. 7 19. 13(x + 33)

3.3 1. 7 2. −4 3. −3 4. 8 5. 4 6. 4 7. 6 8. 9 9. 22 10. 23 11. −13 12. 0 13. −20 14. 7 15. 48 16. −3 17. −38 + x = −21;17

18. 2 ( 7 − x ) = 14;0

19. −3 ( 4 + x ) = −33;7

20.

E-104

120 = 2 x;6 10

Chapter 3

Additional Exercises

Answers (cont’d)

3.4 1. x + 6 = 9 2. x − 7 = 13 3. −3x = 21 4. 20 − 2 x = 10 5. −19 6. 2 7. 5 8. 22 9. −40 10. 16 11. −363 12. −10 13. 38 14. −1 15. −3 16. 27 17. Girls, 134; boys, 67 18. 7 mph 19. 63 votes, 101 votes 20. $70

E-105

Chapter 4

Additional Exercises

Answers

4.1

4.2 1. 3 ⋅ 7 2. 2 3 ⋅ 3 2 3. 2 ⋅ 3 ⋅ 5 ⋅ 7 4. 2 ⋅ 3 ⋅ 31 5. 5 ⋅ 23 6. 2 ⋅ 7 ⋅ 17 7. 54 8. 23 ⋅ 32 ⋅13

4.3

5 1. 8 3 2. 4 3. 1 4. –6 5. 1 6. 6 7. 0 8. undefined 13 9. 4 19 10. 5 18 11. 7 53 12. 3 5 13. 2 8 5 14. 3 9 1 15. 1 7 16. 8 19 17. 11 172 2 18. 10 11 5 19. 8 5 20. 27

E-106

8 17

2x 10. 3 9 11. 20 y 8x 12. 3 2x 13. 5 3c 14. 5b 1 15. − 2y 16. yes 17. no 18. yes 7 19. mile 132 1 20. day 3

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

5 12 1 24 15 a

2 3 x y

1 3y 1 2 3 4 4 8x 15

2 3

12. 1

1 14

1 8 1 14. 1 9 15. yes 16. no 17. $216 18. 12 in 6 19. square m 25 7 20. square inches 24 13. −

Chapter 4

Additional Exercises

Answers (cont’d)

4.4

4.5

4.6

8 1. − 21 3x − 2 2. 4 5 3. 12 10 x + 11 4. 35 9 z + 10 5. 24 3 6. 2 5 1 7. − 16 1 8. 10 9x + 5 9. 20 3x + 35 10. 5x 7 11. 12 3 12. 4 1 13. 12 1 14. 1 3 11 15. 12 16. = 17. > 18. = 3 19. 5 5 20. 70

4 9 8x 2. 21 10 3. 2 17 5x 4. 14 8 5. 7 9 15 6. 15 16 9 7. 32 32 8. 225 1 9. 12 5 10. 16 9 11. 16 1 12. − 6 1 13. −1 3 1 14. 4 2 15. − 3 16. −2 2 17. 3 3 18. 8 19. true 20. false

1. 2.

1 2 3 5

5x 7 3 4. − z 2 5. 5 4 6. 13 2 7. 13 8 8. 15 14 9. 19 1 10. 5 3 11. 4 1 12. 5 13. 30 14. 4y 9 15. 24 16a 16. 40a 8 17. 1 meters 17 18. 1 yard 19. 1 cup 20. 5 cups 3.

−

E-107

Chapter 4

Additional Exercises

4.7

4.8

1. 2. 3. 4. 5. 6. 7. 8. 9.

2 3

4 5 1 5 3 1 3 7 15 31 3 35 13 1 15 1 13 3 1 1 4

10. 13 5 8 7 12. 6 20 14 13. 2 15 23 14. 30 5 15. 6 6 5 16. 9 9 1 17. 16 2 15 18. 10 24 5 19. 3 square feet 9

11. 7

7 8

E-108

1 17 1 2. − 2 3. 15 27 2 4. − or −5 5 5 3 1 5. − or −1 2 2 1 6. 22 1 7. −1 3 8. 8 9. 8 10. 3 3x − 7 11. 21 9 − 4x 12. 3 29 x 13. 36 7x 14. 10 20 + 3 x 15. 5 3 16. 4 1 17. 4 18. 2 19. 11 20. 4

20. 1

Answers (cont’d)

in.

Chapter 5

Additional Exercises

Answers

5.1 1. two and ninety-five hundredths 2. three and six hundred twenty-one thousandths 3. negative thirty-seven thousandths 4. thirty-six and four hundred one thousandths 5. 21.35 6. –43.05 7. 9.202 8. 0.0035 9. 12.0012 2 10. 25 41 11. 200 1 12. 10 2 1 13. 4 4 3 14. −50 125

5.2 1. 6.58 2. 11.3 3. 1.1346 4. 33.795 5. 64.434 6. 4.59 7. 1.88 8. –64.32 9. –15.349 10. 2.3 11. 4.75 12. 3 13. $1174.32 14. $0.29 15. 43.8 yds 16. $1.91 17. 2.34x 18. 1.69 19. 11.81 20. 0.87x

5.3 1. 3.43 2. 1.85 3. 6.4116 4. −20.72 5. −1.512 6. 38.9 7. 102 8. 501.3 9. 0.0035 10. 0.32 11. 8π cm; 25.12 cm 12. 3.5π feet; 10.99 feet 13. 20π cm; 62.8 cm 14. 6 g 15. 47.244 in 16. 2.275 17. 3.6 18. $251.20 19. 12.8 yd 20. 11.28 sq ft

15. < 16. > 17. > 18. > 19. $107.99 20. 786 miles

E-109

Chapter 5

Additional Exercises

Answers (cont’d)

5.4 1. 0.6 2. 0.51 3. 300 4. 40.2 5. 0.71 6. 1.79 7. 11.765 8. 600 9. 159 10. 327.6 11. 165.33 12. 0.3652 13. 0.00836 14. 730 15. Yes 16. Yes 17. 52 months 18. 24.1 points 19. 125 20. 0.02

5.5 1. 0.125 2. 0.6 3. 0.875 4. 0.67 5. 0.14 6. 0.43 7. 6.15 8. 15.73 9. 0.706 10. 0.596 11. > 12. > 13. < 14. 0.72152, 0.725, 0.7252 15. 0.0025 16. 408 17. 2.24 18. 0.4 19. 11.13 sq in 20. 2.24 sq m

5.6 1. 1.8 2. −12.55 3. 6 4. 16.425 5. 0.125 6. −0.6 7. 0.72 8. −37 9. −6.2 10. 18.37 11. 5.676 12. 9.36 13. 30 14. 0.165 15. 11.44 16. −50 17. 90 18. $9.66 19. $120.85 20. $1.58

E-110

Chapter 5

Additional Exercises

Answers (cont’d)

5.7 1. true 2. true 3. true 4. 21.2, 19, no mode 5. 8, 8.75, 8.9 6. 557, 444, no mode 7. 34, 28.5, 21 8. 0.8, 0.7, 0.7 9. 89, 88, 88 10. 7.4, 7.45, 6.3 11. 3.00 12. 947.4 ft 13. 288.8 m 14. 70.2 floors 15. 921 ft 16. 281 m 17. 69.5 floors 18. 1046 ft 19. 319 and 248 m 20. no mode

E-111

Chapter 6

Additional Exercises

Answers

6.1

6.2

6 1. 17 1 2. 3 2 3. 3 7 4. 20 2 5. 3 1 6. 2 4 7. 1 7 8. 3 39 9. 43 17 10. 24 1 11. 5 3 12. 4 2 13. 7 3 14. 14 5 15. 9 5 16. 14 17. 13 calories/ounce 18. 62 miles/hour 19. $3.79 20. 100 aspirin

3 12 = 1. 6 24 0.5 3 = 2. 5 30 1 3 = 1 3. 20 60 4. true 5. false 6. true 7. false 8. true 9. true 10. 9 11. 6 12. 18 13. 3 14. 2 1 15. 40 16. 12 17. 9 18. 20 19. 6 3 20. 8

6.3 1. 168 miles 2. 34.8 gallons 3. 35 bags 4. 80 cu ft 5. 750 sq ft 6. 2 dogs 7. 67.5 miles 8. 5 in 9. $300 10. $34.38 11. 7 people 12. 8 people 13. 20 gallons 14. 14 cups 15. 150 minutes 1 16. 7 cups 2 17. 420 miles 18. 6 eggs 19. 135 minutes 20. 32 pages

E-112

Chapter 6

Additional Exercises

6.4 1. 1 2. 10 3. 6 4. 3 7 5. 8 1 6. 4 7. 13 8. 12 9. 15 10 10. 11 5 11. 7 12. 9 13. 2.828 14. 4.472 15. 4.243 16. 13.342 17. 8.602 ft 18. 13.856 cm 19. 9.165 ft 20. 13 ft

6.5

Answers (cont’d)

2 1 1 2. 3 1 3. 4 1 4. 5 1 5. 5 1 6. 2 7. 4 8. 26.25 9. 5 10. 51.2 11. 8 12. 8.7 13. 12 14. 5 15. 5.116 16. 16 ft 17. 48 ft 18. 12.5 ft 19. 45 ft 20. 4.8, 12.75

E-113

Chapter 7

Additional Exercises

Answers

7.1 1. 72% 2. 61% 3. 0.034 4. 0.006 5. 1.79 6. 0.45 7. 49% 8. 0.3% 9. 72.4% 10. 460% 6 11. 25 7 12. 500 1 13. 1 5 1 14. 4 15. 52% 16. 20% 17. 325%

7.2

7.3 1. 0.62 ⋅ 80 = x 2. x = 0.08 ⋅ 75 3. 0.25x = 40 4. 7.2 = 0.15x 5. 2.5 = 25x 6. 15 = 50x 7. 32 8. 760 9. 80 10. 135 11. 75% 12. 40% 13. 1.5 14. 49.5 15. 5% 16. 250 17. 22 18. 10% 19. 150% 20. 30

19 500 73 19. 0.584; 125 20. 6.5%

18. 0.038;

E-114

x 12 = 1. 42 100 x 30 2. = 50 100 18 60 = 3. x 100 x 30 4. = 100 120 5. 162 6. 51 7. 90 8. 50 9. 20% 10. 25% 11. 6.25 12. 6 13. 50 14. 10,000 15. 2.5% 16. 200% 17. 62.5% 18. 16.64 19. 40 20. 17.2

Chapter 7

Additional Exercises

Answers (cont’d)

7.4 1. 1600 2. $7250 3. 5% 4. 42,000 5. $36,800 6. 33.3% 7. 21.4% 8. 25% 9. 20.7% 10. 50% 11. 6.5% 12. $21 13. 210, 46.7% 14. 1000 mg 15. 20% 16. 25% 17. 9, 54 18. 20% 19. $74.36 20. 126

7.5 1. $11.20 2. $18 3. $665.63 4. $21,525 5. $311.23 6. 5% 7. 6% 8. $3.16 9. $4797 10. $42 11. 0.4% 12. $1780 13. 0.4% 14. $55.25 15. $240 16. $3750 17. $47.70 18. $129.90 19. $4300 20. $1260; $1338.75

7.6 1. $48 2. $270 3. $720 4. $735 5. $39 6. $64 7. $2480 8. $1635 9. $2250 10. $20 11. $12,167 12. $10,511.62 13. $609.20 14. $3307.50 15. $4686.64 16. $5002.27 17. $127,801.15 18. $5546.16 19. $1256.56 20. $750.50

E-115

Chapter 8

Additional Exercises

Answers

8.1 1. 30,000 2. decrease 77 3. 74 4. 745,000 5. decreased 6. 19,000 7. increase 24 8. 43 9. 22% 10. 14% 11. 21% 12. same percent 13. 4.2% 14. 5% 15. 2009 16. 2006 17. 1999 and 2005 18. 2004 19. About the same 20. 60,000

8.2

8.3 1.

1. 2. 3.

12 1 1 1 25 36

(3, 4) (-3, 2)

(-4, 0) (0, -1)

4. High School Enrollment

2. y

Seniors 25%

Freshmen 35%

Juniors 20%

Sophomore s 20%

5. 6. 7.

(3, 2) (0, 2)

4 5 1 4 4 7

3. 4.

8. Cookie Preference Coconut 4% Fig 4%

Date Nut 4%

5. 6. 7. 8. 9. 10.

(2, -1)

(-3, -2)

A (2, 3), B (2, 0), C (−3, −2), D (−2, 2) A (3, 2), B (0, −2), C (−3, 2), D (−2, −3) no yes no no no (-3, 4)

Oatmeal Raisin 20%

(0, 2)

Chocolate Chip 68%

(3, 0) x

1 5 1 10. 17 1 11. 25 12. false, over half 13. 12 14. 35 15. 10 23 16. 5 7 17. 10 18. 25 19. 33 20. 13

E-116

11. y

(0, 2)

(-1, -1)

(-2, -4)

12. (−1, 1), (2, 0), (5, −1) 13. (−1, −4), (0, −3), (2, −1) 14. (0, −3), (1, −2), (3, 0) 15. (2, 1), (1, −1), (0, −3) 16. (0, −1), (1, 2), (2, 5) 17. (0, −4), (−2, 0), (2, −8) 18. no 19. yes 20. no

Chapter 8

Additional Exercises

Answers (cont’d)

8.4

8.4 (cont’d)

1. (−1, −1), (0, 0), (1, 1)

11. y

2. (−1, 4), (0, 4), (1, 4)

12.

y y

13.

3. (−2, 2), (0, 0), (2, −2)

14.

4. (0, −6), (2, 0), (4, 6)

y y

x x

15.

10.

y y

x x

E-117

Chapter 8

Additional Exercises

Answers

8.4 (cont’d) 16.

8.5

8.5 (cont’d) 16.

2. x

3. 4. 5.

17. vertical 18. neither 19. horizontal 20. neither

6. 7. 8. 9. 10. 11. 12. 13. 14.

2 7 1 3 2 7 3 7 2 17 2 17 3 17 1 2 1 2 1 5 1 4 1 52 1 6 1 2

Red

Blue

Green

Red 1

Red 2

Red 3

Blue 1

Blue 2

Blue 3

Green 1

Green 2

3 Green 3 17. The event will never occur. 18. The event will always occur. 19. No, all probabilities must be less than 1. 20. 1 chance out of two that the event will occur.

15. H

TT 4 outcomes

E-118

Chapter 9

Additional Exercises

Answers

9.1 1. ray 2. angle 3. line 4. 75° 5. 105° 6. 90° 7. obtuse 8. acute 9. right 10. 156° 11. 5° 12. 55° 13. 25° 14. 54° 15. νABD and νEBC, νDBE and νEBF 16. < HAO and < OAP <OAP AND <PAE <PAE AND <EAH <EAH AND <HAO 17. x = 58°, y = 122°, z = 58° 18. 66° 19. 31° 20. a = 50° b = 80° c = 50° d = 130° e = 130° f = 130°

9.2 1. 26 ft 2. 39 ft 3. 19.2 m 4. 11 yd 5. 72 inches 6. 1200 ft 7. 32 cm 8. 60 m 9. 52 mm 10. 42 in 11. 24 in 12. 13 cm 13. 45 feet 14. $70 15. 121π ≈ 37.68 ft 16. 7π ≈ 21.98 m 17. 14π ≈ 43.96 cm 18. 16π ≈ 50.24 feet 19. 3 laps 20. 12 feet

9.3 1. 11 sq m 2. 31.5 sq ft 3. 256π ≈ 803.84 sq mi 4. 17.5 sq mi 5. 16π ≈ 50.24 sq m 6. 40.5 sq yd 7. 25.2 sq ft 8. 63 sq ft 9. 91 sq in 10. 40 sq m 11. 93.5 sq in 12. 1808.64 sq in 13. 3600 cu mm 14. 2250 cu m 15. 603.43 cu cm 16. 100.57 cu m 17. 905.14 cu cm 18. 21.952 cu ft 19. 251.43 cm 20. 36,800 cu cm

E-119

Chapter 9

Additional Exercises

Answers (cont’d)

9.4 1. 12,320 ft 2. 1.5 mi 3. 5 yd 4. 3 yd 2 ft 5. 13 yd 6. 100 in 7. 11 yd 1 ft 8. 3 ft 5 in 9. 11 yd 3 10. 9 yd or 9.75 yd 4 11. 7.5 mi 12. km 13. 0.342 dm 14. 83 mm 15. 85.5 cm 16. 5.1 m or 510 cm 17. 1170 m or 0.117 km 18. 68.8 m 19. 5.3 m 20. 16 bows

9.5 1. 3 lb 2. 7000 lb 3. 60 oz 4. 0.7 tons 5. 4 lb 4 oz 6. 34 lb 4 oz 7. 4 tons 1900 lb 8. 12 lb 15 oz 9. no 10. 3 ton 1000 lb 11. 0.48 kg 12. 5700 mg 13. 0.17 g 14. 0.0065 kg 15. 9100 g 16. 5.27 g or 5270 mg 17. 7.8 kg or 7800 g 18. 4.506 kg 19. 0.25 mg 20. 64.6 g

9.6 1. 3 pt 2. 8 pt 1 3. 1 gal 2 4. 13 qt 5. 10 gal 1 qt 6. 4 gal 1 qt 7. 3 qt 8. 5 c 4 oz 9. 3 gal 1 qt 10. 6 cars 11. 6200 ml 12. 7.3 liters 13. 1.8 l liters 14. 500 liters 15. 0.025 kl 16. 3600 ml or 3.6 liters 17. 5200 l or 5.2 kl 18. 840 ml 19. 3.4 l 20. 1820 ml or 1.82 liters

E-120

Chapter 9

Additional Exercises

Answers (cont’d)

9.7 1. 20.29 oz 2. 21.16 oz 3. 40 kg 4. 9.84 feet 5. 12.7 cm 6. 147.85 ml 7. 5.91 in 8. 72.73 kg 9. 88.71 km/hr 10. 30 kg 11. 8.1 km 12. 175.26 cm 13. 1.7° 14. 11.1° 15. 35° 16. 77° 17. 86° 18. 212° 19. 374° 20. 40.6°

E-121

Chapter 10

Additional Exercises

Answers

10.1 1. −3x − 9 2. −2x + 4 3. − x 2 + 2 x + 2 4. 4x + 1.9 5. −3x − 8 6. 3x 2 − 2 x + 1 7. 9 x3 + 2 x 2 − 5 x − 10 8. y + 10 9. 5 x 2 − 20 x + 9 10. 3x 2 + 7 x − 5 11. − x3 + 5 x 2 + 4 x + 1

10.2 1. x12 2. a8 3. −6.8x3

10.3 1. 3x − 15 2. 8 y − 16

12. 11y 2 − 7.7 y + 3.5 13. −8 x 2 + 4.7 x + 6.3 14. 56 15. −14 16. −14 17. 1 18. 144 feet 19. 1600 feet 20. 576 feet

6 x3 + 15 x 2 − 21x

−18x5 y 3

5 y 4 − 15 y 3 + 10 y 2

3y 9

18 x 4 − 24 x3 + 48 x 2

6. 7. 8.

6b 6

5. 6.

x13

y 2 + 3 y − 28

3 y 2 + 14 y + 15

y 24

9. 6 x 2 + 17 x + 5 10. 15 x 2 − 32 x − 7 11. 9 x 2 − 30 x + 25

10. 27x 6 11. 16a 20 b 4 12. a15b9 13. x11 y 7 14. 512x9 y12 15. 12x6 y 7 16. x17 17. −32x5 18. x 6 y 3 19. 2 x 2 − 5 x 20. x 2

E-122

x 2 − 11x + 30

12. y 2 − 6 y + 9 13. 16 x 2 − 40 x + 25 14. x 2 − 16 15. y 2 + 10 y + 25 16. 9 y 2 − 30 y + 25 17. a3 + a 2 − 5a + 3 18. x3 − x 2 − 18 x + 8 19. 2a 3 − 5a 2 − 15a − 6 20. 3 x3 + x 2 − 8 x + 4

Chapter 10

Additional Exercises

Answers (cont’d)

10.4 1. 12 2. 5 3. 6 4. 9 5. y 6. 2x5 7. x3 8. 3x 2 9. 4x 10. xy 2

11. ab 2 12. 3y 13. 2 x 2 (3x + 1)

(

)

14. 5 x 3 x 2 − 2 x + 1

(

15. 4 x 2 + 3x − 4 16. 7 y (1 − 4 y )

)

(

)

17. 2 x 2 − 2 x + 1

(

) 19. 9b ( 2 − b ) 20. 3 ( 2 − 5 x − 4 x )

18. 8 x 2 x3 + 1 4

E-123

Names:

Date: Section:

Instructor:

Group Activity A Chapter 1 Top Grossing Movies

TITLE Titanic The Lord of the Rings: The Return of the King Pirates of the Caribbean: Dead Man’s Chest Star Wars Episode I: The Phantom Menace

TOTAL GROSS 1,834,779,000

U.S. GROSS 600,779,000

COST 200,033000

1,129,219,252

377,019,252

94,392,393

1,060,332,628

423,032,628

225,395,382

968,657,891

431,065,444

110,023,398

HINT: Gross is the total amount of money taken in for each movie, but the profit is total gross – cost. 1. How much profit did Titanic make?

1. __________________________

How much more total gross did Titanic make than The Lord of the Rings: The Return of the King?

2. __________________________

How much did Titanic make in foreign markets?

3. __________________________

Which movie made the most profit?

4. __________________________

Did The Lord of the Rings: The Return of the King or Star Wars Episode I make more profit?

5. __________________________

Did Pirates of the Caribbean: Dead Man’s Chest or Star Wars Episode I: The Phantom menace make more profit?

6. __________________________

7. How much more profit did Titanic make than The Lord of the Rings: The Return of the King in the United States?

7. __________________________

G-1

Names:

Date: Section:

Instructor:

Group Activity B Chapter 1 Top Grossing Movies

TITLE Titanic The Lord of the Rings: The Return of the King Pirates of the Caribbean: Dead Man’s Chest Star Wars Episode I: The Phantom Menace

TOTAL GROSS 1,834,779,000

U.S. GROSS 600,779,000

COST 200,033000

1,129,219,252

377,019,252

94,392,393

1,060,332,628

423,032,628

225,395,382

968,657,891

431,065,444

110,023,398

U.S. GROSS

COST

Round each number to the nearest million.

TITLE Titanic The Lord of the Rings: The Return of the King Pirates of the Caribbean: Dead Man’s Chest Star Wars Episode I: The Phantom Menace

G-2

TOTAL GROSS

Names:

Date: Section:

Instructor:

Group Activity A Chapter 2 I. 1. Use order of operations to evaluate: 3 ⋅ 4 + 2 – 7

1. __________________________

2. Insert one or two sets of parentheses to make the statement true. 3 ⋅ 4 + 2 – 7 = 11

2. __________________________

3. Insert one or two sets of parentheses to make the statement true. 3 ⋅ 4 + 2 – 7 = –3

3. __________________________

4. Use order of operations to evaluate: 4 – 8 ⋅ 2 + 5

4. __________________________

5. Insert one or two sets of parentheses to make the statement true. 4 – 8 ⋅ 2 + 5 = –3

5. __________________________

6. Insert one or two sets of parentheses to make the statement true. 4 – 8 ⋅ 2 + 5 = –17

6. __________________________

7. Use order of operations to evaluate: 2 + 5 – 3 ⋅ 2

7. __________________________

8. Insert one or two sets of parentheses to make the statement true. 2 + 5 – 3 ⋅ 2 = 6

8. __________________________

9. Insert one or two sets of parentheses to make the statement true. 2 + 5 – 3 ⋅ 2 = 8

9. __________________________

II.

III.

IV. Write your own expression using four numbers and any operation (+, –, ×, ÷) and no parentheses. Evaluate the expression. Then insert parentheses to get two different answers.

G-3

Names:

Date: Section:

Instructor:

Group Activity B Chapter 2

Lowest Temperature Recorded for Five States 12

20 0

Alaska

Montana Minnesota

Texas Hawaii

-20 -23

-40 -60 -80

-60 -80

-70

-100

1. What was the lowest temperature recorded for theses five states?

1. __________________________

2. What was the lowest temperature recorded for Minnesota?

2. __________________________

3. What is the difference between the lowest temperature of Alaska and the lowest temperature for Texas?

3. __________________________

4. What is the difference between the lowest temperature of Alaska and the lowest temperature for Montana?

4. __________________________

5. What is the difference between the lowest temperature of Montana and the lowest temperature for Minnesota?

5. __________________________

6. What is the difference between the lowest temperature of Alaska and the lowest temperature for Hawaii?

6. __________________________

7. What is the difference between the lowest temperature of Montana and the lowest temperature for Hawaii?

7. __________________________

G-4

Names:

Date: Section:

Instructor:

Group Activity A Chapter 3 1. Solve and check: x + 7 = 2x – 9

1. __________________________

2. Write a new equation with the same answer as you got in problem 1. Solve your equation to check that the solution is the same as 1.

2. __________________________

3. Solve and check: 2(x – 6) + 3(2x – 7) = 3(2x + 7)

3. __________________________

4. Write a new equation with the same answers as you got in problem 3. Solve your equation to check that the solution is the same as 3.

4. __________________________

5. Solve and check:

5. __________________________

x +6 =8 4

6. Write a new equation with the same answers as you got in problem 5. Solve your equation to check that the solution is the same as 5.

6. __________________________

G-5

Names:

Date: Section:

Instructor:

Group Activity B Chapter 3 I. Solve each equation. Write the soution in the box that corresponds to the variable in the equation. 6a – 3 = 4a + 5

3b + 8 = 35

3c + 4 = 2c + 3c

4(d – 3) = 2d – 6

5(e – 6) + 8 = e – 2

f +1 = 2 7

6(g – 2) + g = 8g – 20

3(h – 3) – 2 = 2(h – 5)

3(i + 2) = 4i

II. Study the numbers in the box. Do you notice any pattern?

G-6

Names:

Date: Section:

Instructor:

Group Activity A Chapter 4 Find the indicated fractional portion of each word and record it in the corresponding blank. Compose a hidden message by reading down the columns when you have finished.

Last

1 of fourth 3

________________

Last

1 of step ________________ 2

First

3 of reed 4

________________

First

3 of race ________________ 4

First

1 of hide 2

________________

Last

4 of lattice ________________ 7

Last

3 of punts 5

________________

First

3 of prattle ________________ 7

First

3 of forward ________________ 7

Last

1 of pact 2

First

1 of leader 2

________________

1 of ice 1

Last

1 of turn 2

________________

Last

3 of land 4

Last

3 of running ________________ 7

First

1 of pie 3

________________

First

1 of mathematical ________________ 3

Last

4 of tract 5

________________

1 of part ________________ 2

Last

3 of rice 4

________________

Middle

________________

The message is: ______________________________________________________________________________

G-7

Names:

Date: Section:

Instructor:

Group Activity B Chapter 4 Shade the indicated fractional part. Fill in the missing part of the fractions. 1 4

1 = 4 8

3 4

3 = 4 8

3 5

3 = 5 10

G-8

Names:

Date: Section:

Instructor:

Group Activity A Chapter 5 Find the balance of the checking account after each activity.

Action

Amount

Beginning Balance Pay Rent Deposit Paycheck Pay Electric Bill Pay Telephone Bill Write a check for Cash Buy Groceries

Balance $468.36

$400.00 $745.32 $69.39 $53.28 $100.00 $96.78

G-9

Names:

Date: Section:

Instructor:

Group Activity B Chapter 5 Justice runs around a circular path. A sign says that each lap is 1000 feet. 1. If she runs four laps, how far does Justice run?

1. __________________________

2. Find the radius of the circular path. Use 3.14 to approximate pi. Round to the tenth of a foot.

2. __________________________

r r

Justice’s friend, Aisha, runs with her but she runs slower. She runs six feet to Justice’s side, inside the path. 3. Find the radius of the circle that Aisha runs. Round to the tenth.

3. __________________________

4. How far does Aisha run on each lap? Round to the tenth.

4. __________________________

5. If she runs four laps, how far does Aisha run?

5. __________________________

← Justice’s path Aisha’s path →

G-10

Names:

Date: Section:

Instructor:

Group Activity A Chapter 6 1. Find the missing parts of the triangles. The triangles are similar.

5 2 4

x = _______________________

y = ______________________________

1 inch on a map represents 75 miles. Find out how much is represented by each length. 4

1 inches = _______________________ 2

3 inches = _____________________ 4

3. 4 pounds of peaches sell for $2.20. What would 5 pounds sell for? ________________________

G-11

Names:

Date: Section:

Instructor:

Group Activity B Chapter 6 Use the Pythagorean Theorem to find the missing side of each right triangle. Round to the nearest hundredth when necessary. 1. ____________

2. ____________

10 6 x 3 4

3. ____________

4. ____________

13 x 32

5. ____________ 76

x x

G-12

Names:

Date: Section:

Instructor:

Group Activity A Chapter 7 Have one member of the group fill in the hours for the following questions. I. Estimate the best that you can

Hours

Percent

How hours are you in class each week?

___________

____________

How many hours do you study each week?

___________

How many hours do you work each week?

___________

How many hours do you sleep each week?

___________

____________

How many hours do you spend on all other activities?

___________

____________

(Hours should total 168)

___________

____________

TOTAL

II. Find the percent of time spent each week on each activity.

III. Can the group recommend way to make better use of the time?

G-13

Names:

Date: Section:

Instructor:

Group Activity B Chapter 7 An accountant of a major movie studio estimates the expenses of making a certain movie fit the following pattern. If the movie cost $12,000,000 to make, find the amount spent on each category. Percent

Amount

Actor’s Salaries

30%

___________________________

Director’s Salaries

___________________________

Other personnel

10%

___________________________

Sets and studio costs

27%

___________________________

Costumes

___________________________

Computer graphics

20%

___________________________

G-14

Names:

Date: Section:

Instructor:

Group Activity A Chapter 8 Go to a parking lot near your building. Count the number of cars, motorcycles, pickup trucks, and SUV’s in the lot. Make a pie chart using this information.

G-15

Names:

Date: Section:

Instructor:

Group Activity B Chapter 8 Use the bar graph to answer the following questions.

Number of Students

Miles Driven to School for a Sample of Students 100 90 80 70 60 50 40 30 20 10 0

0-9

10 10 - 19 20 - 29 30 - 39 40 - 49 50 - 59 Miles

1. How many students in this sample drove 20 – 29 miles?

1. __________________________

2. What range of miles was driven by the most students?

2. __________________________

3. How many students drove 30 – 39 miles?

3. __________________________

4. What range of miles was driven by the least number of students?

4. __________________________

5. What percent of the students drove 30 – 39 miles? Estimate the the nearest tenth.

5. __________________________

G-16

Names:

Date: Section:

Instructor:

Group Activity A Chapter 9 A school is making a planter by building a circular brick wall three-feet high and filling it with loam and planting ivy. Find the total cost of the project. The radius of the planter will be 6 feet. Use 3.14 to approximate pi. Round answers to the nearest tenth of a foot or nearest cent.

1. Find the circumference of the planter.

1. __________________________

2. A brick wall 3 feet high costs $2.75 a foot to make. Find the cost for building the wall.

2. __________________________

3. Find the area of the planter.

3. __________________________

4. Loam will be used to fill the planter 2 feet high. How much loam will be needed? Give answer as a whole number.

4. __________________________

5. Loam costs $1.50 a cubic. What will be the cost of the loam?

5. __________________________

6. One ivy plant is to be planted every square foot. How many plants are needed? Give answer as a whole number.

6. __________________________

7. The plants cost $0.75 each. What will the cost of the plants be?

7. __________________________

8. What will be the total cost of the planter?

8. __________________________

G-17

Names:

Date: Section:

Instructor:

Group Activity B Chapter 9 A nurse has to do the following conversions in one day. Round answers to the nearest tenth. 1. 2 gallons of solutions need to be prepared. The solution formula requires the use of ml. Convert 2 gallons to L and then to ml.

1. __________________________

2. A prescription is for 50 mg of drug for every 1 kg the patient weighs. The patient weighs 176 pounds. Convert the pounds to kg.

2. __________________________

3. 1.6 L of solution is available. Change this to quarts.

3. __________________________

4. A prescription calls for 3 grams of drug. The bottle is labeled in mg. Change 3 grams to mg.

4. __________________________

5. 75 ml needs to be converted to fluid ounces.

5. __________________________

6. Normal temperature for a patient is 98.6° Fahrenheit. Convert to centigrade.

6. __________________________

G-18

Names:

Date: Section:

Instructor:

Group Activity A Chapter 10 What must be added to the first expression to get the second expression? 1. (5x − 3), (−2x + 3)

1. __________________________

2. (3x – 5), (5x − 5)

2. __________________________

3. (3x2 – 5x + 8),

4. (7x – 8),

(4x2 − 3x – 7)

(5x + 12)

3. __________________________

4. __________________________

What must be subtracted from the first expression to get the second expression? 5. (9x − 8), (−2x + 3)

5. __________________________

6. (x – 5), (x − 8)

6. __________________________

7. (7x2 – 4x + 2),

8. (9x – 5),

(8x2 − 3x – 2)

(7x – 4)

7. __________________________

9. __________________________

What must be multiplied by first expression to get the second expression? 9. (5x − 3), (5xy – 3y)

9. __________________________

10. (2x – 7), (4x3 – 14x2)

10. __________________________

11. (3x2 – 5), (12x3y − 20xy)

11. __________________________

(x2 – 16x + 64)

12. __________________________

12. (x – 8),

G-19

Names:

Date: Section:

Instructor:

Group Activity B Chapter 10 Write an algebraic expression to represent the perimeter and area of each figure. Simply each expression by adding or multiplying. 1. P = ____________

A = ____________

2x + 4 x+2

2x − 8

2. P = ____________

3. P = ____________

A = ____________

x−3

4. P = ____________

5. P = ____________

A = ____________

A = ____________ 2x + 3 x+6

x+4

2x − 1

G-20

2x + 5

Name: Instructor:

Date: Section:

Chapter 1 Pretest Form A 1. Determine the place value of the digit 5 in the whole number 38,459.

1. _______________________

2. Write the standard form for “two thousand, six hundred two.”

2. _______________________

3. Add: 86 + 49

3. _______________________

4. Multiply: 289 × 23

4. _______________________

5. Find the perimeter of the rectangle.

5. _______________________

3 cm 8 cm 6. Subtract 386 from 627.

6. _______________________

7. Round 2,587 to the nearest thousand.

7. _______________________

8. Round each number to the nearest hundred and estimate the sum. 392 382 + 592

8. _______________________

9. Find the perimeter of a rectangle that is 8 feet long and 4 feet wide.

9. _______________________

10. Find the area for a square with sides of 6 cm.

10. _______________________

11. Divide. 432 ÷ 16

11. _______________________

12. Divide and check.

9 72,584

12. _______________________

13. Evaluate 32 ⋅ 4

13. _______________________

14. Write in exponential notation: 3 ⋅3 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2

14. _______________________

15. Simplify: 22 + [8 + 3(5 + 2)]

15. _______________________

16. Simplify:

16. _______________________

30 − 42 + 1 24 − 3 ⋅ 5

17. Evaluate 2x + 6 if x = 2 18. Evaluate

x+4 2x

17. _______________________

if x = 4.

18. _______________________

19. Find the average of the following numbers: 7, 9, 15, 18, 21

19. _______________________

20. Write as a mathematical expression using x for “a number.” Seven less than twice a number.

20. _______________________

T-1

Name: Instructor:

Date: Section:

Chapter 1 Pretest Form B 1. Determine the place value of the digit 6 in the number 26,891.

1. _______________________

2. Write the standard form for “ten thousand, five hundred thirty-two.”

2. _______________________

3. Add: 39 + 92 + 74

3. _______________________

4. Multiply: (328)(39)

4. _______________________

5. Find the perimeter of a rectangle that is 11 feet long and 8 feet wide.

5. _______________________

6. Subtract

593 − 298

6. _______________________

7. Round 27,387 to the nearest thousand.

7. _______________________

8. Round each number to the nearest hundred and estimate the sum. 896 314 194 + 462

8. _______________________

9. Subtract 14 from 25.

9. _______________________

10. Find the area of the square with sides of length 12 m.

10. _______________________

11. Find the product of 92 and 4.

11. _______________________

7 3684

12. _______________________

13. Evaluate: 22 ⋅ 4 ⋅ 32

13. _______________________

14. Write in exponential notation: 2⋅6⋅6⋅6⋅8⋅8

14. _______________________

15. Simplify: 2 + 4(6 – 4)2

15. _______________________

16. Simplify: 3(5 – 2)2 + 23

16. _______________________

17. Evaluate: 5(x + 2) if x = 3

17. _______________________

18. Twenty tickets to the ballet cost $540. How much does a single ticket cost?

18. _______________________

12. Divide

19. Find the average of the following numbers: 42, 36, 32, 40, and 25

19. _______________________

20. Write the following as a mathematical expression, use x for “a number.” The product of a number and six.

20. _______________________

T-2

Name: Instructor:

Date: Section:

Chapter 1 Test Form A Evaluate: 1. Write in words: 25,489

1. __________________________

2. Write in standard form: three thousand, six hundred four

2. __________________________

Perform the indicated operations. 3. 36 + 91

3. __________________________

4. 780 – 325

4. __________________________

193 × 74

5. __________________________

6. 1445 ÷ 17

6. __________________________

7. 0 ÷ 18

7. __________________________

8. 37 ÷ 37

9. __________________________

9. 493(24)

9. __________________________

Simplify. 10. __________________________

10. 18 + 4 ÷ 4 ⋅11 − 3 11. 5 ⋅ (3 – 2)2

11. __________________________

12. Round 32, 654 to the nearest thousand.

12. __________________________

Estimate each sum or difference by rounding each number to the nearest hundred. 13. 285 + 569 + 1375

13. __________________________

14. 3624 – 3284

14. __________________________

15. Evaluate

3x + 2 y if x = 2 and y = 3. 3

16. Evaluate 2(x3 – 4) if x = 2.

15. __________________________

16. __________________________

T-3

Name: Instructor:

Date: Section:

Chapter 1 Test Form A (cont’d) Find the perimeter and area of each figure. 17. square

17. __________________________

3 centimeters.

18. __________________________

18. rectangle

2 yards

4 yards Translate the following phrases into mathematical expressions. Use x to represent “a number.” 19. The quotient of a number and seven.

19. __________________________

20. Twice a number added to seventeen.

20. __________________________

21. Admission to a movie costs $8.50 per ticket. A student and six friends are attending a movie together. What is the total cost of the tickets?

21. __________________________

22. A college instructor wants to buy a new car . One car costs $12,500 and another is $13,995. How much more expensive is the higher priced car?

22. __________________________

23. There are 12 people going to a musical. The total cost of the tickets is $312. What is the cost of each ticket?

23. __________________________

24. Find the average of 24, 36, 55, 18, and 42.

24. __________________________

25. Is 5 a solution of the equation 3x + 4 = 19?

25. __________________________

T-4

Name: Instructor:

Date: Section:

Chapter 1 Test Form B Evaluate: 1. Write in words: 96,362

1. __________________________

2. Write in standard form: six thousand, four hundred eight

2. __________________________

Perform the indicated operations. 3. 87 + 31

3. __________________________

4. 721 – 284

4. __________________________

217 × 53

5. __________________________

6. 2064 ÷ 8

6. __________________________

7. 25 ÷ 0

7. __________________________

8. 21 ÷ 21

8. __________________________

9. 0 ÷ 82

9. __________________________

Simplify. 10. __________________________

10. 43 ⋅ 32

11.

( 2 − 2) ⋅ 2

11. __________________________

12. Subtract 15 from 29.

12. __________________________

13. Round 15,632 to the nearest hundred.

13. __________________________

T-5

Name: Instructor: Chapter 1 Form B (cont’d)

Date: Section:

Estimate each sum or difference by rounding each number to the nearest hundred. (Then add or subtract.) 14. 385 + 1209 + 1682

14. __________________________

15. 3857 – 2218

15. __________________________

16. Evaluate

3x + 6 4y

for x = 2 and y = 3

16. __________________________

. 17. Evaluate 3 + 2 ⋅ (x – 2) for x = 3.

17. __________________________

Find the perimeter and area of each figure. 18.

18. __________________________ square

4 in.

19. __________________________

19. rectangle

2 ft.

7 ft. Translate the following phrases into mathematical expressions. Use x to represent “a number.” 20. The sum of three times a number and five.

20. __________________________

21. The quotient of a number and twelve.

21. __________________________

22. An amusement park charges $11 for a child’s ticket. A 4th grade class of 27 students plans to go to the amusement park. What will be the total cost of the childrens’ tickets?

22. __________________________

23. The temperatures for a week in a southern city were 96°, 95°, 101°, 92°, 86°, 93°, and 88°. What was the average daily temperature?

23. __________________________

24. Five people went to a concert. The tickets cost $240. What was the cost for each ticket?

24. __________________________

25. Is 3 a solution of the equation 2x − 3 = −9?

25. __________________________

T-6

Name: Instructor:

Date: Section:

Chapter 1 Test Form C Evaluate: 1. Write in words: 10,371

1. __________________________

2. Write in standard form: thirteen thousand, four hundred ninety-six

2. __________________________

Perform the indicated operations. 3. 93 + 88

3. __________________________

4. 472 – 173

4. __________________________

5. 16 ÷ 0

5. __________________________

6. 328 ÷ 8

6. __________________________

7. 346(27)

7. __________________________

192 192

8. __________________________

9. 0 ÷ 5

9. __________________________

Simplify. 10. 23 ⋅ 42

10. __________________________

11. (3 – 2)2 ⋅ 3

11. __________________________

12. (32 − 5)2

12. __________________________

13. Round 3,942 to the nearest thousand.

13. __________________________

Estimate each sum or difference by rounding each number to the nearest hundred. (Round, then add or subtract.) 14. 903 + 496

14. __________________________

15. 5758 – 3243

15. __________________________

T-7

Name: Instructor: Chapter 1 Form C (cont’d)

Date: Section:

16. Evaluate 2x + 3y for x = 3 and y = 5

16. __________________________

17. Evaluate 2(x + y) for x = 2 and y = 4

17. __________________________

Find the perimeter of each figure. 18.

18. __________________________ rectangle

3 cm

7 cm 19. __________________________ 19. square

3 ft.

Translate the following phrases into mathematical expressions. Use x to represent “a number.” 20. Five less than twice a number.

20. __________________________

21. The product of three times a number decreased by nineteen.

21. __________________________

22. A medium pizza costs $12. A parent needs six pizzas for a birthday party. Find the total cost for the pizzas.

22. __________________________

23. A book order totaled $312. A total of 12 books were ordered. Find the average cost of each book.

23. __________________________

24. Find the average of 92, 95, 85, 87, and 76.

24. __________________________

25. Is 5 a solution of the equation 8x − 8 = 7?

25. __________________________

T-8

Name: Instructor:

Date: Section:

Chapter 1 Test Form D Evaluate: 1. Write in words: 42,042

1. __________________________

2. Write in standard form: Sixty-one thousand, two hundred five.

2. __________________________

Perform the indicated operations. 3. 128 + 372

3. __________________________

4. 957 – 369

4. __________________________

5. 108 × 72

5. __________________________

6. 1116 ÷ 62

6. __________________________

7. 0 ÷ 19

7. __________________________

93 93

8. __________________________

9. 9 ÷ 0

9. __________________________

Simplify. 10. 52 ⋅ 33

10. __________________________

11. (42 + 2) ⋅ 2

11. __________________________

12. 3(22 – 4)

12. __________________________

13. Round 7,969 to the nearest hundred.

13. __________________________

Estimate each sum or difference by rounding each number to the nearest hundred. (Round, then add or subtract.) 14. 897 + 284 + 312

14. __________________________

15. 652 – 251

15. __________________________

16. Evaluate 3(x + y) for x = 3 and y = 5.

16. __________________________

T-9

Name: Instructor: Chapter 1 Form D (cont’d)

Date: Section:

17. Evaluate (x – y)2 for x = 6 and y = 2.

17. __________________________

Find the perimeter and area of each figure. 18. __________________________

18. square

19.

19. __________________________ Rectangle

Translate the following phrases into mathematical expressions. Use x to represent “a number.” 20. Twice the sum of a number and seven.

20. __________________________

21. Four less than the sum of a number and six.

21. __________________________

22. Math books cost $118 each. Find the total cost for a math class which has 22 students.

22. __________________________

23. A used SUV is on sale for $15,490. A used sedan is on sale for $12,995. What is the difference in the two prices?

23. __________________________

24. The total cost for a dinner for twelve people was $216. Find the average cost of each person.

24. __________________________

25. Is 5 a solution of the equation 15 – 2x = 5?

25. __________________________

T-10

Name: Instructor:

Date: Section:

Chapter 1 Test Form E Evaluate: 1. Write in words: 16,295

1. __________________________

2. Write in standard form: one hundred three thousand, nine hundred forty-one

2. __________________________

Perform the indicated operations. 3. 96 + 32

3. __________________________

4. 500 – 396

4. __________________________

5. 172 × 33

5. __________________________

6. 9308 ÷ 26

6. __________________________

7. 48 ÷ 48

7. __________________________

8. 0 ÷ 13

8. __________________________

Simplify.

(5 − 6) ⋅ 2

9. __________________________

10. 32 ⋅ 51 − 24

10. __________________________

2 2 11. 3 ( 5 − 3) + (14 ÷ 2 )   

11. __________________________

12. Round 29,842 to the nearest thousand.

12. __________________________

Estimate each sum or difference by rounding each number to the nearest hundred. (Round, then add or subtract.) 13. 3,620 + 4,895 – 5,036

13. __________________________

14. 6902 – 4375

14. __________________________

15.

Evaluate 2 + x(x + y) for x = 2 and y = 4.

15. __________________________

16.

Evaluate 2x − y for x = 5 and y = 3.

16. __________________________

T-11

Name: Instructor:

Date: Section:

Chapter 1 Test Form E (cont’d) Find the perimeter and area of each figure. 17. __________________________

17. square

7 ft.

18.

18. __________________________ rectangle

2 yds.

8 yds. Translate the following phrases into numerical expressions. Use x to represent “a number.” 19.

The product of a number and three less than the number.

19. __________________________

20. The difference between a number and six.

20. __________________________

21. A bank’s phone center receives an average of 4,832 calls per day. How many calls do they receive in an average 5-day work week?

21. __________________________

22. A homeowner needs a new stove. One stove costs $486. Another stove costs $415. How much more expensive is the higher priced stove?

22. __________________________

23. Four teens attend a hockey game. Each ticket is $24. What is the total cost of the tickets?

23. __________________________

24. The high temperatures during one week in July were 92°, 90°, 93°, 95°, 90°, 98°, and 100°. Find the average daily temperature.

24. __________________________

25. Is 6 a solution of the equation 18 − 2 x = 6 ?

25. __________________________

T-12

Name: Instructor:

Date: Section:

Chapter 1 Test Form F 1. Add: 296 + 149 a. 345

b. 445

c. 443

d. 343

b. 212

c. 200

d. 218

b. 10,221

c. 10,301

d. 10,321

b. 49 R 12

c. 52

d. 43 R 6

b. 7776

c. 72

d. 216

b. 5

c. 0

d. 14

b. 25

c. 10

b. 8

c. 1

d. undefined

b. 0

c. 1

d. undefined

b. 0

c. 27

d. 149

c. 131

d. 59

c. 36

d. 32

c. 27,800

d. 27,000

2. Subtract: 297 – 89 a. 202 3. Multiply: 243 × 47 a. 11,421 4. Divide: 1044 ÷ 18 a. 58 5. Evaluate: 32 ⋅ 23 a. 36 6. Evaluate: 2 + 3(22 − 4) a. 2 7. Evaluate:

( 8 − 3) 2

a. 55

8. Divide: 0 ÷ 8 a. 0 9. Divide: 19 ÷ 19 a. 19 10. Divide: 27 ÷ 1 a. 1

11. Simplify: 3[3 + 2(4 + 5)] – 4 a. 44

b. 23

12. Simplify: 2[15 – (7 – 4)2] a. 12

b. 18

13. Round 27,830 to the nearest thousand. a. 28,000

b. 27,900

T-13

Name: Instructor:

Date: Section:

Chapter 1 Test Form F (cont’d) 14. Estimate the sum by rounding each number to the nearest hundred: 362 + 157 + 209 + 80 a. 600

b. 800

c. 700

d. 1000

c. 4

d. 5

b. 30

c. 65

d. 75

b. 45

c. 225

d. 900

15. Find the solution to 3 x − 10 = 2 . a. 2

b. 3

16. Evaluate 3(x + y)2 if x = 2 and y = 3 a. 39 17. Evaluate 5y2 if y = 3 a. 30

18. Find the area for a rectangle that is 12 feet long and 8 feet wide. a. 48 feet

b. 96 ft. 2

c. 40 feet

d. 9216 ft. 2

19. Find the perimeter of a square whose side is 26 inches. a. 676 in.

b. 104 in.

c. 52 in.

d. 670 in.

Translate the following phrases into mathematical expressions. Use x to represent “a number.” 20.

The sum of two times a number and 7

a. 2(x + 7)

b. 7x + 2

c. 2x + 7

d. 2x – 7

c. 5 + x – 7

d. 7 – 5 + x

21. Seven less than the product of a number and five. a. 7 – 5x

b. 5x – 7

22. Find the total number of aspirin in a case if one case has 36 bottles, and each bottle holds 250 pills. a. 9,000

b. 2,250

c. 11,000

d. 2,450

23. Find the average temperature for a week if daily highs are 35°, 29°, 56°, 52°, 50°, and 42°. a. 308°

b. 62°

c. 56°

d. 44°

24. Admission to a movie costs $9. A student wants to invite five friends to a movie. What is the total cost for the entire group to attend the movie? a. $45

b. $54

c. $35

d. $52

25. Fifteen tickets to a triple-A baseball game cost $195. What is the cost of a single ticket? a. $21

b. $14

c. $15

T-14

d. $13

Name: Instructor:

Date: Section:

Chapter 1 Test Form G 1. Add: 368 + 97 a. 456

b. 576

c. 465

d. 566

b. 49

c. 149

d. 59

b. 6626

c. 6436

d. 6236

b. 356

c. 382

d. 371

c. 12

d. 14

c. 512

d. 16

b. 45

c. 25

d. 21

b. 9

c. 6

d. 121

b. 36

c. 128

d. 24

b. 0

c. 21

d. 1

b. 11

c. 6

d. 24

c. 22,000

d. 22,580

2. Subtract: 921 – 872 a. 151 3. Multiply: 238 × 27 a. 6426 4. Divide: 16,732 ÷ 7 a. 420 R 44

5. Simplify: 2 + 4(8 − 5) a. 2 b. 18 6. Simplify: 4 ⋅ 42 a. 256

b. 64

7. Evaluate: 3 + 2(4 + 5) a. 16 8. Evaluate:

( 7 − 4 )2

a. 33 9. Evaluate:

2(42 + 4)

a. 40 10. Divide:

0 ÷ 21

a. undefined 11. Evaluate: a. 9

3(8 − 2) 2

12. Round 22,486 to the nearest thousand. a. 22,500

b. 23,000

13. Estimate the sum by rounding each addend to the nearest hundred. 215 + 684 + 318 + 79 a. 1,300 b. 1,400 c. 1,100 d. 1,200

T-15

Name: Instructor:

Date: Section:

Chapter 1 Test Form G (cont’d) 14. Estimate the sum by rounding each addend to the nearest hundred. 138 + 193 + 249 + 567 a. 1500

b. 1200

15. Evaluate 2 + x(x + 4) a. 19 16. Evaluate

c. 1300

d. 1100

c. 35

d. 23

c. 3/4

d. 7/6

if x = 3 and y = 2.

b. 15

3x − 7 if x = 3 and y = 2. 4y − 6

a. 1

b. 0

17. Write a mathematical expression for “seven increased by the difference of a number and twelve.” a. 7 + (x – 12)

b. 7x – 12

c. 7(x – 12)

d. 7 + x – 12

18. Write a mathematical expression for “twice the sum of a number and seven.” a.

2+ x 7

2x 7

c. 2x + 7

d. 2(x + 7)

19. Find the area of a rectangle with length of 12 inches and width of 5 inches. a. 17 in.

b. 34 sq. in.

c. 60 sq. in.

d. 144 sq. in.

20. Find the perimeter of a square whose side has length 10 feet. a. 44 feet

b. 20 feet

c. 100 feet

d. 40 feet

21. Vitamin C can be purchased in a bottle of 500 tablets. Each case contains 24 bottles. How many tablets of vitamin C are in a case? a. 12,000

b. 524

c. 7000

d. 120

22. Admissions to a museum costs $6 each. Find the total cost if the third grade class of 24 students attends a movie. a. $123

b. $30

c. $144

d. $132

23. A student earns $8 per hour for the first 40 hours of work in a week, and $9 for any hours above 40 hours. If the student works 43 hours this week, what will be the total pay? a. $347

b. $344

c. $387

d. $329

24. A marching band has 6 rows with 14 members in each row. How many members are in the band? a. 20

b. 84

c. 79

d. 74

25. State High School has 380 seniors, 215 juniors, 212 sophomores, and 182 freshmen. Find the total number of students at SHS. a. 889 T-16

b. 989

c. 998

d. 898

Name: Instructor:

Date: Section:

Chapter 1 Test Form H 1. Add: 67 + 137 a. 211

b. 204

c. 194

d. 214

b. 394

c. 316

d. 294

b. 1348

c. 2072

d. 2088

b. 65

c. 64

d. 74 R 20

b. 5

c. 49

d. 16

c. 10

d. 50

c. 4

d. 12

b. 51

c. 42

d. 75

b. 35

c. undefined

b. 0

c. undefined

b. 72

c. 36

d. 144

c. 21

d. 27

c. 43,000

d. 41,000

2. Subtract: 762 – 478 a. 284 3. Multiply 74 × 28 a. 1794 4. Divide: 2368 ÷ 37 a. 78 R 32 5. Simplify: (32 – 2)2 a. 77 6. Simplify:

3 + 4 (8 − 6 )

a. 11

b. 14

7. Simplify: 8 + 16 ÷ 4 ⋅ 2 a. 16

b. 10

2 2 8. Simplify: 3 ( 6 − 3) + ( 5 − 1)   

a. 66 9. Divide:

35 ÷ 0

a. 0 10. Divide:

0 ÷ 21

a. 21 11. Evaluate: 32 ⋅ 24 a. 1296

12. Simplify: 3 + 4(4 + 2) a. 42

b. 30

13. Round 42,518 to the nearest thousand. a. 42,000

b. 40,000

T-17

Name: Instructor: Chapter 1 Test Form H (cont’d)

Date: Section:

Estimate each sum or difference by rounding each number to the nearest hundred. (Round, then add or subtract.) 14. 782 + 658 + 478 a. 1900

b. 2000

c. 2100

d. 1800

b. 600

c. 400

d. 560

c. 15

d. 49

c. 44

d. 48

15. 897 – 337 a. 500

16. Evaluate x + (x2 + 3) if x = 4. a. 53

b. 23

17. Evaluate x + 3(x + 4) if x = 4. a. 28

b. 60

18. Find the perimeter of a rectangle with length 16 feet and width 24 feet. a. 40 ft. 19.

b. 384 ft.

c. 80 ft.

d. 60 ft.

Find the area of a rectangle with length 15 cm and width 12 cm.

a. 27 cm2

b. 180 cm2

c. 54 cm2

d. 152 cm2

Translate the following phrases into mathematical expressions. Use x to represent “a number.” 20.

The quotient of five times a number and twenty-three.

a. 5 x ÷ 23

b. 5 x + 23

c. 5 ( x ÷ 23)

d. 5 x (23)

c. (7⋅3) – 5

d. 5 −

21. Five less than the quotient of seven and three. a. 22.

7−5 3

7 −5 3

7 3

The difference between three times a number and seven.

a. 3x + 7

b. 3x – 7

c. (3x)7

3x 7

23. A grocery store sold 74 cases of soda. Each case contains 24 cans. How many cans of soda were sold? a. 98

b. 1,728

c. 1,628

d. 1,776

24. A bank balance on Monday is $192. On Tuesday, $318 was deposited into the account, and checks for $27 and $312 were written. What is the account balance after the transactions? a. $261 b. $181 c. $171 d. $251 25. An employee earns $28,740 a year. Find their average monthly income. a. $2,395 b. $23,915 c. $2,391 d. $2,386

T-18

Name: Instructor:

Date: Section:

Chapter 2 Pretest Form A 1. A coal miner works 1400 feet underground. Represent as an integer.

1. _______________________

2. The high school football team gained 7 yards on a play. Represent this quantity as an integer.

2. _______________________

3. Insert < or > to make the statement true. |−7| −|5|

3. _______________________

4. Insert < or > to make the statement true. |4| |−6|

4. _______________________

5. Simplify: | −5|

5. _______________________

6. Add: −9 + (−3)

6. _______________________

7. Add: (–5) + 6 + (−12) + 3

7. _______________________

8. Evaluate: x + y if x = 2 and y = −3

8. _______________________

9. Subtract: 9 − (−4)

9. _______________________

10. Subtract: −5 − (−3) – 8

10. _______________________

11. Subtract −7 from −2.

11. _______________________

12. Multiply: (−5)(−3)

12. _______________________

13. Multiply: −2(6)0

13. _______________________

14. Simplify: (−3)3

14. _______________________

15. Multiply: −2(6)(−3)

15. _______________________

16. Divide: 35 ÷ (−7)

16. _______________________

17. Evaluate: xy if x = −4 and y = −3

17. _______________________

18. A football team lost 3 yards on each of three plays. What is the total number of yards lost?

18. _______________________

19. Simplify: 2(−3) + (– 2)2

19. _______________________

20. Simplify: |3 – 12| ⋅ (4) ÷ 4

20. _______________________

T-19

Name: Instructor:

Date: Section:

Chapter 2 Pretest Form B 1. The Colts (high school football team) gained 8 yards on a play.

1. _______________________

Represent this quantity as an integer. 2. The Dow Jones stock market average fell 196 points in one day. Represent this as an integer.

2. _______________________

3. Insert < or > to make the statement true. −(−18) 15

3. _______________________

4. Insert < or > to make the statement true. –|−8| |8|

4. _______________________

5. Simplify: −(−5)

5. _______________________

6. Add: 14 + (–3)

6. _______________________

7. Add: − 8 + 3 + (− 7) + (− 5)

7. _______________________

8. Evaluate: x + y if x = −3 and y = −9

8. _______________________

9. Subtract: −4 − (−12)

9. _______________________

10. Subtract: −6 – (−2) – 3

10. _______________________

11. Subtract 5 from −19.

11. _______________________ 12. _______________________

12. Multiply: 4(−7) 13. Multiply: −3(−8)

13. _______________________

14. Simplify: (−2)5

14. _______________________

15. Multiply: 7(−4)11

15. _______________________

16. Divide: −48 ÷ (−8)

16. _______________________

17. Evaluate: xy if x = −4 and y = −7

17. _______________________

18. The Eagles football team lost 5 yards on each of three plays. What is the total number of yards lost?

18. _______________________

19. Simplify: 2(−6) + 23

19. _______________________

20. Simplify: |8 – 4| ⋅ (−3) + 6

20. _______________________

T-20

Name: Instructor:

Date: Section:

Chapter 2 Test Form A Simplify each expression. 1. −6 + 11

1. __________________________

2. 20 – 32

2. __________________________

3. 6 ⋅ (−3)

3. __________________________

4. (−12) + (−9)

4. __________________________

5. 63 ÷ (−9)

5. __________________________

6. (−3)(−8)

6. __________________________

−54 −6

7. __________________________

8. |−15|

8. __________________________

9. −(−52)

9. __________________________

10. |5| ⋅ |−2|

10. __________________________

11. (−18) + 3 ⋅ (−4)

11. __________________________

12. (−7) + 2 + (−5) – (−3)

12. __________________________

13. (−2)2 – 12 ÷ (−4)

13. __________________________

14. 3 – 2(8 – 5)2

14. __________________________

15.

−12 + 5(2) − 10 23 − 2

15. __________________________

T-21

Name: Instructor:

Date: Section:

Chapter 2 Test Form A (cont’d) 16.

10(−1) − (−3)(−2) −2

16. __________________________

−12 6 Evaluate each expression for x = 0, y = −3, and z = 4.

17.

17. __________________________

18. 2x + y

18. __________________________

19. |y| + |x + z|

19. __________________________

20. 3 – y

20. __________________________

21. Carlos has $318 in his checking account. He deposits $207 and writes checks for $212 and $57. What is his new balance?

21. __________________________

22. The mountain peak is 1580 feet above sea level. Express this with a positive or negative notation.

22. __________________________

23. Find the average of −6, −8, 4, and 6.

23. __________________________

Solve. 24. −3x = 21

24. __________________________

25. x + 3 = 12

25. __________________________

T-22

Name: Instructor:

Date: Section:

Chapter 2 Test Form B Simplify each expression. 1. −5 + 10

1. __________________________

2. 14 – 35

2. __________________________

3. −4(−3)

3. __________________________

4. (−2) + (−9)

4. __________________________

5. (−24) ÷ (3)

5. __________________________

6. −11 – (−4)

6. __________________________

7. (−12)(−3)

7. __________________________

−25 −5

8. __________________________

9. −|−5|

9. __________________________

10. −(−12)

10. __________________________

11. (5)|−2|

11. __________________________

12. (−5)(2) ÷ (−2)

12. __________________________

13. (−7) + (−3) – 3 – 12

13. __________________________

14. (−2)3 – 2(−3)

14. __________________________

15. 8 – (5 + 2)2

15. __________________________

16.

( −2)( −5) − ( −2)( −7) 22

16. __________________________

T-23

Name: Instructor:

Date: Section:

Chapter 2 Test Form B (cont’d) 17.

−6 6

Name: __________________________ 17. __________________________

Evaluate each expression for x = 3, y = −2, and z = −3. 18. 3x − y

18. __________________________

19. |y| + 2(z)

19. __________________________

20. xy + z

21. __________________________

21. Akiko has $541 in her checking account. She wrote a check for $214 and a second check for $27. What is her new balance?

23. __________________________

22. A gambler wins $150 the first night and loses $185 the second night. Represent these wins and losses with positive or negative numbers. What is the overall outcome?

22. __________________________

23. Find the average of −12, −18, 5, 7, and 3.

23. __________________________

Solve. 24. x − 11 = −4

24. __________________________

25. −4x = −36

25. __________________________

T-24

Name: Instructor:

Date: Section:

Chapter 2 Test Form C Simplify each expression. 1. −7 + 11 − 4

1. __________________________

2. 26 – 42

2. __________________________

3. (−5)(−1)(3)

3. __________________________

−54 9

4. __________________________

5. −2 + 9 – (−6)

5. __________________________

6. −|−2|

6. __________________________

7. 25(−3)(0)

7. __________________________

8. (−3)4

8. __________________________

9. (−3)(−5)

9. __________________________

10. (3 – 5)3

10. __________________________

11. (−1)7

11. __________________________

12. (3)(−12) ÷ (−2)2

12. __________________________

13. (−3)3 – 2(−2)2

13. __________________________

14. 8 – 5(3 – 6)

14. __________________________

15. (5 – 8)3

15. __________________________

16. 28 ÷ (−7) + 23

16. __________________________

T-25

Name: Instructor:

Date: Section:

Chapter 2 Test Form C (cont’d) 17.

−5 −(−5)

17. __________________________

Evaluate each expression for x = −2, y = 3, and z = 1. 18. (x − y)2

18. __________________________

19. 3z + x

19. __________________________

20. x2 − y(z)

20. __________________________

21. The high on Saturday was 7°, the low of Saturday night was 4° below zero. Find the difference between the high and low temperature.

21. __________________________

22. Kwasi wins $50 Friday night playing cards. He loses $63 on Saturday night. What are Kwasi’s overall winnings or losses.

22. __________________________

23. Find the average of −12, 8, 16, and 24.

23. __________________________

Solve. 24. x + 15 = 9

24. __________________________

25. 7x = –21

25. __________________________

T-26

Name: Instructor:

Date: Section:

Chapter 2 Test Form D Simplify each expression. 1. −9 + 17

1. __________________________

2. 12 – 25

2. __________________________

3. 4 ⋅ (−20)

3. __________________________

4. (−30) ÷ 5

4. __________________________

5. (−17) + (−8)

5. __________________________

6. −5 – (−12)

6. __________________________

7. (−8) ⋅ (11)

7. __________________________

−33 −11

8. __________________________

9. 18 − |−5|

9. __________________________

10. |3| ⋅ |−7|

10. __________________________

11. (−2) + 12 ÷ (−2)

11. __________________________

12. −5 + (−2) – 7 – 4 + 3

12. __________________________

13. (−2)3 ⋅ (3)2

13. __________________________

14. (−3)2 – 5(3 – 7)

14. __________________________

15. (6 – 9)3

15. __________________________

16. 28 ÷ (−4) + 2(−5)

16. __________________________

T-27

Name: Instructor:

Date: Section:

Chapter 2 Test Form D (cont’d) 17. 25 ÷ (−5)2

17. __________________________

Evaluate each expression for x = 2, y = −3, and z = −4. 18. 3x + 2y

18. __________________________

19. |x| + |y| + |z|

19. __________________________

20. xy + |z|

20. __________________________

21. A mountain climber is at an elevation of 12,280 feet. He moves down the mountain 250 feet. Represent his final elevation as a sum and find the sum.

21. __________________________

22. The temperatures for a week were 5°, −2°, 0°, −4°, 3°, −1°, and 6°. Find the average temperature.

22. __________________________

Solve. 23. 3x = 2x + 5

23. __________________________

24. x + 5 = 17

24. __________________________

25. −4x = −24

25. __________________________

T-28

Name: Instructor:

Date: Section:

Chapter 2 Test Form E Simplify each expression. 1. −15 + 9

1. __________________________

2. 21 – 35

2. __________________________

3. (−5) ⋅ (–9)

3. __________________________

4. 85 ÷ (−17)

4. __________________________

5. (−5) + (−3)

5. __________________________

6. −7 – (−4)(2)

6. __________________________

38 −2

7. __________________________

8. (−6) + 12 ÷ (−3)(−2)

8. __________________________

9. (−4)2(−1)3

9. __________________________

10. |−2| ⋅ |−5|

10. __________________________

11. −7 + (−32) − 4

11. __________________________

12. (−2)3 – 24 ÷ (3)

12. __________________________

13. 3 – (2 – 8)2

13. __________________________

14. 15 ÷ (−3) ⋅ 5 + (−2)

14. __________________________

15. |−15| + (−15)

15. __________________________

16. (5 – 9)2

16.__________________________

T-29

Name: Instructor:

Date: Section:

Chapter 2 Test Form E (cont’d) 6 42 − 3 8

17.

17. __________________________

Evaluate each expression for x = 0, y = −2, and z = 3. 18. |x| − |y|

18. __________________________

19. (3x + y)2

19 __________________________

4y 2z − y

20. __________________________

20.

21. Find the average of the following numbers: 5, −3, 12, −7, 8

21. __________________________

22. Navi has $214 in her checking account. She deposits $88 and writes checks for $47, $29, and $150. What is her new balance?

22. __________________________

Solve. 23. x − 32 = 17

23. __________________________

x =8 4

24. __________________________

25. ⋅4x = 3x − 1

25. __________________________

24.

T-30

Name: Instructor:

Date: Section:

Chapter 2 Test Form F Simplify each expression and choose the correct response. 1. (−8) + (−4) a. 12

b. −12

c. 4

d. −4

b. 52

c. 26

d. −26

b. −4

c. −40

d. 40

b. −5

c. 125

d. 15

b. −14

c. 0

d. 1

b. 20

c. 4

d. −4

b. −40

c. 45

d. −45

b. −1

c. 0

d. −16

b. −3

c. 7

d. −7

b. −12

c. −3

d. 3

b. −4

c. −8

d. 8

b. −40

c. 40

d. 20

2. − 13 – 39 a. −52 3. (−20)(2) a. 4 4. (−25) ÷ (5) a. 5 5. −7 + (−7) a. 14 6. −8 + 12 a. −20 7. (−5)(−8) a. 40 8.

−8 8

a. 1 9. |5| − |−2| a. 3 10.

−18 − −6

a. 12 11. (−5) + 15 ÷ 5 a. −2 12. −10 + (−29) – 7 + 6 a. 52

T-31

Name: Instructor:

Date: Section:

Chapter 2 Test Form F (cont’d) 13. (−2)2 – 16 ÷ (−4) a. 3

b. 8

c. −8

d. 0

b. 64

c. −64

d. 32

b. −8

c. 4

d. 8

b. 17

c. −17

d. −20

b. 25

c. −2

d. 2

b. −10

c. 10

d. −2

b. 0

c. 25

d. undefined

b. −13

c. 1

d. 13

14. (4 – 8)2 ⋅ (3 – 5)2 a. −32 15. (3 – 5)2 + (−4) a. 0 16. 10 – (5 – 2)3 a. 20 17. −5 + (−45) ÷ (−15) a. −25 18. 6 − 42 a. 2 19. 0 ⋅ (−5)(−5) a. −25 20.

−6(−2) + 14 −2(−3 + 4)

a. −1

21. A mountain climber is at an elevation of 5,280 feet and moves down the mountain a distance of 420 feet. Her final elevation is: a. 5700 feet

b. −5700 feet

c. −4860 feet

d. 4860 feet

22. The Dow Jones Industrial Average fell 210 points on Monday. Represent by a signed number the total number of points it fell on Tuesday if it fell two times as much as on Monday. a. −420 points

b. 420 points

Solve. 23. Find the average of −20, 6, 9, 8, and 12. b. 15 a. −5

c. 210 points

d. −210 points

c. 3

d. −3

24. x + 5 = −7 a. −12

b. −2

c. 2

d. 12

25. −4x = 48 a. 12

b. −12

c. −44

d. −52

T-32

Name: Instructor:

Date: Section:

Chapter 2 Test Form G Simplify each expression and choose the correct response. 1. 36 – 39 a. 39

b. −3

c. −39

d. 3

b. 9

c. 45

d. −45

b. 150

c. −15

d. 15

b. 10

c. 1000

d. −1000

b. 6

c. 8

d. −8

b. −23

c. 23

d. 7

b. −28

c. 11

d. −11

b. −9

c. 20

d. −20

b. −7

c. 14

d. −14

b. −2

c. 12

d. −12

b. −5

c. −45

d. −3

b. −8

c. 8

d. 3

2. (6 – 9)2 a. −9 3. (−30) ⋅ (−5) a. −150 4. (−100) ÷ (10) a. −10 5. (−2)3 a. −6 6. –8 – (−15) a. −7 7. (−4) ⋅ (−7)(1) a. 28 8.

−18 −2

a. 9 9. |−28| + (−14) a. 7 10. 7 − |−5| a. 2 11.

−15 − −3

a. 5 12. −5 + 12 ÷ (−4) a. −3

T-33

Name: Instructor:

Date: Section:

Chapter 2 Test Form G (cont’d) 13. (−4) + (−8) – 12 + 2 a. 22

b. −22

c. 2

d. −2

b. −9

c. −26

d. −34

b. −96

c. −48

d. 96

b. −8

c. undefined

d. 8

b. −1

c. 12

d. −12

b. 1

c. −4

d. −22

b. −54

c. 4

d. −4

b. 9

c. −3

d. −9

14. −25 ÷ (−5) + (22) a. 9 15. (4 – 8)2 ⋅ (3 – 9) a. 48 −8 0

16. a. 0

17. (3 – 5)2 + (−3) a. 1 18. −1 + (14) ÷ (7) a. 3 19. 10 − 82 a. 54 20.

−5( −1) + 22 −3

a. 3

21. Two divers are exploring the bottom of a trench in the Atlantic Ocean. Jim is 146 feet below the surface of the ocean. Represent Richard’s depth by a signed number if he is two times as deep as Jim. a. 144 feet

b. 148 feet

c. 292 feet

d. −292 feet

22. Rashid has $250 in his checking account. He writes one check for $99 and another for $22, and deposits $150. His final balance is: a. $177 b. $377 c. $187 d. $279 23. Find the average of −12, −15, −30, and 5. a. 52 b. −52

c. 13

d. −13

24. Solve. –3x = −6 a. 2

b. −3

c. 3

d. −2

25. Solve. x + 12 = −18 b. 6 a. −30

c. −6

d. 30

T-34

Name: Instructor:

Date: Section:

Chapter 2 Test Form H Simplify each expression and choose the correct response. 1. −12 + 8 + (−3) a. 7

b. −7

c. 23

e. −17

b. 18

c. −18

d. −4

b. −100

c. 100

d. 25

b. 128

c. −36

d. 8

b. −14

c. 28

d. 14

b. 16

c. 0

d. 1

b. −44

c. 15

d. −15

b. 3

c. 6

d. −12

b. −25

c. 29

d. −29

b. −14

c. 14

d. −24

b. 85

c. −85

d. 12

b. 4

c. −17

d. −4

c. 0

d. 1

2. 7 – (−11) a. 4 3. (−50) ⋅ (2) a. −25 4. (−32) ÷ (−4) a. −8 5. −21 + 7 a. −28 6. −8 – (−8) a. −16 7. (11)(−4) a. 44 8.

−9 −3

a. −3 9. |−27| + (2) a. 25 10. 19 − |−5| a. 24 11. |17| ⋅ |−5| 17 a. 5 12. (8) + 12 ÷ (−3) a. 17

13. −3 + (−4) – 3 + (–10) a. 20 b. −20

T-35

Name: Instructor:

Date: Section:

Chapter 2 Test Form H (cont’d) 14. (2)2 – 21 ÷ (−3) a. 11

b. 14

c. −14

d. −51

b. −16

c. 8

d. −8

b. −20

c. 27

d. −27

b. 12

c. undefined

d. −12

b. 5

c. −5

d. −23

b. 14

c. 2

d. −14

b. 15

c. −15

d. −1

15. (1 – 3)2 + (−3)(−4) a. 16 16. 7 – (9 – 6)3 a. 20 0 − −12

17. a. 0

18. −9 + (−21) ÷ (−3)(2) a. 23 19. −8 – (−6) a. −2 20.

−2(−7) + 8(2) −2(4 − 5)

a. 1

21. A photographer took a picture of two mountains that had a difference in elevation of 1200 feet. If the taller mountain was 5450 feet, find the elevation of the other mountain. a. 4250 feet

b. 4450 feet

c. 6650 feet

d. 6850 feet

22. A card player lost $24 for each of four nights. Find her total losses. a. −96

b. 28

c. −28

d. 96

b. −5

c. −5

d. −20

b. 9

c. 4

d. −4

b. 5

c. −19

d. 19

23. Find the average of −8, −12, 10, and 10. a. 0 24. Solve. −3x = –12 a. 15 25. Solve. x − 7 = 12 a. −5

T-36

Name: Instructor:

Date: Section:

Cumulative Test Form A Chapters 1 and 2 Simplify each expression. 1. −8 − (−12)

1. __________________________

2. (−15) ÷ (−3)

2. __________________________

3. |−15| + (−7)

3. __________________________

−25

4. __________________________

−(−5)

5. −7 + (−21) – 12 + 6

5. __________________________

6. (2 – 5)2 ⋅ (3 – 5)2

6. __________________________

7. __________________________

− −7

8. 22 ⋅ 33

8. __________________________

9. (24 – 3) ⋅ 5

9. __________________________

Evaluate when x = 1, y = −2, and z = −3. 10. 3y − z

10. __________________________

11. |xy| + |z|

11. __________________________

12. 14 – 7x

12. __________________________

3x − 5 2x

13. __________________________

13.

14. 5(y2 – 2)

14. __________________________

T-37

Name: Instructor:

Date: Section:

Cumulative Test Form A Chapters 1 and 2 (cont’d) 5 cm 15. Find the perimeter of the square.

15. __________________________ 17 yds.

16. Find the perimeter of the rectangle.

5 yds.

16. __________________________

Translate the following into mathematical expressions. Use x to represent “a number.” 17. The product of three and a number.

17. __________________________

18. Twice a number subtracted from eleven.

18. __________________________

Evaluate. 19. −(−5)2 ÷ 5 ⋅ (−3)

19. __________________________

20. −|−9|

20. __________________________

21. −(−17)

21. __________________________

22.

0 24

22. __________________________

23.

−20 20

23. __________________________

24. A marching band has 15 rows with 8 members per row. How many members are in the band?

24. __________________________

25. A new washing machine costs $429. A new dryer costs $318. How much more expensive is the washer than the dryer?

25. __________________________

T-38

Name: Instructor:

Date: Section:

Cumulative Test Form B Chapters 1 and 2 Simplify each expression. 1. −3 – 27 a. 24

b. −24

c. −30

d. 30

b. 7

c. 5

d. 6

b. −16

c. 40

d. −40

b. −17

85 5

d. 15

b. 26

c. 14

d. −14

b. 16

c. 24

d. 48

b. −105

c. 105

d. −32

b. 144

c. 48

d. 56

b. 14

c. 28

d. −14

b. 4

c. 7

d. −4

11. (2x + y)2 a. 16

b. −16

c. 0

d. 8

12. 5 – 2x + z a. −3

b. 12

c. 9

d. −12

2. −45 ÷ 9 a. −5 3. −|12| + (−28) a. 16 4.

− −85 − ( −5)

a. 17 5. −12 + (−8) + 6 a. −26 6. (3 – 7)2 ⋅ (4 – 5)3 a. −16 7. |35| ⋅ |−3| a. 32 8. 23 ⋅ 32 a. 72 9. (23 – 4) ⋅ 7 a. −28

Evaluate when x = −2, y = 0, and z = 3. 10. 2x – 3y a. −7

T-39

Name: Instructor:

Date: Section:

Cumulative Test Form B Chapters 1 and 2 (cont’d) 3x + y z a. 2

13.

b. 0

c. 3

d. −2

b. 19

c. −16

d. 16

14. 5x – z2 a. −19

15. Find the perimeter of a square with sides of length 15 inches. a. 225 in.

b. 45 in.

c. 60 in.

d. 30 in.

16. Find the perimeter of a rectangle with width 7 cm and length 14 cm. a. 21 cm.

b. 98 cm.

c. 56 cm.

d. 42 cm.

Translate the following into mathematical expressions. Use x to represent “a number.” 17. The product of seven and a number. a. x – 7

b. 7 – x

c. 7x

d. −7x

c. 3x(6)

d. 3x – 6

b. 2(x + 4)

c. 4x + 2

d. (4x)2

b. 41

c. 109

d. 59

b. 39

c. 16

d. −10

0 37 a. 37

b. 0

c. undefined

d. 1

−42 −42 a. 0

b. 1

c. 31

d. −1

18. Six less than three times a number. a. 6 – 3x

b. 3(x) + 6

19. Two more than the sum of a number and four. a. (x + 4) + 2 Evaluate. 20. −(−3)2 + 2(5)2 a. 29 21. −|−13| ⋅ 3 a. −39 22.

23.

24. A bottle of Vitamin C has 250 pills. How many Vitamin C pills are in five bottles? a. 1250 pills b. 255 pills c. 1200 pills d. 120 pills 25. A mountain peak has an elevation of 1850 feet. A second peak has an elevation of 1208 feet. How much higher is the taller peak than the second peak? b. 642 feet c. 3058 feet d. 3050 feet a. 650 feet T-40

Name: Instructor:

Date: Section:

Chapter 3 Pretest Form A Simplify by combining like terms. 1. 8x – x + 3x

1. _______________________

2. 3x – 2(5x + 5) + 4

2. _______________________

3. Multiply: −3(5x)

3. _______________________

4. Multiply: −4 (8x – 7)

4. _______________________

5. Find the perimeter of the rectangle shown.

2x cm.

5. _______________________

(3x + 1) cm. 6. Is −5 a solution to the equation x + 12 = −7 + x – 5?

6. _______________________

Solve. 7. x – 8 = –12

7. _______________________

8. 4x – 3x = 16 + (−1) + 3

8. _______________________

9. 5x + 12 – 4x = 8 + 14

9. _______________________

10. 5y = 2(2y – 3)

10. _______________________

11. 7x = −21

11. _______________________ 12. _______________________

12. −8x = −32 13. −2x = –12

13. _______________________

14. 6x – 8 = 4x + 2

14. _______________________

15. 4(x – 3) = 8

15. _______________________

Translate each phrase into a mathematical expression. If needed, use x to represent “a number.” 16. Seven less than the product of eight and a number.

16. _______________________

17. The quotient of fifteen and three times a number.

17. _______________________

Write each sentence as an equation using x as the variable, if needed. 18. The sum of −27 and 13 is −14.

18. _______________________

19. The product of a number and −7 is 21.

19. _______________________

Solve. 20. Seven less than twice the sum of a number and three is 11.

20. _______________________

T-41

Name: Instructor:

Date: Section:

Chapter 3 Pretest Form B Simplify by combining like terms. 1. −5x + 3x – x

1. _______________________

2. 5x – 3 + 4(x – 5)

2. _______________________

3. Multiply: −2(8y)

3. _______________________

4. Multiply: −2(5x + 3)

4. _______________________

5. Find the area of the rectangle shown.

3 ft.

5. _______________________

(2x – 5) ft. 6. Is −2 a solution to the equation 3x – 2x = −2?

6. _______________________

Solve. 7. y – 8 = 12

7. _______________________

8. x +18 = 13

8. _______________________

9. 5x + 17 – 4x = 17

9. _______________________

10. 3x = 2(x – 12)

10. _______________________

11. −9x = 18

11. _______________________ 12. _______________________

12. 5x = −30

13. _______________________

13. 5y + 18 + 3y = −14 + 8 Translate into a mathematical expression. Use x to represent “a number.” 14. A number increased by 12.

14. _______________________

15. The product of 3 and a number subtracted from 31.

15. _______________________

Solve and check. 16. 4(2x − 3) = 3(2x + 6)

16. _______________________

17. 3(x + 5) = −30

17. _______________________

Write each sentence as an equation. 18. The difference between 38 and −5 is 43.

19. _______________________

19. Seven less than 20 is 13. Solve. 20. Three times the sum of a number and two is thirty-nine.

T-42

18. _______________________

20. _______________________

Name: Instructor:

Date: Section:

Chapter 3 Test Form A Simplify. 1. 3x – 8 – 4x + 4

1. __________________________

2. −2 – x + 2y – 4y + 5x

2. __________________________

3. 3x – 8 + 7x + 9

3. __________________________

4. 2x + 5y – x – (−5y)

4. __________________________

5. −3(2x − 4)

5. __________________________

6 − (3 x − 7 )

6. __________________________

7. 2(5x + 2) – x

7. __________________________

8. 3(2x – 4) – 2(x − 4)

8. __________________________

Solve. 9. x + 6 = −16

9. __________________________

10. 8y = −16

10. __________________________

11. 3(x + 4) = 2x – 5

11. __________________________

12. 2x + 3x = 4x + 8

12. __________________________

13. −3x + 7 = 19

13. __________________________

14. 10y – 1 = 3y + 20

14. __________________________

15. 6 + 2(2x – 3) = 12

15. __________________________

16. 4(2x + 3) = 2(5x + 2)

16. __________________________

T-43

Name: Instructor:

Date: Section:

Chapter 3 Test Form A (cont’d) 17. Find the perimeter.

17. __________________________

(2x + 5) cm

(2x + 5) cm (x + 7) cm

18. Find the perimeter.

18. __________________________ 2x inches x inches

19. The sum of three times a number and five is twenty-three. Find the number.

19. __________________________

20. In a competition, there are 56 more professionals than amateurs. Find the number of amateurs participants if the total number of participants is 138.

20. __________________________

21. A rectangular room measures 10 feet by 12 feet. How much trim is needed to go around the ceiling?

21. __________________________

Find the area of the rectangle. 22.

22. __________________________ 5 feet (3x – 2) feet

23. Is 3 a solution to the equation 4(2x – 5) = 7?

23. __________________________

Translate each into an equation and solve.

24. Five times a number added to twelve is twenty-seven.

24. __________________________

25. Sixteen subtracted from twice a number is equal to the sum of the number and ten.

25. __________________________

T-44

Name: Instructor:

Date: Section:

Chapter 3 Test Form B Simplify. 1. 3x – (−2x) + 7x

1. __________________________

2. 11a – b + 3a – (−7b)

2. __________________________

3. 3x – (−2x) + 5

3. __________________________

4. −a + 3b + 6a – 3b

4. __________________________

5. 2(4x – 3)

5. __________________________

6. −(x – 7)

6. __________________________

7. 2(x − 3) – (x – 7)

7. __________________________

8. 3(2x – 5) + 6x

8. __________________________

Solve. 9. x – 7 = −8

9. __________________________

10. −4y = 12

10. __________________________

11. 2(x – 4) = x – 10

11. __________________________

12. 3x − 2x = 4

12. __________________________

13. −6x + 4 = − 8

13. __________________________

14. 7y + 1 = 5y + 19

14. __________________________

15. 3 + 2(x – 4) = 13

15. __________________________

16. 2(x + 3) = 3(3x − 5)

16. __________________________

T-45

Name: Instructor:

Date: Section:

Chapter 3 Test Form B (cont’d) 17. Find the perimeter of the rectangle.

17. __________________________ (x + 2) inches

(3x – 5) inches 18. Find the perimeter of a square with sides of length (3x + 2) centimeters.

18. __________________________

19. Is 3 a solution to −2(x + 5) = −x – 12?

19. __________________________

20. Is –1 a solution to 5 – (x + 3) = –2x + 1

20. __________________________

Write each sentence as an equation.

21. Six subtracted from a number is twenty-two.

21. __________________________

22. The quotient of a number and four is equal to sixteen.

22. __________________________

Solve.

23. The product of twice a number and three is 36. Find the number.

23. __________________________

24. Three times the sum of a number and five is negative forty-five. Find the number.

24. __________________________

25. In the high school band, there are 27 more brass players than percussionists. Find the number of percussionistss if the band has a total of 97 members.

25. __________________________

T-46

Name: Instructor:

Date: Section:

Chapter 3 Test Form C Simplify. 1. 3a − 2 + 4a – 7

1. __________________________

2. 2y + 8 + 3y + 6

2. __________________________

3. 2(y – 3) – 4 + 2y

3. __________________________

4. −x – 9x − y + 3y

4. __________________________

5. −2(x + 4)

5. __________________________

6. 3(3a − b)

6. __________________________

−5 ( 2 x + 3 ) − ( x − 2 )

7. __________________________

8. 3(2y – 1) – 4(y – 1)

8. __________________________

Solve. 9. x + 3 = −4

9. __________________________

10. −x = −4

10. __________________________

11. −4x = −12

11. __________________________

12. 7x − 6x = 9

12. __________________________

13. x + 3 = 2x – 7

13. __________________________

14. 2y + 1 = 5(y − 4)

14. __________________________

15. 3 − 5 + 2x = x – 6

15. __________________________

16. 2(x − 1) = x + 11

16. __________________________

T-47

Name: Instructor:

Date: Section:

Chapter 3 Test Form C (cont’d) 17. 2x – 5x = 18

17. __________________________

18. 4y + 2 = 3y – 13

18. __________________________

19. Is 8 a solution to the equation −4 x + 20 = −4 − x

19. __________________________

20. Find the perimeter of the rectangle.

20. __________________________ (x – 3) feet

(2x + 7) feet 21. Find the perimeter of a square with sides of length (2x + 5) meters.

21. __________________________

Solve. 22. The difference between a number and seven is thirteen. Find the number.

22. __________________________

23. The product of three and a number is eighty-one. Find the number.

23. __________________________

24. Twice the sum of a number and eight is the same as the number minus four.

24. __________________________

25. Philia has twice as much money as Zimo. If they have a total of $27, find out how much Philia has.

25. __________________________

T-48

Name: Instructor:

Date: Section:

Chapter 3 Test Form D Simplify. −2 x − x + 7 x

1. __________________________

2. 4b – 3b + 6b − 5

2. __________________________

8 a − 4b + 2 a + 6 b

3. __________________________

4. 19x – 2 + 11x + 9

4. __________________________

5. −4(3y − 8)

5. __________________________

6. 8(2x − 7)

6. __________________________

7. 3(2x + 5) – x – 7

7. __________________________

8. 3(5y + 3) – 9y – 13

8. __________________________

Solve. 9. x – 7 = −3

9. __________________________

10. 3(y + 1) = 6

10. __________________________

11. 7(7x + 5) = 6(8x + 3)

11. __________________________

12. −5x + 4x = 16 + 7

12. __________________________

13. 7x − x = 42

13. __________________________

14. 5x – 1 = 24

14. __________________________

15. 2(5x + 4) = 3(2x + 8)

15. __________________________

16. 12x – 7 – 11x = 6 + (−3)

16. __________________________

T-49

Name: Instructor:

Date: Section:

Chapter 3 Test Form D (cont’d) 17. 4x – 5 = 3x + 17

17. __________________________

18. Twice the sum of a number and eight is twenty. Find the number.

18. __________________________

19. The difference between a number and seven is seventeen. Find the number.

19. __________________________

20. It costs $5 to join a video rental club. How many videos did a patrion rent if they spent $23, and it costs $3 to rent each video?

20. __________________________

21. The length of a rectangle is three times the width. If the perimeter is 96 inches, find the length and the width.

21. __________________________

22. Is 8 a solution to 3(x + 2) = 2x + 15?

22. __________________________

23. Is −2 a solution to −3(x – 5) = 21?

23. __________________________

24. Find the perimeter of a square with sides of length 3x

24. __________________________

25. Find the perimeter of the triangle.

25. __________________________

3x – 5

x 2x + 3

T-50

Name: Instructor:

Date: Section:

Chapter 3 Test Form E Simplify. 1. 3x – 8 + 2x − 5

1. __________________________

2. 6y – 2 + 8y − 7

2. __________________________

3. 4x – 7y + x + 10y

3. __________________________

4. 4x − 2 − 7x + 5

4. __________________________

5. −5(3x − 7)

5. __________________________

6. −2(3y + 8)

6. __________________________

7. 2(x − 3)

7. __________________________

8. 6 x − 2 ( x − 8 )

8. __________________________

Solve and check. 9. x − 17 = −2

9. __________________________

10. −3y = −15

10. __________________________

11. 4x + 9 = −3

11. __________________________

12. 5x − 9 = −44 + 4x

12. __________________________

13. 15x + 4 − 16x = 33

13. __________________________

14. 3(2x + 9) = 3

14. __________________________

15. 9 + 2(7y – 4) = −27

15. __________________________

16. −11x = −121

16. __________________________

T-51

Name: Instructor:

Date: Section:

Chapter 3 Test Form E (cont’d) 17. 11x – 3 = 2(5x + 3)

17. __________________________

18. Find the area of the rectangle.

18. __________________________

7 cm (x + 2) cm 19. The sum of −11 and a number is 37. Find the number.

19. __________________________

20. Three times the differnce of a number and eighteen is ninety-six. Find the number.

20. __________________________

21. A bicyclist travels twice as fast as someone who is walking. The combined speed is 18 miles per hour. Find the speed of the bicyclist.

21. __________________________

22. The difference of 7 times a number and 4 times the same number is 21. Find the number.

22. __________________________

23. Is 9 a solution to 2x + 3 = 21?

23. __________________________

24. Is −5 a solution to x + 11 = 2x + 7?

24. __________________________

T-52

Name: Instructor:

Date: Section:

Chapter 3 Test Form F Simplify by combining like terms. 1. 5x − x + 4x a. 10x

b. 8x

c. 9x

d. 16x

b. 6a – 2b

c. 5a – 3b

d. 6a + 4b

b. 10x – 4

c. 10x + 4

d. 14x + 4

b. 24y – 6x + 3

c. 24y + 8x + 3

d. 20y – 6x − 3

b. 7x + 8

c. 12x + 32

d. 7x + 32

b. − x − 11

c. −5 x + 5

d. −5 x + 19

b. 2x + 17

c. x + 11

d. x + 17

b. 8

c. −8

d. −24

5 3

c. −5

d. 3

b. 5

c. 7

d. −7

b. −2

c. −1

d. 2

b. 12

12 7

d. 2

b. 12

c. –28

d. 3

2. a – 3b + 5a – b a. 6a – 4b 3. 12x – 2x – 3 + 7 a. 10x – 10 4. 22y – 7x + 3 – 2y + x a. 20y – 6x + 3 5. 4 ( 3x + 8 ) a. 12x + 8 6. −3 ( x − 4 ) − ( 2 x − 7 ) a. − x − 19 7. −(x + 3) + 2(x + 7) a. −3x + 10 Solve. 8. w + 16 = −8 a. 24 9. 3x + 5 = −10 a. 5

10. 7y = −35 a. −5 11. 2(x – 5) = 6x – 6 a. 1 12. 6x – 7x = 12 a. −12 13. 4(x – 5) = 3x − 8 a. –3

T-53

Name: Instructor: Chapter 3 Test Form F (cont’d)

Date: Section:

14. 2x + 5 = 3x – (−6) a. 1

b. −1

c. 5

d. 11

b. 2

c. −2

d. −3

15. 2(x – 5) + 22x = 38 a. 3

16. Write an expression for “twice the sum of a number and eight.” a. 2x + 8

b. 8 + x

c. 2(x + 8)

d. 16

17. Write an expression for “six less than three times a number.” a. x – 18

b. 6 – 3x

c. 3x – 6

d. 6 − x

18. Solve: Three less than twice a number is thirteen. Find the number. a. 5

b. 8

c. 16

d. 10

19. Solve: Twice the sum of a number and five is equal to ten. a.

5 2

b. −10

c. 0

d. 10

20. Juan and his brother collect baseball cards. Juan has seventeen more cards than his brother. Together they have 267 cards. How many cards does Juan have? a. 125

b. 250

c. 147

d. 142

21. A gambler lost some money on cards on Friday. On Saturday, she lost twice as much money. After the two nights of cards, she has lost $81. How much did she lose on Friday? a. $27

b. $40

c. $25

d. $81

22. A mountain climber was at 632 feet elevation. How many feet did the climber descend after climbing down to an elevation of 430 feet? a. 1062 feet

b. 202 feet

c. 230 feet

d. 232 feet

23. Is 3 a solution to the equation −3(x + 5) = −27? a. yes

b. no

24. Is −5 a solution to the equation 2x – 15 = 5x? a. yes

b. no

25. Find the perimeter of the rectangle.

2x 3x + 5 a. 5x + 5 T-54

b. 6x + 5

c. 10x + 20

d. 10x + 10

Name: Instructor:

Date: Section:

Chapter 3 Test Form G Simplify. 1. 12x – 3x + 2x a. 9x + 2

b. −9x + 2

c. 15x + 2

d. 11x

b. 18x

c. 12x

d. 16x

b. 14a – 3b

c. 15a + 3b

d. 15a – 9b

b. 5x – 5y

c. 5x – 4y

d. 5x – 6y

b. 7xy

c. −7xy

d. −10x + y

b. 5x + 15

c. 5x + 10

d. 5x – 5

b. −24x + 9

c. −24x + 7

d. −24x + 5

b. −5

c. 15

d. −15

b. 4

c. −4

d. −20

b. −9

c. 30

d. −30

b. 4

c. −4

b. −15

c. 1

d. 15

2. 9x –x + 8x a. 17x 3. 14a − 3b + a + 6b a. 15a – 3b 4. 2x – (−3x) + y – 5y a. −x – 4y 5. −2(5x) + 3y a. −10x + 3y 6. 2(3x + 5) – (x – 5) a. 5x 7. −2(12x) + 7 a. −24x – 14

Solve. 8. x − 7 = −12 a. −19 9. 5x + 12 – 4x = 8 a. 20 10. −3x = 27 a. 9 11. 6x + 7 = 3x − 5 a.

3 2

4 3

12. 3(5x – 7) = 2(7x – 3) a. 4

T-55

Name: Instructor: Chapter 3 Test Form G (cont’d)

Date: Section:

13. −2(3x) = −18 a. −1

b. 1

c. 3

d. −3

b. 16

c. −16

d. −10

b. 30

c. 34

d. 44

b. 0

c. −4

d. −10

b. 45

c. −8

d. 40

14. x –3 = 13 a. 10 15. 2x + 5 – (x + 7) = 32 a. 20 16. 6x + 5 = 5x + 5 a. 10 17.

15 x = 40 3

a. 8

18. The sum of a number and six is equal to negative fourteen. Find the number. a. 8

b. −20

c. −8

d. 20

19. Twice the sum of a number and ten is equal to six. Find the number. a. −2

c. −7

b. 8

d. 7

20. The length of a rectangle is twice the width. The perimeter is 72 meters. Find the length. a. 12 meters

b. 24 meters

c. 36 meters

d. 18 meters

21. Write an expression for “five less than twice a number.” a. 2x – 5

b. 2(x – 5)

c. 5 – 2x

d. 5(−2x)

22. Write an expression for “three times the sum of a number and five.” a. 3(x + 5)

b. 3x + 5

c. 3 + 5x

d. 15x

23. A farmer needs 70 feet of fencing material to make a pen. The length is to be 5 feet longer than the width. Find the width. a. 35 feet

b. 30 feet

c. 15 feet

24. Is −2 a solution to the equation 5x + 6 = 4x – (−4)? a. yes

b. no

25. Is 8 a solution to the equation 3x – 2 = 20 – (−2)? a. no

T-56

b. yes

d. 25 feet

Name: Instructor:

Date: Section:

Chapter 3 Test Form H Simplify. 1. 8 y − y + 3 y a. 11y

b. 5y

c. 10y

d. 12y

b. 8x – 12y

c. 2x – 12y

d. 2x + 12y

b. 20 x − 4 y

c. 20 x + 4 y

d. 10 x − 4 y

b. –8x

c. –10x

d. 8x

b. 8x – 5 + 7y

c. −2x – 5 + 7y

d. −2x + 7y – 15

b. −11x − 5

c. −11x – 15

d. −11x + 15

b. 4x + 14

c. 26x + 14

d. 4x + 7

b. 14x + 6

c. −40x + 6

d. −14x + 6

b. 14

c. −14

d. −40

b. 43

c. −43

d. −84

b. 6

c. −5

d. 5

b. 50

c. 28

d. −28

2. 5x – 12y + 3x a. 8x + 12y 3. 15 x − 2 y − ( −5 x ) + 6 y a. 10 x + 4 y 4. x – 4x –5x a. −7x 5. −(5x) + 3(x – 5) + 7y a. 8x + 5 + 7y 6. −3(2x – 5) – 5x a. x – 5

7. 13x – (−11x) + 2(x + 7) a. 24x + 14 8. 27x – (−13x) + 6 a. 40x + 6

Solve. 9. w − 13 = 27 a. 40 10. 2p = −86 a. −88 11. 2x – 12 = 3x – 17 a. 29 12. x – 11 – 2x = 39 a. −50

T-57

Name: Instructor:

Date: Section:

Chapter 3 Test Form H (cont’d) 13. 5(x – 7) = 3(x + 8) + x a. 43

b. 59

c. −59

d. −43

b. 26

c. 40

d. −40

b. −9

c. 9

d. −3

5 2

c. −4

d. 4

b. 15

c. 51

d. −51

14. 2x – (−7) = 3(x + 11) a. −26 15. 7(x – 3) = 42 a. 3 16.

8x − 3 = 13 2

a. −5

17. x − 18 = 33 a. −15

18. Solve: The sum of a number and twelve is equal to thirty-one. a. 43

b. 19

c. −19

d. −43

19. Solve: Three times the difference of a number and forty-five is equal to twice the same number. a. 135

b. 45

c. 9

d. −45

20. Find the length of a rectangle if the length is four less than twice the width and the perimeter is 40 cm. a. 12 cm

b. 8 cm

c. 20 cm

40 cm 3

21. Write an expression for “Eight less than four times a number.” a. 8 − 4x

b. 8 − 4

c. 8 x − 4

d. 4 x − 8

22. Write an expression for “five times the difference of a number and nine.” a. 45x

b. 5 x + 9

c. 5 ( x − 9 )

d. 5 x − 9

23. Hao and Luis collect baseball cards. Hao has twice as many cards as Luis. Together they have 1,263 cards. Find the number of cards Luis has. a. 421 cards

b. 420 cards

c. 842 cards

24. Is −2 a solution to the equation 11x – 4 = 7x – 14? a. yes T-58

b. no

d. 631 cards

Name: Instructor:

Date: Section:

Chapter 3 Test Form H (cont’d) 25. Is 5 a solution to the equation 4x = 3x + 5? a. yes

b. no

T-59

Name: Instructor:

Date: Section:

Chapter 4 Pretest Form A 1. Simplify:

3 0

1. _______________________

2. Simplify:

−15 15

2. _______________________

3. Write the prime factorization for 96.

3. _______________________

4 as an improper fraction. 5

4. _______________________

4. Write 3

5. Multiply and write your answer in simplest form.

3 10 ⋅ 8 15

 1  1  6. Multiply:  3   1   2  7 

6. _______________________

7. Divide and write your answer in simplest form.

8. Evaluate: x ÷ y if x = −

9. Write

5. _______________________

1 7 ÷ 3 x 9 xy

2 7 and y = − 5 6

5 as an equivalent fraction with 49 in the denominator. 7

7. _______________________

8. _______________________

9. _______________________

3 1 10. Add: 2 + 1 4 5

10. _______________________

3x 2 y + 2 3

11. _______________________

11. Add:

12. Subtract:

T-60

8 x 3x − 5 5

12. _______________________

Name: Instructor:

Date: Section:

Chapter 4 Pretest Form A (cont’d) 3  5  13. Add: − +  −  7  21 

14. Add: −

13. _______________________

1 5 + 4 8

15. Evaluate: 2x – y

16. Solve: −

14. _______________________

if x =

1 3 and y = 3 5

15. _______________________

2 7 +y=− 5 10

16. _______________________

7 17. Simplify the following complex fraction: 5 2x 15

18. Solve:

17. _______________________

x 2x = −3 5 5

18. _______________________

19. How many quarter-pound hamburgers can be made from 12

3 4

19. _______________________

pounds of hamburger?

3 feet long is cut from a 6 foot board. What is the length 5 of the piece that is left.

20. A piece 2

20. _______________________

T-61

Name: Instructor:

Date: Section:

Chapter 4 Pretest Form B 1. Simplify:

0 5

1. _______________________

2. Simplify:

54 6

2. _______________________

3. Write the prime factorization for 240.

3. _______________________

2 as an improper fraction. 3

4. _______________________

4. Write 13

5. Multiply and write your answer in simplest form. 

3   7  ⋅−   28   15 

 7  1  6. Multiply:  1   1   9  5 

6. _______________________

7. Divide and write your answer in simplest form.

8. Evaluate: x − y if x =

3 9 ÷ 2 x 4 xy

1 2 and y = − 4 3

3 3 9. Solve: − z = 5 25

10. Is

2x x + 3 2

12. Subtract:

T-62

7. _______________________

8. _______________________

9. _______________________

1 1 a solution to the equation 5 z = 3 + ? 2 2

11. Add:

5. _______________________

10. _______________________

11. _______________________

9 3 − 10 5

12. _______________________

Name: Instructor:

Date: Section:

Chapter 4 Pretest Form B (cont’d)  1  13. Add: −2 x +  − x   2 

13. _______________________

2 1 14. Add: 4 + 1 5 3

14. _______________________

15. Evaluate: x(y)

16. Write

if x =

4 −5 and y = 3 6

5 as an equivalent fraction with 48 in the denominator. 12

17. Simplify the following complex fraction:

18. Solve:

15. _______________________

1 4 8 x

16. _______________________

17. _______________________

x 3x = +2 7 14

18. _______________________

19. Find the perimeter of a rectangle with length 2

1 1 and width 1 . 5 3

3 feet long is cut into 4 equal pieces to make a book 5 shelf. What is the length of the shelves?

20. A board 13

19. _______________________

20. _______________________

T-63

Name: Instructor:

Date: Section:

Chapter 4 Test Form A Write each mixed number as an improper fraction. 1. 3

4 5

1. __________________________

2. 4

1 2

2. __________________________

Write each improper fraction as a mixed number or a whole number. 3.

76 7

3. __________________________

21 5

4. __________________________

Write each fraction in simplest form. 9 24

5. __________________________

45 185

6. __________________________

50 175

7. __________________________

5. −

Perform the indicated operations. Write answers in simplest form. 8.

3x 5 x + 2 4

8. __________________________

3 10 ⋅ 8 12

9. __________________________

10.

3 2 + 5 4

10. __________________________

11.

−2 4 ÷ 3 9

11. __________________________

1 2 12. 3 ⋅ 5 3 7

12. __________________________

13.

T-64

5  1  − − 6  18 

13. __________________________

Name: Instructor:

Date: Section:

Chapter 4 Test Form A (cont’d) Simplify the complex fraction. 2 3 x 14. 4 x

14. __________________________

13 xy 26 x 3y

15. __________________________

16.

2 1 x=− 3 3

16. __________________________

17.

1 5 x+ x = 3 12

17. __________________________

4 1 x = x+− 15 5

18. __________________________

3 1 7 x− x = 4 12 12

19. __________________________

15.

Solve.

18. −

19.

3 20. Evaluate −7y when y = − . 5

21. Evaluate xy

when x =

20. __________________________

1 1 and y = 2 . 2 3

21. __________________________

1 4 feet from a 8 foot long board. 3 5 What is the length of the remaining piece?

22. A carpenter cuts 1

22. __________________________

1 cups of sugar. How 3 much sugar is needed if the recipe is doubled?

23. A recipe for cookies asks for 2

24. A sports coat regularly sells for $120. It is on sale for

23. __________________________

1 3

24. __________________________

off. What is the new sale price? 25. Find the perimeter of the triangle.

25. __________________________ 3 ft. 8

3 ft. 8 1 ft. 8

T-65

Name: Instructor:

Date: Section:

Chapter 4 Test Form B Write each mixed number as an improper fraction. 1. 2

5 7

1. __________________________

2. 4

5 9

2. __________________________

Write each improper fraction as a mixed number or a whole number. 3.

93 8

3. __________________________

37 4

4. __________________________

Write each fraction in simplest form. 4 20

5. __________________________

85 130

6. __________________________

282 648

7. __________________________

5. −

Perform the indicated operations. Write answers in simplest form. 8.

3 2 + 5 3

8. __________________________

3 15 ⋅ 10 18

9. __________________________

2 2x + 3 9

10. __________________________

3 3x 11. − − 5 10

11. __________________________

 1 2 12.  1  ⋅    5  3

12. __________________________

4 x 3x − 5 7

13. __________________________

10. −

13.

T-66

Name: Instructor:

Date: Section:

Chapter 4 Test Form B (cont’d) Simplify the complex fraction. 2 3 14. 5 6

14. __________________________

2x 5 4x 15

15. __________________________

1 3 x=− 4 8

16. __________________________

x x +2= 13 13

17. __________________________

18.

3 x = 15 5

18. __________________________

19.

x = −24 8

19. __________________________

20.

1 9 − 2x = − 10 10

20. __________________________

15.

Solve. 16.

17. −

21. Evaluate x − y

when x = −

22. Evaluate 2x + y

if x =

3 4 and y = − . 5 5

1 4 and y = . 3 5

1 off the regular price. Find the 5 sale price of a sweater if the regular price is $65.

23. A sale has sweaters for

1 cups of water. How much water is 2 needed if you are making 4 times the recipe?

24. A recipe asks for 1

25. Find the area of a rectangle with width 1

4 ft. and length 7

21. __________________________ 22. __________________________

23. __________________________

24. __________________________

25. __________________________

2 feet. 7

T-67

Name: Instructor:

Date: Section:

Chapter 4 Test Form C Write each mixed number as an improper fraction. 1. 4

5 9

1. __________________________

2. 3

1 7

2. __________________________

Write each improper fraction as a mixed number or a whole number. 3.

38 5

3. __________________________

90 7

4. __________________________

45 31

5. −

5. __________________________

Write each fraction in simplest form. 6.

6 8

6. __________________________

45 125

7. __________________________

93 123

8. __________________________

8. −

Perform the indicated operations. Write answers in simplest form. 9. 2 −

2 5

9. __________________________

10. −

3 x −4 x + 7 7

10. __________________________

 1  3  11.  3     3  7 

11. __________________________

3 1 12. 1 ÷ 4 4 5

12. __________________________

13.

T-68

5 1 − 9 3

13. __________________________

Name: Instructor:

Date: Section:

Chapter 4 Test Form C (cont’d)  3  5  14.      7  6 

14. __________________________

Simplify the complex fraction.

15.

3 4x 5 2x

15. __________________________

16.

3x 8 2x 24 xy

16. __________________________

1 3 x= 4 4

17. __________________________

Solve. 17.

18. 5 x + 4 =

18. __________________________

x 12 = 4 5

19. __________________________

3 1 x = x+ 7 3

20. __________________________

1 3 x+2= 7 7

21. __________________________

19. −

20. −

21.

2 3

x y

if x =

3 1 and y = − . 5 3

22. __________________________

23. Evaluate 3x

if x = −

3 . 7

23. __________________________

22. Evaluate

2 feet long is cut into 3 equal pieces. What is 3 the length of the pieces?

24. A board 12

25. A dress which regularly sells for $85 is on sale for

1 off. 5

24. __________________________

25. __________________________

Find the new price of the dress.

T-69

Name: Instructor:

Date: Section:

Chapter 4 Test Form D Write each mixed number as an improper fraction. 1. 4

6 7

1. __________________________

3 8

2. __________________________

2. −2

Write each improper fraction as a mixed number or a whole number. 3.

65 9

3. __________________________

−51 5

4. __________________________

17 11

5. __________________________

Write each fraction in simplest form. 6.

18 72

6. __________________________

10 85

7. __________________________

8. −

111 240

8. __________________________

Perform the indicated operations. Write answers in simplest form. 9.

3 2x + 5 5

9. __________________________

10.

5 3 − 2x 2x

10. __________________________

 1  1  11.    1   6  5 

11. __________________________

7 5 ÷ 13 26

12. __________________________

x 14 ⋅ 7 20

13. __________________________

12. −

13.

T-70

Name: Instructor:

Date: Section:

Chapter 4 Test Form D (cont’d) 2  1 14. − −  −  3  5

14. __________________________

Simplify the complex fraction.

15.

1 2x 5 4x

15. __________________________

16.

7 12 4 3x

16. __________________________

4 x = −20 5

17. __________________________

18.

1 2 1 x+ x = 3 5 15

18. __________________________

19.

x = −5 3

19. __________________________

20.

5 1 = x− x 6 3

20. __________________________

Solve. 17. −

21. Evaluate x ÷ y if x =

3 1 and y = − . 5 3

22. Evaluate x – y if x = −

21. __________________________

1 2 and y = . 4 3

22. __________________________

1 3 feet piece is cut from a board that is 12 feet 5 5 long. What is the length of the remaining piece?

23. A 2

24. Find the area of a rectangle with width

3 cm. and length 4

23. __________________________

24. __________________________

1 1 cm. 5 1 cups of nuts is called for in a cookie recipe and we 2 want to double the recipe, how many cups of nuts will be needed?

25. If 1

25. __________________________

T-71

Name: Instructor:

Date: Section:

Chapter 4 Test Form E Write each mixed number as an improper fraction. 1. 5

4 9

1. __________________________

2 11

2. __________________________

2. −6

Write each improper fraction as a mixed number or a whole number. 3.

34 7

3. __________________________

45 8

4. __________________________

92 9

5. __________________________

Write each fraction in simplest form. 49 6. 70 7.

6. __________________________

35 175

8. −

7. __________________________

6 522

8. __________________________

Perform the indicated operations. Write answers in simplest form. x 3

9. __________________________

1 4 2 10. 2 + 1 + 3 3 5 3

10. __________________________

11.

5a 24 ⋅ 8 45a

11. __________________________

12.

−16 −32 ÷ 5 45

12. __________________________

3  4 13. − −  −  5  7

13. __________________________

9. 3 x 2 ÷

T-72

Name: Instructor:

Date: Section:

Chapter 4 Test Form E (cont’d)  1  1  14.    −   3  2 

14. __________________________

Simplify the complex fraction.

15.

1 5x 3 10 xy

15. __________________________

16.

3x 5 x 5

16. __________________________

Solve. 17.

3 2 = x 5 5

18.

x + 1x = −12 3

18. __________________________

19.

2  1  + − =x 3  12 

19. __________________________

20.

x x 3 + = 9 3 9

20. __________________________

17. __________________________

21. Evaluate xy

if x = −

1 2 and y = . 5 5

21. __________________________

22. Evaluate x4

if x = −

1 . 3

22. __________________________

23. Find the perimeter of a rectangle with width 1

1 yards 2

23. __________________________

1 and length 3 yards. 4

24. A suit regularly sells for $252. It is on sale for

1 off. 4

24. __________________________

What is the savings on this suit? 1 miles for each gallon of gasoline. How 2 many miles can you expect to travel on 18 gallons?

25. A car gets 22

25. __________________________

T-73

Name: Instructor:

Date: Section:

Chapter 4 Test Form F Write each mixed number as an improper fraction. 1. 3

2 3

33 3

2. 4

35 3

4 5

12 3

22 5

11 3

20 5

2 5

6 5

Write each improper fraction as a mixed number. 3.

79 24

a. 4

1 8

b. 3

7 24

1 3

d. 2

c. 190

d. 189

1 2

d. 11

c. 3

1 3

950 5

a. 109 138 12 2 a. 34 3

b. 19

3 5

7 12

c. 11

1 5

c. −5

d. 5

17 34

c. 2

b. 11

Write each fraction in simplest form. 65 13 1 a. − 3

6. −

17 34 1 a. 3

b. −

T-74

1 2

5 12

Name: Instructor:

Date: Section:

Chapter 4 Test Form F (cont’d) Perform the indicated operations. Write answers in simplest form.  1   5   12  8.        3   6   18 

5 12

9. −

5 9

5 27

40 54

2 3

14 3

d. −

4 9

c. 8

2 3

d. 11

5 9

39 95

c. 16

1 2

d. 16

4 5

3 4

1 4

9 30

9 60

3 10

7 21 ÷ 12 24 2 3

b. −

7 6

 2  1  10.  2   4   3  3 

a. 8

2 9

b. 11

 3  1 11.  2  ÷  6   5  3

247 15

12. 5 − 4

a. 1

13.

3 4

1 4

b. 1

1 5 3 + + 3 12 15 19 20

2 3 14. 5 + 7 5 10 4 a. 12 5

b. 12

2 5

c. 12

7 10

d. 12

1 2

T-75

Name: Instructor:

Date: Section:

Chapter 4 Test Form F (cont’d) Simplify the complex fraction. 3 4 15. 5 8

15 32

16.

x−2 x+2 2 x+2

x x+2

5 12

c. 1

1 5

x−2 2

1 2

c. −5

3 5

x x−2

d. 2( x − 2)

Solve.

17. x − 5 = −

a. 5

18.

1 2

b. 4

1 2

9 2

1 x+5 = 2 3

a. –1

19. −

a. −

T-76

1 2

b. 1

c. –9

d. 9

−9 20

d. 1

3 3 x= 4 5

4 5

7 20

Name: Instructor:

Date: Section:

Chapter 4 Test Form F (cont’d) 20. 3 x −

1 x = 15 5

b. −42

a. 42

21.

Evaluate 2x2⋅y if x = −

2 12

22. Evaluate (−x)3

1 27

75 14

25 14

1 9

1 6

1 1 and y = . 2 3

1 9

c. −

if x = −

1 . 3

b. −

1 27

23. Miss Lucero’s 3rd grade class read

d. −

c. 9

1 9

1 1 of a book on Monday and on Tuesday. How much of 2 4

the book have they read?

1 6

24. If

2 6

3 4

1 4

3 of 63 boxes were unpacked at a toy store, how many boxes were unpacked? 7

a. 27

25. A recipe calls for

b. 19

c. 16

d. 18

3 cup of nuts and we want to triple the recipe. Find the number of cups of nuts 4

needed. a.

9 cup 12

b. 2

1 cup 4

c. 3 cup

1 cups 4

T-77

Name: Instructor:

Date: Section:

Chapter 4 Test Form G Write each mixed number as an improper fraction. 1. 1

2 3

5 3

2. 3

4 5

19 5

12 3

13 3

10 3

7 5

20 5

23 5

3 4

d. 7

1 4

2 3

Write each improper fraction as a mixed number. 3.

31 4

a. 7

4. −

7 8

5 8

c. 7

2 3

3 2

d. −

1 13

13 27

d. 1

9 6

a. −1

b. 7

1 2

b. −1

27 13

a. 2

b. 2

14 13

Write each fraction in simplest form. 51 54 32 a. 36

17 18

8 9

68 72

1 4

24 39

12 39

120 195

T-78

8 13

Name: Instructor:

Date: Section:

Chapter 4 Test Form G (cont’d) Perform the indicated operations. Write answers in simplest form. 8.

2  −5   12  3  12   18 

−5 27

10 54

5 12

15 12

1 2

2 5

d. −

2 5

20 x 12

7 14 9. − ÷ 5 25

1 2

a. −2

10. −

12. a.

7 3x ( 5)   12  5 

105 60

11. −

b. 2

7x 4

c. −

7x 4

d. −

1 3

8 39

4 16

3 5

d. 5

13 36

2 1 + 3 6

1 2

b. −

1 2

c. −

b. −

10 39

b. 6

1 3

c. 5

2 13

3  1  − − 13  39 

10 39

1 1 1 +2 +3 2 2 3 1 a. 6 2

13.

1 1 14.   +    2 3 1 a. 5

3 7

5 6

T-79

Name: Instructor:

Date: Section:

Chapter 4 Test Form G (cont’d) Simplify the complex fraction. x 3 15. 2x 27

2 x2 135

16.

1 3y 12 15 x

5x 12 y

3x 135

9 2

9 27

4 15xy

12 45xy

1x 3y

8 5

c. 1

3 5

1 5

1 2

c. 18

1 20

c. –

Solve.

17. 4 x −

1 3 = 5 5

1 10

18.

x x + =9 3 6

9 2

19. −4 x = −

a. −

T-80

4 5

d. 6

1 5

1 20

d. −

5 4

Name: Instructor:

Date: Section:

Chapter 4 Test Form G (cont’d) 20. −

3 x = 7 − (−8) 2

2 3

a. −

21. Evaluate xy

if x = −

1 5

a. −

2 3

c. −10

d. 10

1 3

d. 9

1  3 and y = −  −  . 3  5

9 15

22. Evaluate 4x – yz

if x =

a. 3

b. 1

1 1 , y = , and z = −3. 2 3

c. −1

d. 4

1 1 1 books the first week, 1 books the second week, and of a book this week. 2 3 4 How many books has she read?

23. Susie read 1

a. 3

1 12

b. 2

3 9

c. 2

3 12

d. 3

3 of 120 boxes in a Christmas toy shipment has been received, how many boxes have been 5 received?

24. If

a. 80

b. 360

c. 72

25. Find the perimeter of a rectangle with width 1

a. 7

13 feet 21

b. 3

2 feet 10

d. 24

2 1 feet and length 2 feet. 3 7

c. 3

17 feet 21

d. 6

2 feet 5

T-81

Name: Instructor:

Date: Section:

Chapter 4 Test Form H Write each mixed number as an improper fraction. 1. 8

33 4

2. 5

1 4

63 4

60 4

18 4

12 9

45 9

50 9

5 9

52 9

Write each improper fraction as a mixed number or a whole number. 3.

0 8

a. undefined

b. 1

c. −1

b. 9

c. 8

d. 0

 47  4. −  −   5 

a. 9

2 5

7 5

d. 8

Write each fraction in simplest form. 5.

15 135 5 45

57 942 19 a. 314

b. 9

1 9

b. 16

1 16

3 27

17 314

Perform the indicated operations. Write answers in simplest form. 7.

1  2 5 + − + 3  3  6

T-82

2 3

1 2

4 3

d. 1

Name: Instructor:

Date: Section:

Chapter 4 Test Form H (cont’d) 8.

13 5 − 18 6

a. −9

1 9

2 9

d. 9

b. 242

1 242

d. −

35 5

21 5

21 6

30 75

3 7

7 15

3 4

3 16

12 16

35 180

d. −

3x x−4

b. −

 1  3  4  9.  −   −     8   11   33 

1 44

1 242

 1  3  10.  2   3   9  5 

38 5

 2  4 11.  1  ÷  3   3  7 120 a. 21

3 32 ÷ 8 64 24 a. 4

12.

 5  7  13.    −   12   15  25 a. − 21

b. −

Simplify the complex fraction. 3 x−4 14. x x−4 3 x a. b. x 3

7 12

c. −

7 36

3+ x x−4

T-83

Name: Instructor:

Date: Section:

Chapter 4 Test Form H (cont’d)

15.

1 5 4 3

15 4

4 15

3 20

5 6

c. −

5 10

20 3

5 6

d. −

5 21

d. −

5 21

5 27

d. −

5 27

b. 10

44 5

Solve. 16. −

3 5 x=− 11 33

5 9

1 3 17. 2 x − x = 5 7

1 2

1 5 18. − x − x = 8 24

a. −

5 21

19. x +

1 10

T-84

x = 12 5

11 6

Name: Instructor:

Date: Section:

Chapter 4 Test Form H (cont’d) 20. Evaluate xy

1 9

21.

if x = −

1 5 and y = − 5 9

b. 9

Evaluate (y)2 if y = −

a. −4

5 9

9 5

1 4

d. −

1 . 2

b. 4

1 4

1 feet to make a shelf. The piece was originally 6 feet long. What length 3 is left after the shelf is cut?

22. A carpenter cuts off 1

a. 6

2 3

b. 4

1 3

c. 4

2 3

d. 5

2 3

1 2 cups of four, 1 cup of sugar and cup of water. What is the total of the 3 3 mixture needed (flour, sugar and water)?

23. A recipe needs 2

a. 5

2 3

b. 3

2 3

d. 3

c. 4

24. A sports coat normally sells for $150. It is now on sale for

a. $100

b. $30

a. 10 yards

1 off. What will the savings be? 3

c. $120

25. Find the perimeter of a square with sides of length 2

b. 5 yards

1 2

d. $50

1 yards. 2

1 c. 8 yards 2

d. 4

1 yards 2

T-85

Name: Instructor:

Date: Section:

Cumulative Test Form A Chapters 3 and 4 1. Simplify by combining like terms: 4x − 5x + 2 − 5

1. __________________________

2. Multiply: −4(5xy)

2. __________________________

3. Multiply: 5(6x – 2)

3. __________________________

4. Is −2 a solution to the equation 13x + 5 – (2x + 5) = 9x – 4?

4. __________________________

5. Solve and check: −6x = 42

5. __________________________

6. Translate into a mathematical expression. Use x to represent “a number.” Twice the sum of a number and seven.

6. __________________________

7. Simplify:

75 120

7. __________________________

8. Write the prime factorization for 360.

9. Simplify:

35 xy 42 x

10. Multiply:

9. __________________________

1  9  5  3  40   27 

1 11. Evaluate:   3

8. __________________________

10. __________________________

11. __________________________

 3  4  12. Multiply:      7  5 

12. __________________________

 2  1 13. Divide:  2  ÷  1   3  3

13. __________________________

T-86

Name: Instructor: Cumulative Test Form A Chapters 3 and 4 (cont’d) 1 3 14. Subtract: 1 − 3 5

15. Evaluate x(y) if x =

14. __________________________ 1 4 and y = − 3 5

15. __________________________

1 x−6 = 4 3

16. __________________________

3  5 1 + − + 13  13  13

17. __________________________

16. Solve:

17. Add:

Date: Section:

18. Add: −

3 2 + 11 33

18. __________________________

4 5

19. __________________________

19. Subtract: 7 − 2

1 2 1 4

20. Simplify the complex fraction

20. __________________________

21. The perimeter of an equilateral triangle (all sides equal) is 1 10 cm. Find the length of the sides. 5

22. Find the perimeter of a square whose side is 2

3 feet. 4

21. __________________________

22. __________________________

 1  1  23. Multiply:  3   2   2  5 

23. __________________________

3  1 24. Subtract: 5 −  −2  5  5

24. __________________________

4 of the regular price. 5 What is the regular price of the suit?

25. A suit is on sale for $120. This is

25. __________________________

T-87

Name: Instructor:

Date: Section:

Cumulative Test Form B Chapters 3 and 4 1. Simplify by combining like terms: 3x – (−2x) + 4x a. −5x

b. x

c. −3x

d. 2x

b. 6x

c. 9x

d. −x

b. −6x + 8

c. −5x – 6

d. −5x − 8

2. Multiply: −3(−2x) a. −6x

3. Multiply: −2(3x + 4) a. −6x – 8

4. Is 5 a solution to the equation 5x – 3 = 2(x + 7)? a. yes

b. no

5. Solve and check 4x = −3x + 6 a.

6 7

c. −6

b. 6

d. −

6 7

6. Translate into a mathematical expression. Use x to represent “a number.” Seven less than five times a number. a. −35x

b. 7 – 5x

c. −7(5x)

d. 5x – 7

7. Translate into a mathematical expression. Use x to represent “a number.” The product of a number and four decreased by that same number. a. 4x2

b. x – 4x

c. 4x – x

d. −4x2

c. 3 ⋅ 3 ⋅ 23

d. 3 ⋅ 3 ⋅ 3 ⋅ 7

8. Write the prime factorization for 135. a. 3 ⋅ 3 ⋅ 3 ⋅ 5

T-88

b. 3 ⋅ 61

Name: Instructor: Cumulative Test Form B Chapters 3 and 4 (cont’d) 9. Simplify:

Date: Section:

11xy 33x

a. 3

y 3

3 y

d. 3y

7 23

 1   2   21  10. Multiply:  −     −   2   3   23 

42 183

 1 11. Evaluate:  −   2

a. −

1 4

c. 4

2 7

1 21

7 30

6y 17

18 y 17

2y 17

1 6

c. 2

3 17 ÷ 5 x 30 xy

3y 17

14. Solve: −

a. −

1 6

d. −4

11 1 + 3  5 2 

2 30

13. Divide:

7 23

1 4

12. Multiply:

1 3

b. −

4 2 x= 5 15

d. −2

T-89

Name: Instructor: Cumulative Test Form B Chapters 3 and 4 (cont’d) if x = −

15. Evaluate: x – y 1 3

a. −

b. −8

a. −

41 45

20.

1 4

c. −88

d. 8

5 11

6 22

41 45

c. −

31 45

3 4

c. 3

3 4

d. 2

1 4

1 y

6 44

4  1 + − 5  9 

19. Subtract: 7 − 3

a. 3

3 2 1 + + 11 22 11

3 11

18. Add:

1 3

c. −1

3 x + 18 = −15 8

a. 88

17. Add:

2 1 and y = 3 3

b. 1

16. Solve:

Date: Section:

1 4

b. 2

Simplify the complex fraction: 2 y y+2 y

T-90

2 y+2

y+2 2

c. y

Name: Instructor: Cumulative Test Form B Chapters 3 and 4 (cont’d)

Date: Section:

 1  2  Multiply:  2   2   4  3 

21.

a. 6

1 6

d. 4

7 15

d. 5

c. 4

1 12

1  2 22. Subtract: 8 −  −2  5  3

26 3

b. 10

13 15

c. 6

7 15

1 cups of sugar for 2 dozen cookies. We want to make 6 dozen cookies. How 2 much sugar is needed?

23. A recipe needs 1

a. 4

24.

1 cups 2

A suit is on sale for

a. $100

25.

a. 9

b. 6 cups

d. 1

1 cups 2

1 off the regular price of $125. What is the savings? 5

b. $25

c. $150

Find the area of a rectangle with width 2

1 ft. 2 2

c. 2 cups

b. 8

3 ft.2 50

d. $105

1 3 feet, and length 4 feet. 5 10

c. 6

1 2 ft. 2

d. 9

23 2 ft. 50

T-91

Name: Instructor:

Date: Section:

Chapter 5 Pretest Form A 1. Determine the place value of the 5 in the number 62.0357.

1. _______________________

2. Insert <, >, or = to make a true statement: 0.0252

2. _______________________

0.0251

3. Write 0.48 as a fraction in lowest terms.

3. _______________________

4. Round 283.295 to the nearest tenth.

4. _______________________

5. Add:

72.09 + 3.392 + 23

5. _______________________

6. Subtract: 384.39 – 139.99

6. _______________________

7. Simplify by combining like terms: −6.81 + 4.87x – 2.65x + (−8.19)

7. _______________________

8. Multiply: (2.7)(−3.81)

8. _______________________

9. Multiply: 5.83(100)

9. _______________________

10. Evaluate: xy

if x = 2.1 and y = −3.5

10. _______________________

11. Divide: 3.672 ÷ 0.72

11. _______________________

12. Divide: 29.36 ÷ 100

12. _______________________

13. Evaluate: x ÷ y

13. _______________________

if x = 0.9 and y = 0.003

14. Evaluate: xy if x = 3.1 and y = 0.4

14. _______________________

15. Estimate the perimeter of the rectangle with length 12.6 feet and width 8.4 feet by rounding each to the nearest foot.

15. _______________________

16. Simplify: (0.4)3

16. _______________________

17. Write

5 as a decimal. 8

17. _______________________

18. Find the area of a triangle with a height of 9 inches and a base of 10 inches.

18. _______________________

19. Find the mean and median of the following list of numbers: 18, 61, 36, 55, 40.

19. _______________________

20. Find the circumference of a circle with diameter 12 cm. Use 3.14 for π and the formula C = πd.

20. _______________________

T-92

Name: Instructor:

Date: Section:

Chapter 5 Pretest Form B 2.5702

1. _______________________

2. Determine the place value of the 8 in the number 6341.7528.

2. _______________________

3. Write 4.8 as a fraction or mixed number in lowest terms.

3. _______________________

4. Round 38.345 to the nearest hundredth.

4. _______________________

5. Add: 23.243 + 3.009 + 11.1

5. _______________________

6. Subtract: 53 – 9.92

6. _______________________

7. Simplify by combining like terms: 3x – (−0.52x) + 6.37y

7. _______________________

8. Multiply: (0.3)(0.004)

8. _______________________

9. Multiply: 52.58(10)

9. _______________________

1. Insert <, >, or = to make a true statement: 2.572

if x = −0.02 and y = −2.5.

10. Evaluate xy

10. _______________________

11. Divide: 0.692 ÷ 0.31 to the nearest thousandth

11. _______________________

12. Divide: 68.24 ÷ 1000

12. _______________________

13. Determine the mean, median and mode of the following list of numbers: 49, 22, 59, 78, 22, 35, 43.

13. _______________________

14. Evaluate (0.5)

14. _______________________

15. Perform the indicated operations: (0.3)2 – 4(1.8)

16. _______________________

16. Simplify: (−0.03)3 17. Simplify:

18. Write

15. _______________________

0.28 + 4.32

17. _______________________

0.02

7 as a decimal. 8

18. _______________________

19. Find the area of a triangle with a height of 5.2 inches and a base of 6.4 inches.

19. _______________________

20. A novel that costs $12.99 has $.52 sales tax added. Find the final cost of the novel.

20. _______________________

T-93

Name: Instructor:

Date: Section:

Chapter 5 Test Form A Write in words. 1. 37.234

1. __________________________

2. 13.0128

2. __________________________

Write in standard form: 3. “two hundred twenty-seven and four hundred two thousandths” 5 to a decimal number. Round to the nearest 7 thousandth if necessary.

4. Convert

3. __________________________

4. __________________________

Perform the indicated operation. 5. 2.61 + 3.95 + 0.412

5. __________________________

6. −7.43 – 0.901

6. __________________________

7. 4.82(3.3)

7. __________________________

8. 0.085 ÷ (0.25)

8. __________________________

9. (−0.2)2 + 1.75

9. __________________________

10. 2.4x – 3.2x – 1.8x

10. __________________________

Convert each decimal to a fraction or a mixed number (in lowest terms). 11. 3.25

11. __________________________

12. 16.002

12. __________________________

Insert <, >, or = to make a true statement. 13. 67.029

67.209

13. __________________________

1 12

0.08

14. __________________________

14.

T-94

Name: Instructor:

Date: Section:

Chapter 5 Test Form A (cont’d) Solve and check: 15. 0.4x = 0.24

15. __________________________

16. x − 0.5x = 8.5

16. __________________________

17. 2x + 7 = x – 4.3

17. __________________________

18. x – (2x + 6.9) = 18.6

18. __________________________

x = 1.9 0.5

19. __________________________

20. (0.25 − 0.5)2

20. __________________________

5.20 + 6.38 −0.2

21. __________________________

19. Simplify:

21.

22. Find the mean, median, and mode: 12, 15, 81, 56, 31

22. __________________________

23. Find a student’s grade average if he has test scores of 92, 90, 75, and 87.

23. __________________________

24. Find the circumference of the circle with radius 4 ft. Use 3.14 for π and the formula C = πd.

24. __________________________

25. Find the area of a triangle that has a height of 8.4 centimeters and a base of 10.8 centimeters.

25. __________________________

T-95

Name: Instructor:

Date: Section:

Chapter 5 Test Form B 1. Write 33.001 in words.

1. __________________________

2. Write in standard form “three hundred and nine thousandths.”

2. __________________________

3. Convert 5.24 into a fraction or mixed number in lowest terms.

3. __________________________

4. Convert

5 to a decimal number. 8

4. __________________________

Insert <, >, or = to make a true statement. 5. 76.920

76.92

5. __________________________

6. 27.254

27.245

6. __________________________

7. Round 139.496 to the nearest tenth.

7. __________________________

Perform the indicated operations. 8. 24.12 + 6.279 + 13 + 1.112

8. __________________________

9. 3.21 – 8.4

9. __________________________

10. 6.24 – (−4.3)

10. __________________________

11. (0.6)(3.2)(−1)

11. __________________________

12. 8.02 ÷ (−0.4)

12. __________________________

13. (1.3)3

13. __________________________

Solve. 14. 2.5x + 7.6 = 1.5x – 32.72

T-96

14. __________________________

Name: Instructor:

Date: Section:

Chapter 5 Test Form B (cont’d) x = −3 2.1

15. __________________________

16. 2x − 5.6 = 0.028

16. __________________________

15.

Simplify each expression. Give the result as a decimal.

17.

1 8

17. __________________________

( 30 − 13.2 )

18. (0.25)2 (80 )

18. __________________________

(0.03)2 0.01

19. __________________________

20. 3x + 4.5 = −1.29

20. __________________________

21. 0.5(2x + 1.8) = −3.5

21. __________________________

22. Find the mean, median, and mode of the following list of numbers: 85, 98, 60, 85, 66, 58, 101.

22. __________________________

19. Solve:

23. Find the area of the rectangle with length 6.5 feet and width of 2.5 feet.

23. __________________________

24. Find the circumference of a circular flower bed which has a diameter of 10 feet. Use 3.14 for π and the formula C = πd.

24. __________________________

25. Put the numbers in order from smallest to largest: 0.7, 0.072, 0.72, 0.702.

25. __________________________

T-97

Name: Instructor:

Date: Section:

Chapter 5 Test Form C 1. Write 36.52 in words.

1. __________________________

2. Write in standard form “two hundred seven and forty-three thousandths.”

2. __________________________

3. Convert

3 into a decimal. 5

3. __________________________

4. Convert 13.55 into a fraction or mixed number in lowest terms.

4. __________________________

Insert <, >, or = to make a true statement.

3 4

6. 29.013

5. __________________________

0.74

29.0143

6. __________________________

7. Round 273.495 to the nearest tenth.

7. __________________________

Perform the indicated operations. 8. 5 + 3.9 + 0.004

8. __________________________

9. 21.3 − 19.41

9. __________________________

10. −2.37 – 8.1

10. __________________________

11. (0.05)(2.12)(0.2)

11. __________________________

12. 16.64 ÷ 0.08

12. __________________________

13. (−0.02)3

13. __________________________

14. Put the numbers in order from smallest to largest: 0.18, 0.018, 0.8, 0.08

14. __________________________

T-98

Name: Instructor:

Date: Section:

Chapter 5 Test Form C (cont’d) Solve. 15. −3x = 1.2

15. __________________________

16. 0.3x = 12

16. __________________________

x = 1.06 0.4

17. __________________________

18. 4x = 12.608

18. __________________________

17.

Simplify each expression. Give the result as a decimal. 19.

3 ( 20.1 − 16.6 ) 5

19. __________________________

20. (−0.02)3(1.2)

20. __________________________

17.24 + 3.76 −0.2

21. __________________________

21.

22. Find the mean, median, and mode: 24, 46, 28, 24, 37, 48, 24

22. __________________________

23. The scores on a basic math final exam were as follows: 85, 67, 92, 90, 77, 76, 82, 85, 93, 88, 65, 78, and 80. Find the median score.

23. __________________________

24. Find the area of a rectangle with length 5.6 feet and width 3.2 feet.

24. __________________________

25. Find the exact circumference of a circle that has a radius of 4.2 centimeters. Then use 3.14 for π and approximate the circumference.

25. __________________________

T-99

Name: Instructor:

Date: Section:

Chapter 5 Test Form D 1. Write 120.056 in words.

1. __________________________

2. Write in standard form “two thousand two hundred and twenty-four thousandths.”

2. __________________________

3. Convert

7 into a decimal. Round to the thousandth place. 9

4. Convert 35.72 into a fraction or mixed number in lowest terms.

3. __________________________ 4. __________________________

Insert <, >, or = to make a true statement. 5. 8.069

8.0690

5. __________________________

6. 3.256

3.257

6. __________________________

7. Round 87.925 to the nearest tenth.

7. __________________________

Perform the indicated operations. 8. 8.692 + 3.25 + (−1.87)

8. __________________________

9. 53.2 − 4.638

9. __________________________

10. −3.57 + (−6.9)

10. __________________________

11. (0.14)(0.06)

11. __________________________

12. 93.05 ÷ (2)

12. __________________________

13. (3.6)(0.14)

13. __________________________

14. (−0.2)3(−0.1)

14. __________________________

15. 0.75 ÷ 7.5

15. __________________________

T-100

Name: Instructor:

Date: Section:

Chapter 5 Test Form D (cont’d) 16. Put the numbers in order from smallest to largest: 0.6, 0.26, 0.206, 0.062

16. __________________________

Solve. 17. 1.1x − 0.7 x = 12.8

17. __________________________

18. 3x = 18.9

18. __________________________

19. 42.8 + x = −6.12

19. __________________________

Simplify. Give each answer as a decimal. 2

1 20.   + (0.25) 2 4

20. __________________________

(0.75) + 2.6 0.5

21. __________________________

21.

22. Find the mean, median, and mode: 58, 34, 62, 37, 45, 66, 34

22. __________________________

23. Tashara bought a shirt for $29.89 and a pair of jeans for $68.99. How much change should she receive if she pays with a $100 bill?

23. __________________________

24. Luz and 5 of her friends went for pizza and a movie. The total cost for the evening was $105. What was the average cost for each of the 6 people?

24. __________________________

25. Find the area of a rectangle that is 9.2 centimeters long and 4.1 centimeters wide.

25. __________________________

T-101

Name: Instructor:

Date: Section:

Chapter 5 Test Form E Write in words. 1. 124.36

1. __________________________

2. 15.0102

2. __________________________

3. Write standard form for: “three thousand, two hundred twenty-seven and fifty-one hundredths.”

3. __________________________

5 to a decimal number. (Round to the nearest 7 thousandth.)

4. Convert

4. __________________________

5. Convert 6.25 to a fraction or mixed number in lowest terms.

5. __________________________

6. Round 285.3697 to the nearest thousandth.

6. __________________________

Insert <, >, or = to make a true statement. 7. 8.274

8.275

7. __________________________

1 3

0.334

8. __________________________

Perform the indicated operations: 9. 2.946 + 42.93

9. __________________________

10. −6.35 + 7.9 – (−2.3)

10. __________________________

11. 17 – 8.04

11. __________________________

12. (0.71)(0.4)

12. __________________________

−10.5 −0.7

13. __________________________

14. (0.2)3 – 0.8

14. __________________________

13.

T-102

Name: Instructor: Chapter 5 Test Form E (cont’d)

Date: Section:

Solve. 15. 44.24 − 3 x = 4 x

15. __________________________

16. 0.2x – 0.1x + 8.6 = 12.4

16. __________________________

17. 2x – 17.8 = −8.2

17. __________________________

x = −0.4 0.2

18. __________________________

19. 0.5x = 8.5 – (−1.25)

19. __________________________

18.

Simplify. Give each answer as a decimal. 20. (1.2)3

1 21. (0.25)   2

20. __________________________ 2

21. __________________________

22. Find the mean, median, and mode for the following numbers: 29, 50, 17, 24, 36, 58, 42, 38.

22. __________________________

23. Find the exact circumference of a circle with radius 1.3 yards. Then use 3.14 for π and approximate the circumference.

23. __________________________

24.

24. __________________________

Find the area of a triangle with a base of 12.8 inches and a height of 9.3 inches.

25. Jorge’s annual salary is $36,958.20. Find his monthly salary. (Round to the nearest cent.)

25. __________________________

T-103

Name: Instructor:

Date: Section:

Chapter 5 Test Form F 1. Write 243.095 in words a. Two hundred forty-three and ninety-five thousandths b. Two hundred forty and three and ninety-five hundredths c. Two hundred forty-three ones and ninety-five thousandths d. Two hundred forty-three and nine tens and five hundredths 2. Write standard form for “one hundred four and two tenths.” a. 140.2 b. 104.02 c. 104.2 3. Convert 12.025 into a fraction or mixed number (in lowest terms). 25 1 12025 c. 12 b. 12 a. 100 40 10, 000 4. Convert

3 8

d. 140.02

d. 12

25 1, 000

to a decimal number.

a. 3.75

b. 0.375

c. 2.66

d. 0.266

c. 87.385

d. 87.39

5. Round 87.3859 to the nearest hundredth. a. 87.38

b. 87.389

6. Insert <, >, or = to form a true statement: 19.792 19.789 a. < b. >

c. =

7. Which number is the smallest: 0.05, 0.51, 0.005, 0.5? a. 0.05

b. 0.51

c. 0.005

d. 0.5

b. 23.04

c. 2250

d. 284.4

b. 18.945

c. 18.955

d. −18.955

b. 7.781

c. 0.78

d. 7.8

Perform the indicated operations. 8. 13.52 + 6 + 8.92 a. 28.44 9. 18.95 – 0.005 a. −18.945 10. 32.76 ÷ 4.2 a. 7.788

11. (15.47) ÷ (–3.5) (Round to the nearest thousandth.) a. 4.42

b. 0.442

c. −4.42

d. −0.442

b. 12.90

c. −12.90

d. −1.29

12. (0.06)(−2.15) a. −0.129

T-104

Name: Instructor:

Date: Section:

Chapter 5 Test Form F (cont’d) Solve and check: 13. 2x – 16.50 = 8.26 a. 12.38

b. −12.38

c. 8.24

d. 4.12

b. –0.3

c. 3

d. –1.4

b. 2.9375

c. 1.3125

d. 0.5

b. −20.51

c. 2.93

d. 20.51

b. 16.4

c. 6.56

d. 0.656

b. 0.00008

c. −0.000008

d. −0.06

b. 15

c. 1

d. −0.1

14. 0.5x – 1.2x = −2.1 a. –3 15. 2(2x – 3.25) = 8.5 a. 3.75

16. 4x + 6.82 = 3x – 13.69 a. 6.87 17.

x = 32.8 0.02

a. 1.64 Simplify: 18. (−0.02)3 a. −0.8 19.

0.5 − (−2.5) 0.2

a. 1.5

Use the following numbers for problems 20−22: 25, 48, 54, 35, 68 20. Find the mean. a. 48

b. 54

c. 46

d. not one

a. 48

b. 54

c. 35

d. not one

22. Find the mode. a. 35

b. 54

c. 25

d. not one

21. Find the median.

23. Find the circumference of a circle with diameter 6 feet. Use 3.14 for π and approximate the circumference. a. 18.84 feet b. 9.42 feet c. 188.4 feet d. 94.2 feet 24. A new automobile costs $22,642.86. Find the monthly payment (to the nearest cent) if 60 payments are made. a. $377.38 b. $377.30 c. $377.80 d. $377.39 25. Find the annual salary for Rebecca if she earns $1,250.82 each month. a. $15,090.84 b. $1509.84 c. $15,090.80

d. $15,009.84

T-105

Name: Instructor:

Date: Section:

Chapter 5 Test Form G 1. Write 386.204 in words a. Three eighty-six and two hundred four b. Three hundred eighty-six and two and four thousandths c. Three hundred eighty-six and two hundred four thousandths d. Three hundred eighty-six and two hundred four 2. Write standard form for “two thousand five and two thousandths.” a. 2,500.200

b. 2,005.002

c. 2,050.200

d. 2,050.002

3. Convert 15.03 into a fraction or mixed number (in lowest terms). a. 15

3 10

4. Convert

b. 3 16

153 100

c. 15

3 100

d. 15

30 100

to a decimal number.

a. 0.625

b. 0.1875

c. 0.19

d. 0.188

c. 357.866

d. 357.800

5. Round 357.86529 to the nearest thousandth. a. 357.865

b. 357.86

6. Insert <, >, or = to form a true statement: 385.629 385.6219 a. <

c. =

b. >

7. Find the largest number: 0.08, 0.085, 0.8, 0.85 a. 0.08

b. 0.085

Perform the indicated operations. 8. 8.325 + 12 + 0.265 a. 20.585 b. 205.9

c. 0.8

d. 0.85

c. 28.533

d. 20.59

9. 27 – 13.659 a. 13.341

b. 133.41

c. −13.34

d. 13.59

10. 45 ÷ (0.36) a. 0.008

b. 125

c. 16.2

d. 1620

11. (0.2)2(0.2)3 a. 0.00032

b. 0.0032

c. 0.32

d. 0.024

12. 25.5 ÷ (–0.5) a. –5.1

b. 51

c. 5.1

d. –51

T-106

Name: Instructor:

Date: Section:

Chapter 5 Test Form G (cont’d) Solve and check: 13. 0.1x + 2.7 = 3.86 a. 6.56

b. 1.16

c. 11.6

d. 0.656

b. −6.3

c. 6.3

d. −0.63

b. −2.08

c. 27.2

d. 6.8

b. 0.7

c. −3.5

d. −0.7

b. 0

c. −16.26

d. 16.26

b. −3.1

c. 3.1

d. −2.3

b. 0.9

c. 0.5776

d. 1.444

14. −3x = 18.9 a. −63 15.

x + 12 = 1.6 2

a. −20.8 16. 3x + 1.8 = 2x – 1.7 a. 0.1 17. x – 32.52 = −x a. −32.52 18. 3x + 1.2 = −8.1 a. 2.3 Simplify: 19.

(1.9 )2 (0.4)

a. 0.76

Use the following numbers for problems 20−22: 20. Find the mean. a. 52

58, 52, 47, 62, 38, 47, 74

b. 54

c. 55

d. not one

b. 47

c. 52

d. not one

b. 47

c. 55

d. not one

21. Find the median. a. 54 22. Find the mode. a. 52

23. Find the circumference of a circle with radius 4 cm. Use 3.14 for π and the formula C = πd. a. 1.256 cm. b. 25.13 cm. c. 12.56 cm. d. 2.512 cm. 24. Jeff has $836.15 in his checking account. He pays his rent with a check for $512.00, buys food with a check for $84.18, and pays his car payment of $204.36. What is his new balance? a. $356.10 b. $35.61 c. $3.56 d. $38.16 25. Judy earns $36,586.12 each year. Find her monthly salary. (Round to the nearest cent.) a. $3,048.84 b. $3,048.80 c. $3,048.48 d. $3,048.83 Copyright © 2025 Pearson Education, Inc.

T-107

Name: Instructor:

Date: Section:

Chapter 5 Test Form H 1. Write 127.005 in words a. One twenty-seven hundred and five b. One hundred twenty-seven and five thousandths c. One hundred twenty-five and five tenths d. One twenty-seven hundred and five hundredths 2. Write standard form for “three hundred ten thousandths.” a. 0.301 b. 0.30010 c. 0.310 3. Convert 8

d. 310,000

1 into a decimal. 5

a. 8.2

b. 82

c. 0.082

d. 0.82

2 5

d. 20

5. Round 5.8792 to the nearest hundredth. a. 5.90 b. 5.88

c. 5.89

d. 5.86

6. Insert <, >, or = to form a true statement: 6.3332 6.33312 a. < b. >

c. =

4. Convert 20.04 into a fraction or mixed number. a. 20

40 100

204 100

c. 20

1 25

Find the smallest number: 0.12, 0.2, 0.012, 0.21

a. 0.12

b. 0.2

c. 0.012

d. 0.21

Perform the indicated operations. 8. 0.875 + 2 + 12.62 a. 15.495

b. 14.847

c. 1.549

d. 15.895

b. 26.18

c. 26.80

d. 27.28

b. 2502

c. −2500.2

d. −25.002

b. −417.5

c. 41.75

d. −41.75

b. 0.00675

c. 0.15

d. 0.015

9. 36 – 9.82 a. 27.82 10. 625.05 ÷ (−0.25) a. 25.002 11. −8.35 ÷ −0.02 a. 417.5 12. (0.25)(0.3)3 a. 0.0675

T-108

Name: Instructor:

Date: Section:

Chapter 5 Test Form H (cont’d) 13. (0.2)3 + (0.7)(0.2) a. 1.48 Solve and check: 14. 1 − x = −0.5 x + 16 a. −30

b. 14.8

c. 0.74

d. 0.148

b. 10

c. 34

d. –10

b. 3.68

c. 8.32

d. 4.32

b. −20

c. 2

d. −2

b. 3.759

c. 0.73

d. 2.92

b. −25.08

c. −6.27

d. 6.27

b. 0.69

c. 0.07

d. 0.64

15. 2(x – 8) = 0.64 a. 7.36 16. 3.2x = −64 a. 32

17. 4x – 12.79 = 3x + 13.52 a. 26.31 18. −

x = 12.54 2

a. 25.08 Simplify: 1 + (0.7)2 2 a. 0.99

19.

Use the following numbers for problems 20−22: 20. Find the mean. a. 48 b. 43

32, 45, 58, 28, 59, 36 c. 45

d. not one

b. 40.5

c. 43

d. not one

b. 40.5

c. 45

d. not one

21. Find the median. a. 58 22. Find the mode. a. 36

23. Find the circumference of a circle with diameter 10 yards. Use 3.14 for π and approximate the circumference. a. 31.4 yards

b. 314 yards

c. 15.7 yards

d. 3.14 yards

24. Tomas purchased an automobile for $18,692.59. He wants to pay for his new car in 48 payments. What will the monthly payments be? Round to the nearest cent. a. $390.00

b. $389.00

c. $389.43

d. $389.40 1 25. Leticia wants to make cookies for the school bake sale. The recipe calls for 1 cups of sugar. 2 She wants to triple the recipe. How many cups of sugar will she need? a. 4 cups b. 3 cups 1 1 d. 4 cups c. 3 cups 2

T-109

Name: Instructor:

Date: Section:

Chapter 6 Pretest Form A Write each ratio as a fraction. Do not simplify. 1. 13 to 50

2. 4

1 to 8.75 2

1. _______________________

2. _______________________

Write each rate as a fraction in simplest form. 3. 1.75 to 25

3. _______________________

4. $215 to $35

4. _______________________

5. Write the rate as a fraction in simplest form: $7.20 for 9 pounds

5. _______________________

6. Write as a unit rate: 258 miles for every 10 gallons of gas.

6. _______________________

7. Frozen peas cost $2.88 for 48 ounces. What is the unit cost?

7. _______________________

Write each sentence as a proportion. 8. 4 cups flour is to 12 muffins as 12 cups flour is to 36 muffins.

1 3 inch is to 50 miles as inch is to 75 miles 2 4

8. _______________________

9. _______________________

10. Is

1 4 = a true proportion? 4 20

10. _______________________

11. Is

24 192 = a true proportion? 25 200

11. _______________________

T-110

Name: Instructor:

Date: Section:

Chapter 6 Pretest Form A (cont’d) Solve. (If needed round to the nearest tenth.) 12.

5 x = 6 20

12. _______________________

13.

0.5 x = 6 36

13. _______________________

14.

1 x 2 = 1 12 3

14. _______________________

15.

1 13 2 15

x 7

15. _______________________

16. If 1 inch = 45 miles on a road map scale, find how many miles is represented by 4 52 inches?

16. _______________________

17. Find the ratio of the corresponding sides of the given similar triangles.

17. _______________________

9 6

7 21 18. Given the similar triangles, find x. 2 8 x

19. Find the square root:

18. _______________________ 16

100

19. _______________________

20. Find the square root. Round to the nearest thousandth

205

20. _______________________

T-111

Name: Instructor:

Date: Section:

Chapter 6 Pretest Form B Write each ratio as a fraction. Do not simplify. 1. _______________________ 1. 28 to 54 2. 3.2 to 42.4

2. _______________________

Write each rate as a fraction in simplest form. 3. 306 to 51

3. _______________________

4. 480 to 400

4. _______________________

5. 8 hours for 2 assignments

5. _______________________

6. Write as a unit rate: $115 for 20 hours

6. _______________________

7. A certain juice costs $4.19 for 32 ounces. Find the unit cost. (Round to the nearest cent.)

7. _______________________

Write each sentence as a proportion. 8. 1 cup of blueberries is to 6 muffins as 1 12 cups of blueberries is to 9

8. _______________________

muffins.

9. 24 hours is to 1 day as 96 hours is to 4 days. 1 in. represents 125 miles on a roadmap, find the distance 2 represented by 3 12 inches.

10. If

11. Is

4 3 = a true proportion? 18 9

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9. _______________________

10. _______________________

11. _______________________

Name: Instructor:

Date: Section:

Chapter 6 Pretest Form B (cont’d) Solve. 12.

0.5 x = 8 10

12. _______________________

13.

1 4 = x 1 8 2

13. _______________________

14.

2 x = 15 35

14. _______________________

15.

−

3 51 =− 4 x

15. _______________________

16. Find the ratio of the corresponding sides of the given similar triangles. 1.5 7.5 3.5 17.5

16. _______________________

17. Given the similar triangles, find x.

17. _______________________

1 12

18. Jim is 6 feet tall and casts a shadow of 8.5 feet. Find the height of a tree if its shadow is 12 feet. (Round to the nearest tenth.)

18. _______________________

19. Find the square root.

19. _______________________

20. Find the square root. Round to the nearest tenth.

295

20. _______________________

T-113

Name: Instructor:

Date: Section:

Chapter 6 Test Form A Write each ratio or rate as a fraction in simplest terms. 1. 250 books to 780 books

1. __________________________

2. $40 to $185

2. __________________________

1 3 to 2 4

3. __________________________

1 3 to 3 5 5

4. __________________________

4. 2

5. 6 inches to 18 inches

5. __________________________

6. 35 miles to 90 miles

6. __________________________

7. 10 computers to 22 students

7. __________________________

8. 6 teachers to 132 students

8. __________________________

9. 280 miles to 10 gallons of gas

9. __________________________

Write each sentence as a proportion. 10. 8 passes completed is to 10 attempts as 16 passes is to 20 attempts.

10. __________________________

11. 2 printers is to 6 computers as 4 printers is to 12 computers.

11. __________________________

Determine whether the proportion is true. 2 6 = 12. 3 12 13.

9 27 = 15 45

12. __________________________

13. __________________________

Write each as a unit rate. 14. 380 miles in 10 gallons of gas

14. __________________________

15. 600 calories in a 5-ounce serving

15. __________________________

T-114

Name: Instructor:

Date: Section:

Chapter 6 Test Form A (cont’d) Solve the proportion. 16.

x 20 = 3 15

16. __________________________

17.

3 x = 8 48

17. __________________________

18.

2 6 3 = 8 x 12

18. __________________________

19.

x 3.4 = 0.5 2

19. __________________________

20. Given the similar triangles, find x. 3

20. __________________________

9 4 x

21. Find the length of a wall represented by 1 12 inches on a

21. __________________________

blueprint if 1 inch represents 10 feet.

22. A law university accepts 2 out of every 7 applications it receives. If 350 applications are received, how many were accepted?

22. __________________________

Find each square root. Round to the nearest thousandth if necessary.

23.

23. __________________________

24.

100 121

24. __________________________

25.

150

25. __________________________

T-115

Name: Instructor:

Date: Section:

Chapter 6 Test Form B Write each ratio or rate as a fraction in simplest terms. 1. 36 books to 126 books

1. __________________________

2. $80 to $155

2. __________________________

3. 0.7 cm to 3.5 cm

3. __________________________

2 5 in. to in. 3 9

4. __________________________

5. 14 printers to 35 computers

5. __________________________

6. 8 computers to 70 students

6. __________________________

7. 75 students to 3 teachers

7. __________________________

8. 200 books to 40 students

8. __________________________

Write each sentence as a proportion. 9. 1 12 cups of sugar is to 24 cookies as 4 12 cups of sugar is

9. __________________________

to 72 cookies. 10. 6 books is to 3 courses as 8 books is to 4 courses.

10. __________________________

Determine whether the proportion is true. 11.

2 6 = 5 20

11. __________________________

12.

3 18 = 8 48

12. __________________________

Write each as a unit rate. 13. 330 miles in 6 hours

13. __________________________

14. 360 calories in a 4-ounce serving

14. __________________________

T-116

Name: Instructor:

Date: Section:

Chapter 6 Test Form B (cont’d) Solve the proportion. 7 x = 13 26

15.

15. __________________________

8 4

16. __________________________

17.

9 x = 36 4

17. __________________________

18.

4 16 = 5 x

18. __________________________

16.

1 2

19. Given the similar triangles, find x. 3

19. __________________________ x

4 8 20. It is suggested that a college classroom should have at least 9 square feet of floor space for each student. Find the minimum floor space for 25 students. 21. A 20 foot by 16 foot room is converted into a classroom. If 9 square feet of floorspace is needed for each student, find the number of students which could occupy this new room. (Round to the nearest ones place.)

22. A quarterback completes 2 passes for each 5 he attempts. How many will he complete if he attempts 30 passes?

20. __________________________

21. __________________________

22. __________________________

Find each square root. Round to the nearest thousandth if necessary. 23.

23. __________________________

24.

4 81

24. __________________________

25.

112

25. __________________________

T-117

Name: Instructor:

Date: Section:

Chapter 6 Test Form C Write each ratio or rate as a fraction in simplest terms. 1. 195 baseball cards to 980 baseball cards

1. __________________________

2. $49 to $35

2. __________________________

3. 19 in. to 76 in.

3. __________________________

4. 36 books to 12 books

4. __________________________

5. 16 printers to 80 students

5. __________________________

6. 9 gallons to 117 miles

6. __________________________

7. 9 hours to 540 miles

7. __________________________

Write each sentence as a proportion. 1 cups of flour is to 36 cookies as 9 cups of flour is to 72 2 cookies.

8. 4

9. 6 books is to 1 class as 36 books is to 6 classes.

8. __________________________

9. __________________________

Determine whether the proportion is true. 10.

7 63 = 9 81

10. __________________________

11.

1 18 = 5 85

11. __________________________

Find each rate as a unit rate. 12. 720 calories in 6 glasses of punch.

12. __________________________

13. 825 miles in 12.5 hours.

13. __________________________

T-118

Name: Instructor:

Date: Section:

Chapter 6 Test Form C (cont’d) Solve the proportion. 14.

7 x = 9 36

14. __________________________

15.

0.5 3.5 = x 49

15. __________________________

16. __________________________

16.

1 2 = x 50 300

17.

7 56 = 12 x

17. __________________________

18.

1 5 = 7 x

18. __________________________

19. Given the similar triangles, find x. 5

19. __________________________

10 8

20. During a basketball game, Chiang made 3 out of 5 freethrows. If he continues at this ratio, how many free throws can he expect to make out of 25 attempts?

20. __________________________

21. How many miles can one expect to travel on 10 gallons of gas if the car gives 25 miles per gallon?

21. __________________________

22. Which is a better buy; a 16 ounce bag of frozen peas costing $1.19 or a 24 ounce bag costing $1.60? (Hint: find unit cost for each first.)

22. __________________________

Find each square root. Round to the nearest thousandth if necessary. 23.

23. __________________________

24.

25 49

24. __________________________

25.

175

25. __________________________

T-119

Name: Instructor:

Date: Section:

Chapter 6 Test Form D Write each ratio or rate as a fraction in simplest terms. 1. 54 books to 42books

1. __________________________

2. $790 to $375

2. __________________________

3. 8.75 km to 1.25 km

3. __________________________

4. 73 bushes to 18 bushes

4. __________________________

5. 9 tutors to 54 students

5. __________________________

6. 17 gallons to 340 miles

6. __________________________

7. 12 people to 24sandwiches

7. __________________________

Write each sentence as a proportion. 8. 2 gallons of milk is to 3 days as 14 gallons is to 21 days.

8. __________________________

9. 4 printers is to 12 students as 8 printers is to 24 students.

9. __________________________

Determine whether the proportion is true. 10.

5 25 = 12 48

10. __________________________

11.

3 18 = 6 36

11. __________________________

Find each rate as a unit rate. 12. $6.45 for 5 sodas.

12. __________________________

13. $20.85 for 15 gallons of gas.

13. __________________________

Solve the proportion. 8 x = 14. 7 28

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14. __________________________

Name: Instructor:

Date: Section:

Chapter 6 Test Form D (cont’d) 15.

8 12 = x 9

15. __________________________

16.

3 x = 10 70

16. __________________________

17.

7 42 = 11 x

17. __________________________

18.

1 x = 2 186

18. __________________________

19. Given the similar triangles, find x. 3

19. __________________________

5 x 20. A Honda can go 450 miles on 12 gallons of gas. How many gallons of gas will be needed for a 1500-mile trip?

20. __________________________

1 inch represents 25 miles. Find the 2 distance between two cities if they are 3 12 inches apart.

21. __________________________

22. A student can type 4 pages in 30 minutes. How long will it take this student to type 16 pages?

22. __________________________

21. On a road map,

Find each square root. Round to the nearest thousandth if necessary. 23.

23. __________________________

24.

25 49

24. __________________________

25.

135

25. __________________________

T-121

Name: Instructor:

Date: Section:

Chapter 6 Test Form E Write each ratio or rate as a fraction in simplest terms. 1. 650 pages to 250 pages

1. __________________________

2. 18 books to 45 books

2. __________________________

3. $276 to $92

3. __________________________

1 7 inches to 3 inches. 2 8

4. __________________________

5. 8 computers to 24 students

5. __________________________

6. 8 teachers to 168 students

6. __________________________

7. 455 miles to 7 hours

7. __________________________

8. 42 ounces to 24 cents

8. __________________________

9. 336 miles to 12 gallons

9. __________________________

4. 2

Write each sentence as a proportion. 10. 1 12 cups of sugar is to 24 cookies as 3 cups of sugar is to 48 cookies. 11. 2 printers is to 8 computers as 4 printers is to 16 computers

10. __________________________

11. __________________________

Determine whether the proportion is true. 12.

6 18 = 9 27

12. __________________________

13.

3 10 = 7 28

13. __________________________

Write each as a unit rate. 14. 700 miles to 28 gallons of gas.

14. __________________________

15. 420 calories in a 12 ounce serving.

15. __________________________

T-122

Name: Instructor:

Date: Section:

Chapter 6 Test Form E (cont’d) Solve the proportion. 16.

5 x = 12 72

16. __________________________

17.

13 52 = 15 x

17. __________________________

1 3 = 10 x 15

18. __________________________

x 3.6 = 1.2 4

19. __________________________

18.

19.

20. Given the similar isosceles triangles, find x. 20. __________________________ 4

3 x 21. A quarterback completes 3 out of 7 passes attempted. At this rate, how many can he expect to complete if he attempts 56 passes?

21. __________________________

22. A car gets 28.5 miles per gallon. How many miles can the car travel on 15 gallons of gas?

22. __________________________

Find each square root. Round to the nearest thousandth if necessary. 23.

23. __________________________

24.

16 49

24. __________________________

25.

220

25. __________________________

T-123

Name: Instructor:

Date: Section:

Chapter 6 Test Form F Write each ratio or rate as a fraction in simplest terms. 1. 912 books to 18 books a.

18 912

912 18

687 35

152 3

3 152

229 7

2. $35 to $687 a.

35 687

c. 652

3. 75 in. to 15 in. a. 5

b. 8

3 25

25 3

29 65

4. 290 miles to 650 miles a.

5 13

65 29

650 290

28 7

1 4

d. 4

15 387

c. 25

c. 31

5. 7 inches to 28 inches a.

7 28

6. $15 to 387 kwh a.

387 15

129 5

7. 5 tutors to 36 students a.

5 36

T-124

36 5

7 1

Name: Instructor:

Date: Section:

Chapter 6 Test Form F (cont’d) 8. 8 gallons to 280 miles a.

8 280

280 8

1 35

140 4

Write each sentence as a proportion. 9. 3

1 cups of flour is to 3 dozen cookies as 7 cups of four is to 6 dozen cookies. 2

1 6 a. 2 = 3 7 3

1 2 =7 3 6

1 2 =3 6 7

1 2 =6 7 3

7 14 = 54 108

10. 7 printers is to 54 computers as 14 printers is to 108 computers. a.

7 14 = 108 54

7 108 = 54 14

7 54 = 54 14

Determine whether the proportion is true. 11.

5 10 = 17 34

a. true

12.

b. false

7 70 = 11 121

a. true

b. false

Solve the proportion. 13.

2 x = 17 34

a. x = 4

b. x =

1 2

c. x = 1

d. x = 2

T-125

Name: Instructor:

Date: Section:

Chapter 6 Test Form F (cont’d) 1 5 = 20 7 x 10

14.

c. x =

7 5

d. x =

b. x = 1

c. x =

1 3

d. x = 2

b. x = 2.8

c. x = 1

d. x = 10

b. x =

a. x = 70

14 5

40 7

1.5 9 = x 6

15.

a. x = 3

7.5 9

280 28 = 10 x

16.

a. x = 28

Write each as a unit rate. 17. 285 miles to 10 gallons a. 28.5 miles per gallon

b. 2850 miles per gallon

2 miles per 57 gallon

57 miles per 2 gallon

18. 14 baskets attempted to 7 completed a. 2 attempted to 1 completed

b. 4 attempted to 1 completed

c. 4 completed to 1 attempted

d. 4 attempted to 7 completed

c. x = 12

d. x = 9

19. Given the similar triangles, find x. 3 12 4 x a. x = 16

T-126

b. x = 4

Name: Instructor:

Date: Section:

Chapter 6 Test Form F (cont’d) 20. Find the distance between 2 cities which are 2 83 inches apart on a road map if

1 in. represents 4

50 miles. a. 480 miles

b. 465 miles

21. Five large trees are planted in each

c. 475 miles

d. 470 miles

1 acre of land. How many trees will need to be planted in 2 4

acres? a. 3 trees

b. 40 trees

c. 20 trees

d. 10 trees.

22. A student can type 5 pages in 45 minutes. How long will it take this student to type 65 pages? a. 480 minutes

b. 485 minutes

c. 585 minutes

d. 580 minutes

Find each square root. Round to the nearest thousandth if necessary. 23.

a. 4

c. 8

d. 16

4 49

24.

b. 2

2 7

25. a. 43

4 7

16 49

8 29

172

b. 86

c. 13.115

d. 13.114

T-127

Name: Instructor:

Date: Section:

Chapter 6 Test Form G Write each ratio or rate as a fraction in simplest terms. 1. 90 pens to 72 pencils a.

72 90

90 72

4 5

5 4

85 395

395 852

452 395

12 5

c. 7

7 12

3 2

2 3

1 4

1 6

36 6

d. 6

3 87

87 261

36 42

2. $395 to $852 a.

852 395

3. 12 miles to 5 miles a.

5 12

4. 8 shelves to 12 shelves a.

12 8

5. 6 tables to 36 chairs a.

6 36

6. 87 blocks to 261 blocks a.

27 87

1 3

7. 42 math classes to 36 English classes a.

7 6

T-128

6 7

4 5

Name: Instructor:

Date: Section:

Chapter 6 Test Form G (cont’d) Write each sentence as a proportion. 1 cup of blueberries is to 8 muffins as 1 cup blueberries is to 16 muffins. 2

1 1 a. 2 = 8 16

1 8 = 2 16

2 1 = 8 16

d. 2 = 1

9. 13 printers is to 25 computers as 26 printers is to 50 computers. a.

26 25 = 13 50

26 1 = 13 2

13 26 = 25 50

2 1 = 1 2

Determine whether the proportion is true. 10.

17 51 = 12 36

a. true

11.

b. false

8 15 = 15 30

a. true

b. false

Solve the proportion. 12.

2.5 x = 4 6.8

a. x = 42.5

13.

b. x = 425

c. x = 0.425

d. x = 4.25

b. x = 15

c. x =

1 15

d. x = 9

1 5 = x 3 15

a. x = 1

T-129

Name: Instructor:

Date: Section:

Chapter 6 Test Form G (cont’d) 0.9 3 = x 81

14.

a. x = 2.43

b. x = 243

c. x = 0.243

d. x = 24.3

b. x = 27.75

c. x = 277.5

d. x = 2.7

b. $0.20/ ounce

c. $0.05/ ounce

d. 20 ounces

b. 68.7 mi./hr.

c. 76.8 mi./hr.

d. 7.68 mi./hr.

b. $76 per hour

c. $9.50 per hour

d. $8.60 per hour

c. x = 30

d. x = 21

37 4 = x 3

15.

a. x = 2.775

Write each as a unit rate. 16. $3.60 for 18 ounces a. $0.10/ ounce

17. 384 miles to 5 hours a. 67.8 mi./hr.

18. $215 to 25 hours a. $7.60 per hour

19. Given the similar triangles, find x. 5 15 7 x a. x =

T-130

5 7

b. x = 12

Name: Instructor:

Date: Section:

Chapter 6 Test Form G (cont’d) 20. A mason is building a wall. The plans on paper show

1 inch for every 10 feet. How long is the 2

wall if the plans show 3 inches? a. 12 feet

b. 60 feet

c. 30 feet

d. 20 feet

21. If you can type 5 pages in 60 minutes, how many pages can you type in 120 minutes? a. 15 pages

b. 2 pages

c. 10 pages

d. 12 pages

22. If a 6-foot tall person casts a shadow 8 feet long, how tall is a tree which casts a 32 foot shadow? a. 27 feet

b. 24 feet

c. 28 feet

d. 32 feet

Find each square root. Round to the nearest thousandth if necessary. 23.

a. 7

24.

1 4

9 49

25.

b. 49

3 49

1 2

d. 24.5

3 7

231

a. 15.100

b. 15.199

c. 115.5

d. 15.198

T-131

Name: Instructor:

Date: Section:

Chapter 6 Test Form H Write each ratio or rate as a fraction in simplest terms. 1. 14 baskets to 24 baskets a.

24 14

14 24

12 7

7 12

12 7

7 12

8.7 3.2

87 32

5 33

33 5

4 3

3 4

4 18

84 1

29 3

1 3

3 1

2. 5 calculators to 7 calculators a.

5 7

b. 2

3. 3.2 cm to 8.7 cm a.

32 87

3.2 8.7

1 1 inches to 2 inches 3 5 3 1 2 5

33 10

5. 9 tennis players to 12 cans of tennis balls a.

3 1

b. 1

2 3

6. 4 hours to18 km a.

18 4

9 2

2 9

7. 3 boxes to 87 books a.

29 1

1 29

8. 9 puppies to 3 children a.

6 3

T-132

b. 6

Name: Instructor:

Date: Section:

Chapter 6 Test Form H (cont’d) Write each sentence as a proportion. 9. 2 cups of nuts is to 24 cookies as 3 12 cups of nuts is to 42 cookies. a.

2 24 = 1 42 3 2

1 3 2 b. = 2 24 42

1 3 2 c. = 2 42 24

1 3 24 d. = 2 2 42

10. $78 is to 2 books as $156 is to 4 books. a.

78 156 = 2 4

2 156 = 4 78

2 4 = 78 156

d. 78 =

156 2

Determine whether the proportion is true.

11.

5 25 = 24 96

a. true

12.

b. false

9 18 = 15 30

a. true

b. false

Find each rate as a unit rate. 13. 180 calories to 5 ounces a. 0.3 calories/oz.

180 calories/oz. 5

5 calories/oz. 180

d. 36 calories/oz.

14. 638 miles in 8.5 hours (Round to the nearest mile.) a. 75 mi./hr.

b. 78 mi./hr.

c. 69.5 mi./hr.

d. 76 mi./hr.

T-133

Name: Instructor:

Date: Section:

Chapter 6 Test Form H (cont’d) Solve. 15.

4 12 = 15 x

a. x = 15

16.

d. x = 3.2

b. x = 28

c. x = 24

d. x = 39

b. x = 24

c. x = 32

d. x = 27

b. x = 4.8

c. x = 48

d. x= 4.6

c. x = 9

d. x = 9.5

9 x = 63 189

a. x = 25

18.

c. x = 5

13 x = 54 108

a. x = 26

17.

b. x = 45

1 1.5 = 32 x

a. x = 46

19. Given the similar triangles, find x. 1.5

1.5

a. x = 4.5

7.5

b. x = 7.5

20. A classroom should have at least 9 square feet of floor space for each student. Find the minimum floor space for 36 students. a. 348 square feet

T-134

b. 388 square feet

c. 324 square feet

d. 342 square feet

Name: Instructor:

Date: Section:

Chapter 6 Test Form H (cont’d) 21. A math class has 5 men and 12 women. If a similar ratio applies to a class with 15 men, how many women can be expected to be in the 2nd class? a. 35 women

b. 34 women

c. 36 women

d. 38 women

22. If 9 out of 15 people prefer vanilla to chocolate ice cream, how many out of 75 people would prefer vanilla ice cream? a. 5

b. 45

c. 30

d. 127

Find each square root. Round to the nearest thousandth if necessary. 23.

a. 3

24.

b. 6

c. 9

d. 18

49 100

25.

a. 6.000

7 10

b. 6.083

7 100

c. 18.5

49 10

d. 6.082

T-135

Name: Instructor:

Date: Section:

Cumulative Test Form A Chapters 5 and 6 Perform the indicated operation. Round the result to the nearest thousandth if necessary. 1. 32.4 + 9.34 + 24

1. __________________________

2. −23.9 + (−46.7)

2. __________________________

3. 3.5 – 1.61

3. __________________________

4. (9.3)(5.7)

4. __________________________

5. (−0.15) ÷ (0.25)

5. __________________________

6. Round 34.6945 to the nearest tenth.

6. __________________________

7. Insert <, >, or = to make a true statement: 39.0205 39.2051

7. __________________________

8. Write 0.034 as a fraction in simplest terms

8. __________________________

2 as a decimal. If necessary round to the nearest 7 thousandths place.

9. Write

9. __________________________

Simplify. 10. (0.4)2 + 12.3

10. __________________________

11. 2.35x – 4.2x + 4.3

11. __________________________

12. Find the square root and simplify:

49 64

12. __________________________

Solve. 13. 2(x + 3.57) = x + 4.31

13. __________________________

14. 6x + 29.3 = 5x – 18.61

14. __________________________

Write each ratio or rate as a fraction in simplest form. 15. 30 men to every 12 women.

15. __________________________

16. 5 inches to 45 inches

16. __________________________

T-136

Name: Instructor:

Date: Section:

Cumulative Test Form A Chapters 5 and 6 (cont’d) Find each unit rate. 17. 290 km. in 10 hours.

17. __________________________

18. 120 computers for 20 printers

18. __________________________

19. Determine if

13 1 = is a true proportion. 39 3

19. __________________________

Solve the proportions. 20.

x 6 = 7 21

20. __________________________

21.

5 x = 1.5 4.8

21. __________________________

22. Given that the following triangles are similar, find the missing length, x.

22. __________________________

4 12 x 18 23. A scale drawing for a sundeck is 2 inches corresponding to 9 feet. Find the length of the deck if a line is 6 inches long. 24. A student can drive their car for 87.75 miles on 2 gallons of gas. How many miles can they drive on one gallon of gas? (Round to the nearest tenth of a mile.)

25. Find the area of the triangle; use the formula A =

1 bh . 2

23. __________________________

24. __________________________

25. __________________________

2.5 cm 4.6 cm

T-137

Name: Instructor:

Date: Section:

Cumulative Test Form B Chapters 5 and 6 Perform the indicated operation. Round the result to the nearest thousandth if necessary. 1. 2.35 + 12.5 + 0.003 a. 3.63

b. 14.88

c. 14.853

d. 14.353

b. 7.29

c. –9.39

d. −7.29

b. 13.41

c. 13.59

d. −13.59

b. 13.876

c. 13.888

d. 13.886

b. 28.241

c. 28.243

d. −28.243

c. 400

d. 356.26

2. 8.34 – (−1.05) a. 9.39

3. 3.21 – 16.8 a. –13.41

4. (8.52)(1.63) a. 13.8

5. (−65.52) ÷ (−2.32) a. −28.241

6. Round 356.2584 to the nearest hundredth. a. 356.3

b. 356.25

7. Insert <, >, or = to make a true statement: a. <

T-138

b. >

8.2535 c. =

8.2525

Name: Instructor:

Date: Section:

Cumulative Test Form B Chapters 5 and 6 (cont’d) 8. Write 3.4 as a fraction or mixed number in simplest form. a.

3 4

9. Write

b. 3

2 5

34 10

d. 3

4 6

7 as a decimal. If necessary, round to the nearest hundredths place. 9

a. 0.777

b. 0.778

c. 0.78

d. 0.7777

b. 28.8

c. 27.927

d. 27.99

c. 12.25x + 20

d. 12.25x – 20

Simplify 10. (0.3)2 + 27.9 a. 27.9

11. 8.5x – 18.2 + 3.75x + 1.8 a. 12.25x – 16.4

b. 12.25x + 16.4

12. Find the square root and simplify

a. 9

9 10

9 100

81 100

3 10

Solve. 13. 5x − 9.36 = 4x – 2.94 a. x = 6.42

b. x = –6.42

c. x = –12.3

d. x = 12.3

T-139

Name: Instructor:

Date: Section:

Cumulative Test Form B Chapters 5 and 6 (cont’d) 14. 0.3x + 2.9 = 0.2x + 5.8 a. 87

b. 8.7

c. 2.9

d. 29

Write each ratio or rate as a fraction in simplest form. 15. 68 puppies to 28 kittens

7 17

7 15

6 17

17 7

4 3

3 4

3 5

16. 24 yards to 18 yards

2 3

Find each unit rate. (Round to the nearest tenth if necessary.) 17. 2472 miles in 34 hours. a. 72.71 mi./hr.

b. 72.5 mi./hr.

c. 72.7 mi./hr.

d. 72.0 mi./hr.

18. $1.44 for 36 ounces (round to the nearest cent) a. $0.03/oz.

T-140

b. $0.04/oz.

c. $0.4/oz.

d. $0.034/oz.

Name: Instructor:

Date: Section:

Cumulative Test Form B Chapters 5 and 6 (cont’d) Solve the proportion. 3 x = 19. 21 7 a. 0

20.

c. 49

d. 21

b. x = 21

c. x = 14

d. x = 30

b. x = 0.2

c. x = 20

d. x = 200

7 x = 18 54

a. x = 28

21.

b. 1

0.25 5.00 = 1.00 x

a. x = 2

22. Find the circumference of a circle with radius 5 cm. Use 3.14 for π and the formula C = πd. a. 15.70 cm

b. 3.14 cm

c. 78.5 cm

d. 31.4 cm

c. 7.3

d. 9.7

23. Given that the triangles are similar, find x. 1.2

x 9

a. 8.5

b. 1.2

24. Find the perimeter of a rectangle with width 12.5 cm and length 24.3 cm. a. 36.8 cm

b. 303.75 cm

c. 73.6 cm

d. 11.8 cm

25. If 13 out of 17 adults are coffee drinkers, how many would we expect to be coffee drinkers out of a group of 136 adults? a. 104

b. 10.4

c. 140

d. 144

T-141

Name: Instructor:

Date: Section:

Chapter 7 Pretest Form A 1. An algebra class has 25 students. Eighteen of the students have dark hair. Find the percent of students with dark hair in this class.

1. _______________________

2. Write 35% as a decimal.

2. _______________________

3. Write 1.23 as a percent.

3. _______________________

4. Write 0.4% as a fraction in simplest form.

4. _______________________

5. Write 2 14 as a percent.

5. _______________________

Translate each question into an equation. Do not solve. 6. 12% of 95 is what number?

6. _______________________

7. 5.6% of what number is 372?

7. _______________________

Translate each question into a proportion. Do not solve. 8. 12% of what number is 10.8?

8. _______________________

9. 18 is what percent of 72? Solve.

9. _______________________

10. 9 is 15% of what number?

10. _______________________

11. 25 is what percent of 250?

11. _______________________ 12. _______________________

12. What number is 400% of 28? 13. Capital High School has 1460 students. 30% of the students are seniors. Find the number of seniors.

13. _______________________

14. Junita’s salary of $600 increased by 5%. Find her new salary.

14. _______________________

15. A car salesperson receives 8.5% commission on each new car sale. If the salesperson sells a car for $23,590, how much will be paid as a commission?

15. _______________________

16. Find the simple interest earned on an investment of $2000 after 2 years. The interest rate is 8%.

16. _______________________

17. $500 is invested at 5% compounded semiannually for 2 years. Find the total amount at the end of 2 years.

17. _______________________

18. Find the simple interest after 5 years on a $1500 investment if the interest rate is 2.5%.

18. _______________________

19. A new coat regularly sells for $360. It is on sale for 25% off. Find the new price.

19. _______________________

20. If $12,500 is borrowed at 8% simple interest for 2 years, find the total amount due at the bank at the end of 2 years.

20. _______________________

T-142

Name: Instructor:

Date: Section:

Chapter 7 Pretest Form B 1. A community college has 8200 female students. If 61% of the students are female, how many of the students are female?

1. _______________________

2. Write 87% as a decimal.

2. _______________________

3. Write 0.003 as a percent.

3. _______________________

4. Write 2.4% as a fraction in simplest form.

4. _______________________

5. Write

5 8

5. _______________________

as a percent.

Translate each question into an equation. Do not solve. 6. 12.8% of 160 is what number?

6. _______________________

7. 17% of what number is 156?

7. _______________________

Translate each question into a proportion. Do not solve. 8. 17% of what number is 34? 9. 30 is what percent of 90?

8. _______________________ 9. _______________________

Solve. (Round to the nearest tenth if needed.) 10. 11 is 22% of what number?

10. _______________________

11. 27 is what percent of 900?

11. _______________________

12. What number is 6.25% of 138?

12. _______________________

13. An instructor received a promotion at work, resulting in an 8% raise in salary. If the old salary was $22,500, what will the new salary be?

13. _______________________

14. Morrow High School has 420 juniors. This is 20% of the total student body. Find the number of students at Morrow High School.

14. _______________________

15. If a salesperson receives 12% commission on each tractor sale, how much will be earned when a tractor priced at $6800 is sold?

15. _______________________

16. Find the simple interest earned on $500 for 5 years if the interest rate is 8%.

16. _______________________

17. Find the total amount in an account after 3 years if $600 is compounded annually. The interest rate is 10%.

17. _______________________

18. Shoes that normally sell for $68 are on sale for $48. Find the discount rate. Round to the nearest tenth.

18. _______________________

19. Find the total amount due after 2 years on a $1200 loan with a 6% simple interest rate.

19. _______________________

20. $1800 is invested at 5.5% simple interest for 4 years. Find the total in this account.

20. _______________________

T-143

Name: Instructor:

Date: Section:

Chapter 7 Test Form A Write each percent as a decimal. 1. 329%

1. __________________________

2. 5.3%

2. __________________________

3. 0.10%

3. __________________________

Write each decimal as a percent. 4. 0.321

4. __________________________

5. 3.9

5. __________________________

6. 0.73

6. __________________________

Write each percent as a fraction or a mixed number in simplest form. 7. 172%

7. __________________________

8. 58%

8. __________________________

9. 0.3%

9. __________________________

Write each fraction or mixed number as a percent. 10.

11.

12.

10. __________________________

11. __________________________

12. __________________________

Solve. 13. What number is 12% of 200?

13. __________________________

14. 5% of what number is 32?

14. __________________________

T-144

Name: Instructor:

Date: Section:

Chapter 7 Test Form A (cont’d) 15. 12 is what percent of 96?

15. __________________________

16. What number is 3.5% of 180?

16. __________________________

17. 16% of what number is 64?

17. __________________________

18. 18 is what percent of 180?

18. __________________________

19. Find the sales tax on a purchase of $2000 if the tax rate is 6.25%.

19. __________________________

20. Find the simple interest earned on $2000 saved for 8 years with interest rate of 3.5%

20. __________________________

21. A sales tax of $8.25 is added to a purchase of $165.00. Find the tax rate. (Round to the nearest tenth of a percent if needed.)

21. __________________________

22. A college student borrowed $500 to buy books. He repaid the loan after 6 months. The interest rate was 4%. Find the total amount repaid.

22. __________________________

23. Juanita invested $500 in an account compounded annually at 8.5%. Find the total amount in the account after 2 years.

23. __________________________

24. A dress is on sale for 25% off the regular price of $120. Find the sale price.

24. __________________________

25. A realtor earns 3% commission for each home sold. Find the commission on a $195,000 home.

25. __________________________

T-145

Name: Instructor:

Date: Section:

Chapter 7 Test Form B Write each percent as a decimal. 1. 193%

1. __________________________

2. 3.5%

2. __________________________

3. 87%

3. __________________________

Write each decimal as a percent. 4. 0.4

4. __________________________

5. 0.397

5. __________________________

6. 3.2

6. __________________________

Write each percent as a fraction or a mixed number in simplest form. 7. 95%

7. __________________________

8. 120%

8. __________________________

9. 12.5%

9. __________________________

Write each fraction or mixed number as a percent. (Round to the nearest tenth if needed.) 10.

11. 3

12.

10. __________________________

3 4

11. __________________________

5 9

12. __________________________

Solve. 13. What number is 15% of 135?

13. __________________________

14. 12% of what number is 138?

14. __________________________

T-146

Name: Instructor:

Date: Section:

Chapter 7 Test Form B (cont’d) 15. 3 is what percent of 8?

15. __________________________

16. What number is 0.5% of 70?

16. __________________________

17. 5% of what number is 50?

17. __________________________

18. 21 is what percent of 70?

18. __________________________

19. Find the sales tax due on a purchase of $1050 if the tax rate is 5%. (Round to the nearest cent.)

19. __________________________

20. Find the simple interest earned on $500 saved for 2 years with interest rate of 7%

20. __________________________

21. A sales tax of $6.18 is added to a purchase of $95.00. Find the tax rate. (Round to the nearest tenth of a percent if needed.)

21. __________________________

22. A college loan of $1500 is borrowed for 1 year with interest rate of 8%. What is the total amount due?

22. __________________________

23. $1000 is invested in an account which is compounded semiannually at 5%. Find the total in the account after 18 months.

23. __________________________

24. A new dress regularly selling for $250 is on sale for 30% off. Find the sale price.

24. __________________________

25. Find the commission on $85,000 sales if the commission rate is 4%.

25. __________________________

T-147

Name: Instructor:

Date: Section:

Chapter 7 Test Form C Write each percent as a decimal. 1. 38%

1. __________________________

2. 3.5%

2. __________________________

3. 0.09%

3. __________________________

Write each decimal as a percent. 4. 3.32

4. __________________________

5. 0.04

5. __________________________

6. 0.063

6. __________________________

Write each percent as a fraction or a mixed number in simplest form. 7. 175%

7. __________________________

8. 35%

8. __________________________

9. 0.25%

9. __________________________

Write each fraction or mixed number as a percent.

10.

10. __________________________

1 16

11.

5 8

11. __________________________

12.

12. __________________________

Solve. 13. What number is 6% of 280?

13. __________________________

14. 8% of what number is 32?

14. __________________________

T-148

Name: Instructor:

Date: Section:

Chapter 7 Test Form C (cont’d) 15. 3 is what percent of 20?

15. __________________________

16. What number is 1.2% of 500?

16. __________________________

17. 6.2% of what number is 31?

17. __________________________

18. 318 is what percent of 159?

18. __________________________

19. Find the simple interest earned on $250 for 5 years with interest rate 6.5%. (Round to the nearest cent.)

19. __________________________

20. Find the sales tax due on groceries totaling $79.32 if the rate is 6.25%. (Round to the nearest cent.)

20. __________________________

21. A sales tax of $14.66 is added to a purchase of $318.59. Find the tax rate. (Round to the nearest tenth of a percent if needed.)

21. __________________________

22. A school loan of $450 is to be repaid in 9 months. The interest rate is 4.5%. Find the total amount due.

22. __________________________

23. $800 is invested in an account that is compounded annually for 4 years. The interst rate is 6.8%. Find the total amount. (Round to the nearest cent.)

23. __________________________

24. A television which regularly sells for $350 is on sale for 15% off. Find the sale price.

24. __________________________

25. Find the commission on sales of $17,500 if the commission rate is 6.2%.

25. __________________________

T-149

Name: Instructor:

Date: Section:

Chapter 7 Test Form D Write each percent as a decimal. 1. 12%

1. __________________________

2. 2.4%

2. __________________________

3. 8%

3. __________________________

Write each decimal as a percent. 4. 0.6

4. __________________________

5. 0.97

5. __________________________

6. 5.3

6. __________________________

Write each percent as a fraction or a mixed number in simplest form. 7. 86%

7. __________________________

8. 1.4%

8. __________________________

9. 135%

9. __________________________

Write each fraction or mixed number as a percent.

10.

11.

12.

10. __________________________

3 5

1 5

11. __________________________

12. __________________________

Solve. 13. What number is 2% of 240?

13. __________________________

14. 1.5% of what number is 82?

14. __________________________

T-150

Name: Instructor:

Date: Section:

Chapter 7 Test Form D (cont’d) 15. 16 is what percent of 128?

15. __________________________

16. What number is 2.5% of 800?

16. __________________________

17. 8% of what number is 62?

17. __________________________

18. 13 is what percent of 200?

18. __________________________

19. Find the sales tax on a washer and dryer totaling $869 if the sales tax rate is 7.25%. (Round to the nearest cent.)

19. __________________________

20. Find the simple interest earned on $2500 saved with a rate of 6.5% for 3 years. (Round to the nearest cent.)

20. __________________________

21. A sales tax of $6.15 is added to a grocery bill of $102.50. Find the tax rate. (Round to the nearest tenth of a percent if needed.)

21. __________________________

22. A college student borrowed $300 to buy books. The loan was repaid after 120 days. The interest rate was 4%. Find the total amount due.

22. __________________________

23. If $2000 is invested in an account paying interest compounded semi-annually at 8%, find the total in the account after 2 years. (Round to the nearest cent.)

23. __________________________

24. Find the cost of a computer if it is 15% off the regular price of $2800.

24. __________________________

25. Find the commission on a $95,000 home if the commission rate is 6%.

25. __________________________

T-151

Name: Instructor:

Date: Section:

Chapter 7 Test Form E Write each percent as a decimal. 1. 390%

1. __________________________

2. 5%

2. __________________________

3. 0.7%

3. __________________________

Write each decimal as a percent. 4. 0.4

4. __________________________

5. 0.003

5. __________________________

6. 0.73

6. __________________________

Write each percent as a fraction or a mixed number in simplest form. 7. 150%

7. __________________________

8. 24.4%

8. __________________________

9. 0.5%

9. __________________________

Write each fraction or mixed number as a percent. (Round to the nearest tenth if needed.)

10.

11.

12.

1 2

10. __________________________

11. __________________________

12. __________________________

7 8

Solve. 13. What number is 2.5% of 138?

13. __________________________

14. 5% of what number is 685?

14. __________________________

T-152

Name: Instructor:

Date: Section:

Chapter 7 Test Form E (cont’d) 15. 85 is what percent of 250?

15. __________________________

16. What number is 42% of 180?

16. __________________________

17. 0.5% of what number is 75?

17. __________________________

18. 114 is what percent of 456?

18. __________________________

19. Find the sales tax due on groceries totalling $127.62 if the tax rate is 5.2%. (Round to the nearest cent.)

19. __________________________

20. Find the simple interest earned on $1800 saved for 4 years with interest rate of 8%.

20. __________________________

21. Sales tax of $28.80 is added to the price of a TV. Find the tax rate if the price of the TV is $480. (Round to the nearest tenth of a percent if needed.)

21. __________________________

22. A loan of $600 is to be repaid in 90 days. The rate is 3%. Find the total amount due.

22. __________________________

23. Find the total in a savings account after 2 years if $2200 is invested in a compounded annually account with interest rate of 10%.

23. __________________________

24. Find the sale price of a automobile that is on sale for 5% off the regular price of $22,500.

24. __________________________

25. A salesperson receives 3.5% commission on sales. Find the commission earned if sales total $685. Round to the nearest cent.

25. __________________________

T-153

Name: Instructor:

Date: Section:

Chapter 7 Test Form F Write each percent as a decimal. 1. 93% a. 93

b. 0.93

c. 9300

d. 9.3

b. 0.8

c. 80

d. 800

b. 0.01

c. 0.001

d. 1

b. 12.7%

c. 0.127%

d. 1.27%

b. 290%

c. 29%

d. 0.029%

b. 50%

c. 0.05%

d. 0.5%

2. 8% a. 0.08 3. 0.1% a. 0.1

Write each decimal as a percent. 4. 0.127 a. 127% 5. 0.29 a. 2.9% 6. 0.005 a. 5

Write each percent as a fraction or a mixed number in simplest form. 7. 3.5% a.

7 20

b. 35 100

7 200

1 2

3 8

3 4

3 50

2 5

35 200

8. 75% a.

15 20

b. 7

9. 6% b.

a. 6

3 5

Write each fraction or mixed number as a percent. 10.

5 8

a. 1.625%

b. 16.25%

c. 162.5%

d. 1625%

3 5 a. 0.6%

b. 6%

c. 60%

d. 600%

11.

T-154

Name: Instructor: Chapter 7 Test Form F (cont’d) 12.

Date: Section:

17 20

a. 85%

b. 8.5%

c. 17.5%

d. 75%

Solve. (Translate into an equation or into a proportion. Round to the nearest tenth if needed.) 13. What number is 2.2% of 60? a. 2.73

b. 1.32

c. 132

d. 13.2

c. 98

d. 200

c. 20%

d. 24%

c. 10500

d. 1680

c. 320

d. 3200

c. 34%

d. 30%

14. 7% of what number is 14? a. 50

b. 220

15. 23 is what percent of 115? a. 27%

b. 22%

16. What number is 25% of 420? a. 105

b. 10.5

17. 0.5% of what number is 16? a. 320.2

b. 32.0

18. 85 is what percent of 250? a. 29.4%

b. 25%

19. Find the sales tax on a purchase of $850 if the tax rate is 5.25%. (Round to the nearest cent.) a. $44.63 20.

b. $4.46

c. $44.00

d. $40.60

Find the simple interest earned on an investment of $350 for 5 years if the interest rate is 8.2%. (Round to the nearest cent.)

a. $140.35

b. $14.35

c. $143.50

d. $143.00

21.

Sales tax of $2.52 is added to a purchase of $43.45. Find the tax rate. (Round to the nearest tenth of a percent if needed.) a. 5.1% b. 5.8% c. 5% d. 8.5%

22. A loan of $1200 is to be repaid in 1 year. The interest rate is 4%. Find the total to be paid using simple interest. a. $48 b. $1248 c. $1152 d. $1680 23. Shoes, which regularly sell for $68, are on sale for $46. What is the percent decrease? a. 32.4% b. 30.4% c. 33% d. 68% 24. A prom dress that sells for $260 is on sale for 25% off. Find the sale price of the dress. a. $165 b. $185 c. $195 d. $65 25. Find the commission on a sale of $89.70 if the commission rate is 3.5%. (Round to the nearest cent.) a. $31.40 b. $3.14 c. $30.40 d. $3.41 Copyright © 2025 Pearson Education, Inc.

T-155

Name: Instructor:

Date: Section:

Chapter 7 Test Form G Write each percent as a decimal. 1. 1.57% a. 15.7

b. 0.157

c. 0.0157

d. 1.57

b. 150

c. 0.15

d. 15,000

b. 1.68

c. 168

d. 16.8

b. 0.972%

c. 97.2%

d. 9720%

b. 32%

c. 0.0032%

d. 3.2%

b. 6%

c. 60%

d. 0.006%

2. 15% a. 1.5 3. 168% a. 0.168

Write each decimal as a percent. 4. 0.972 a. 972% 5. 0.0032 a. 0.32% 6. 0.60 a. 0.6%

Write each percent as a fraction or a mixed number in simplest form. 7. 3.4% a.

34 10

17 50

34 100

17 500

11 20

11 200

1 5

13 1000

c. 1

3 10

d. 13

8. 55% 5 100

a. 5

9. 13% a.

13 100

Write each fraction or mixed number as a percent. Round to the nearest, if needed. 10.

3 20

a. 20%

b. 15%

c. 6 % 3

d. 1.5%

c. 38%

d. 380%

11. 3 54 a. 3.8%

T-156

b. 0.38%

Name: Instructor:

Date: Section:

Chapter 7 Test Form G (cont’d) 12.

7 8

a. 87.5%

b. 0.875%

c. 8.75%

d. 88%

c. 224

d. 22.4

c. 145.8

d. 642.8

c. 20%

d. 38.5%

c. 11.7

d. 117

c. 1000

d. 100.7

c. 19.2%

d. 192%

Solve. 13. What number is 7% of 320? a. 2.24

b. 2240

14. 7.2% of what number is 45? a. 324

b. 625

15. 169 is what percent of 650? a. 26%

b. 38.4%

16. What number is 6.5% of 180? a. 1.17

b. 1170

17. 7.3% of what number is 73? a. 10

b. 100

18. 91 is what percent of 175? a. 52.2%

b. 52%

19. Find the sales tax on a purchase of $218.50 if the tax rate is 6.5%. (Round to the nearest cent.) a. $14.20

b. $1.42

c. $142

d. $14.80

20. Find the simple interest earned on an investment of $1200 for 2 years. The interest rate is 6.2%. (Round to the nearest cent.) a. $148 21.

b. $14.88

c. $148.80

d. $1488

Sales tax of $5.25 is added to a purchase of $102.87. Find the tax rate. (Round to the nearest tenth of a percent if needed.)

a. 5.1%

b. 51%

c. 5.9%

d. 5%

22. A loan of $1800 is to be repaid in 18 months. Find total due if the interest rate is 5%, simple interest. a. $135 b. $1939.89 c. $1935 d. $150 23. Winter coats that sold for $125 are now on sale for $95. Find the percent decrease. a. 32% b. 76% c. 24% d. 25% 24. Swimsuits are included in a 40% off sale. If a suit was originally priced at $55, what is the sale price? a. $28 b. $40 c. $22 d. $33 25. Find the commission on the sale of a $165,000 home if the commission rate is 2.5%. a. $4100 b. $4125 c. $4150 d. $4152 Copyright © 2025 Pearson Education, Inc.

T-157

Name: Instructor:

Date: Section:

Chapter 7 Test Form H Write each percent as a decimal. 1. 90% a. 9

b. 0.9

c. 0.09

d. 900

b. 32.5

c. 3250

d. 3.25

b. 12

c. 0.12

d. 0.0012

b. 375%

c. 37.5 %

d. 3.75%

b. 1.65%

c. 0.165%

d. 1.65%

b. 2638%

c. 2.638%

d. 263.8%

2. 325% a. 0.325 3. 0.12% a. 1.2

Write each decimal as a percent. 4. 0.0375 a. 0.375% 5. 1.65 a. 165% 6. 26.38 a. 0.2638%

Write each percent as a fraction or a mixed number in simplest form. 7. 120% a. 1

1 5

8. 66% 33 a. 500 9. 5% 1 a. 2

12 10

12 100

6 50

33 50

11 25

d. 6

1 20

1 200

d. 2

3 5

Write each fraction or mixed number as a percent. 10.

4 25

a. 20%

b. 1.6%

c. 16%

d. 0.16%

b. 210%

c. 21%

d. 0.21%

b. 0.16%

c. 1.6%

d. 6.25%

1 11. 2 10

a. 2.1% 12.

1 16

a. 16% T-158

Name: Instructor:

Date: Section:

Chapter 7 Test Form H (cont’d) Solve. 13. What number is 8% of 2580? a. 206.4

b. 208.9

c. 208

d. 206

c. 250

d. 32.4

c. 24%

d. 41.6%

c. 89700

d. 0.897

c. 60

d. 6

c. 10%

d. 12%

14. 3.6% of what number is 9? a. 324

b. 25

15. 60 is what percent of 250? a. 15%

b. 22.5%

16. What number is 65% of 1380? a. 8.97

b. 897

17. 0.5% of what number is 30? a. 6000

b. 600

18. 15 is what percent of 1500? a. 10.1%

b. 1%

19. Find the sales tax due on a purchase of $65.80 if the tax rate is 7.2%. (Round to the nearest cent.) a. $4.70

b. $47.40

c. $4.47

d. $4.74

20. Find the simple interest earned on an investment of $800 for 1 12 years. The interest rate is 3.5%. (Round to the nearest cent.) a. $42.20 21.

b. $40

c. $42

d. $40.20

Sales tax of $8.25 is added to a purchase of $149.98. Find the tax rate. (Round to the nearest tenth of a percent if needed.)

a. 5.5%

b. 5%

c. 18%

d. 55%

22. A loan of $586.00 is to be repaid in 9 months. The interest rate is 4%. Find the total due with simple interest. a. $17.58

b. $603.58

c. $55

d. $641

23. A faux leather sofa is on sale for $425. This is a decrease from the regular price of $580. Find the percentage decrease. a. 26.7% b. 2.67% c. 27.6% d. 27% 24. All men’s suits that are regularly priced at $250 are on sale for 30% off. What is the sale price of the suits? a. $75 b. $175 c. $167 d. $166.67 25. Find the commission on the sale of a $2500 T.V. if the commission rate is 6%. a. $150 b. $155 c. $2450 d. $1500 Copyright © 2025 Pearson Education, Inc.

T-159

Name: Instructor:

Date: Section:

% Smokers (18+ age)

Chapter 8 Pretest Form A

30 25 20 15 10 5 0 Alaska

Maine

Utah

Indiana

Kentucky

1. From the graph, which state has the greatest percentage of smokers age 18 and older?

1. _______________________

2. Which state has the smallest percentage of smokers age 18 and older?

2. _______________________

3. What can be said about Alaska and Maine?

3. ______________________

Number of Wrecks

Intersection of Main and Independence Streets

May

Mar

Jan 0

4. What was the number of wrecks in January?

4. _______________________

5. Which month had the least number of wrecks?

5. _______________________

6. What was the total number of wrecks for these six months?

6. _______________________

7. What two months had the same number of wrecks?

7. _______________________

T-160

Name: Instructor:

Date: Section:

Chapter 8 Pretest Form A (cont’d) Top Grossing Movies of 2006 Pirates of the Caribbean: Dead Man’s Chest Night at the Museum

= $50 million

Cars X-Men: The Last Stand The Da Vinci Code Superman Returns

8. Which movie made the most money in 2006?

1. _______________________

9. How much money did Night at the Museum make?

2. _______________________

10. How much money did X-Men: The Last Stand make?

3. _______________________

11. How much more money did Pirates of the Caribbean: Dead Man’s Chest make than Superman Returns?

4. _______________________

12. Is (4, -6) a solution to the equation -2x + y = -14?

5. _______________________

1 13. Complete the ordered-pair solutions of the equation: y = - x - 4 5 (5, ), (-5, ), (0, ) 14. Graph y = 2x - 2.

6. _______________________

14.

T-161

Name: Instructor:

Date: Section:

Chapter 8 Pretest Form A (cont’d) 15. Graph x + 2y = 4.

15.

16. Graph x = 3.

16.

17. If a single die is tossed, find the probability that an even number will appear.

17. _______________________

18. Find the probability that a “1” will appear if a single die is tossed.

18. _______________________

19. From a deck of cards (52 total), what is the probability that an “Ace” of hearts will be selected the first time?

19. _______________________

20. What is the probability that diamond card is chosen?

20. _______________________

T-162

Name: Instructor:

Date: Section:

Chapter 8 Pretest Form B

Thousand students (000)

Enrollment at South Plains College 7 6 5 4 3 2 1 0

Freshmen Sophomores

2002

2003

From the graph: 1. Which year had the lowest enrollment?

1. _______________________

2. What can be said about the enrollment from 2002 to 2003?

2. _______________________

3. In 2002, were there more freshmen or sophomores?

3. _______________________

4. What was the total enrollment for 2002?

4. _______________________

5. What was the total enrollment for 2003?

5. _______________________

6. Is (5, -8) a solution to the equation y = -x – 3? 7. Draw a circle graph showing this data. Nia’s Schedule for one day Class time Study Work Sleep Driving Other

6. _______________________ 7.

6 hrs 5 hrs 4 hrs 5 hrs 1 hr 3 hrs

T-163

Name: Instructor:

Date: Section:

Sales (in dollars)

Chapter 8 Pretest Form B (cont’d)

350 300 250 200 150 100 50 0

5th Grade Magazine Sales

Weeks

8. What was the total in sales in week 3?

8. _______________________

9. Which week had the least sales?

9. _______________________

10. What was the total in sales for the 6 weeks?

10. _______________________

11. The 5th grade was hoping to buy playground equipment costing $1000. Did they meet their goal?

11. _______________________

12. Graph y = 2x - 3

T-164

12.

Name: Instructor:

Date: Section:

Chapter 8 Pretest Form B (cont’d) 13. Graph x - 2y = 5

14. Graph y = -2

13.

14.

15. Complete the ordered pair solutions of the equation x + 4y = 8.

15. _______________________

(4, ), (-4, ), ( , 2) 16. Complete the ordered pair solutions of the equation y = x – 2. (1, ), (-1, ), (0, )

16. _______________________

17. If a single die is tossed, find the probability that a “5” will appear.

17. _______________________

18. Find the probability that a number less than 4 will appear if a single die is tossed.

18. _______________________

19. What is the probability that a “10” is selected from a deck of cards (52 total)?

19. _______________________

20. What is the probability of a club card being selected from a deck of cards?

20. _______________________

T-165

Name: Instructor:

Date: Section:

Chapter 8 Test Form A The bar graph below shows the profits and losses for 5 years of Martin Construction Company. Use the bar graph for problems 1–3.

Martin Construction Company

Profits (in millions of dollars)

5 4 3 2 1 0 -1 -2

Years

1. What is the profit for year 3?

1. __________________________

2. In which year did the construction firm have a loss?

2. __________________________

3. Estimate the profit or loss for the first 3 years.

3. __________________________

Write the ordered pair for each numbered point.

4. __________________________

5 4

5. __________________________

0 -6 -5 -4 -3 -2 -1 -1 0

6. __________________________

-2 -3

7. __________________________

-4 -5 -6

T-166

8. __________________________

Name: Instructor:

Date: Section:

Chapter 8 Test Form A (cont’d) The pie chart shows the most poplar color of cars in America. Use the pie chart for problems 9–13.

Most Poplar Color of Cars in America

Other 20%

White 25%

Brown or Grey 10% Red 12%

Blue ?%

Black 20%

9. What is the most popular color of car in America?

9. _________________________

10. If there are 300 cars in a parking lot, how many will be red?

10. ________________________

11. What percent of the cars are blue?

11. ________________________

12. If there are 300 cars in a parking lot, how many will be brown or grey?

12. ________________________

13. If there are 5000 registered in one town, how many will be white? 13. ________________________ Graph the ordered pairs on the same axis. Label the points 14 – 15. 14. (-2, 2) 15. (-1, 0)

14 and 15.

T-167

Name: Instructor:

Date: Section:

Chapter 8 Test Form A (cont’d) Complete the ordered pair solution for the given equation. 16. y = 2x - 5; (2, ), (0, )

16. __________________________

1 x + 1; (3, ), (0, ) 3

17. __________________________

17. y =

Determine if the ordered pair is a solution for the given equation. 18. (1, 3)

x + 2y = 5

18. __________________________

19. Graph y = 2x - 4

19.

20. Graph y = x - 3

20.

T-168

Name: Instructor:

Date: Section:

Chapter 8 Test Form A (cont’d) 21. Graph y = 2

21.

22. If a die is rolled, what is the probability of an odd number appearing?

22. __________________________

23. What is the probability of a number less “5” appearing on a roll of a die?

23. __________________________

24. What is the probability of a “3” being selected from a deck of cards (52 total)?

24. __________________________

25. What is the probability of a “3 of Hearts” begin selected from the deck of cards?

25. __________________________

T-169

Name: Instructor:

Date: Section:

Chapter 8 Test Form B

Income in thousand dollars ($000)

Alfred's Accounting Service 12 10 8 6 4 2 0 Jan

Feb

Mar

Apr

May

June

The bar graph below shows the income for Alfred’s Accounting Service. 1. In what month was the income the lowest?

1. __________________________

2. Find the total income for the six months.

2. __________________________

Write the ordered pair for each numbered point. 6 5

3. __________________________

4 3

4. __________________________

60 -6 -5 -4 -3 -2 -1 -1 0

-2 -3 -4

5. __________________________

5 6. __________________________

-5 -6

7. __________________________

T-170

Name: Instructor: Chapter 8 Test Form B (cont’d)

Date: Section:

Use the pie chart for problems 8-12.

Number of Breeds of Animals

Dogs* 41%

Horses 48%

Cats** 11% *American Kennel Club Breeds

**Cat Fancier Breeds

8. Which type of animal has the most different kinds of breeds?

8. ________________________

9. Which type of animal has the least types of breeds?

9. ________________________

10. If there are 380 types of breeds in all, how many breeds of dogs are there? Round to a whole number.

10. ________________________

11. If there are 380 types of breeds in all, how many breeds of cats are there? Round to a whole number.

11. ________________________

12. How many more breeds of dogs are than breeds of cats? Round to a whole number.

12. ________________________

Graph the ordered pairs on the same axis. Label the points 13–14. 13. (1, 2) 14. (-3, 2)

13 and 14.

T-171

Name: Instructor:

Date: Section:

Chapter 8 Test Form B (cont’d) Complete the ordered pair solution for the given equation. 15. y = x – 5; (2, ), (-2, )

15. __________________________

16. y = -3x + 2; (3, ), (1, )

16. __________________________

17. 3x – 2y = 10; (4, ), (0, )

17. __________________________

Determine if the ordered pair is a solution for the given equation. 18. (10, 3)

x + 2y = 6

18. __________________________

19. Graph y = 3x - 2

19.

20. Graph x = -2

20.

T-172

Name: Instructor:

Date: Section:

Chapter 8 Test Form B (cont’d) 21. Graph y =

1 x -3 2

21.

22. If a die is rolled, what is the probability of an even number being rolled?

22. __________________________

23. What is the probability of a number less than 3 being rolled? 23. __________________________

24. Out of a deck of cards (52 total), what is the probability of The 4 of clubs being picked from a deck?

24. __________________________

25. What is the probability of a “6” being picked from a deck of cards?

25. __________________________

T-173

Name: Instructor:

Date: Section:

Chapter 8 Test Form C The bar graph below shows the sales of a new company during the first 4 weeks.

Sales in hundred $(00)

Betty's Boutique 9 8 7 6 5 4 3 2 1 0 1

Week

1. How much were sales in the third week?

1. __________________________

2. How much were sales in the second week?

2. __________________________

3. What is the total amount in sales for the 4 weeks?

3. __________________________

Write the ordered pair for each numbered point.

4. __________________________

5 4 3

5. __________________________

2 1 0 -6

-5

-4

-3

-2

-1

6. __________________________

-2 -3

7. __________________________

-4 -5 -6

8. __________________________ 9. __________________________

T-174

Name: Instructor: Chapter 8 Test Form C (cont’d)

Date: Section:

Use the pictograph for problems 10–13. Number of Breeds

Each animal = 50 breeds

Horses

Dogs Cats 10. About how many horse breeds are there?

10. __________________________

11. About how many dog breeds are there?

11. __________________________

12. About how many cat breeds are there?

12. __________________________

13. How many more dog breeds are there than cat breeds?

13. __________________________

Graph the ordered pairs on the same axis. Label the points 14–15. 14. (1, -3) 15. (-3, -2)

14 and 15.

Complete the ordered pair solution for the given equation. 16 y = x – 4; (1, ), (-1, )

16. __________________________

17. 2x – y = 6; (2, ), (-2, )

17. __________________________

18. x – 3y = 0; (3, ), (-3, )

18. __________________________

T-175

Name: Instructor: Chapter 8 Test Form C (cont’d)

Date: Section:

Determine if the ordered pair is a solution for the given equation. 19. (-3, 0) 20. Graph y =

2x + 3y = 6 1 x-2 3

21. Graph y = 4

T-176

19. __________________________ 20.

21.

Name: Instructor: Chapter 8 Test Form C (cont’d)

Date: Section:

22. Graph 2x - 3y = 6

22.

23. Find the probability of a number less than 4 being rolled with a die.

23. __________________________

24. What is the probability that a heart is picked from a deck of 24. __________________________ cards (52 total)?

25. What is the probability of a queen being selected from a deck of cards?

25. __________________________

T-177

Name: Instructor:

Date: Section:

Chapter 8 Test Form D

Sales (in hundreds of $)

Betty's Boutique Sales 3 2.5 2 1.5 1 0.5 0 Necklaces

Bracelets

Rings

1. Estimate the total dollar amount sold for all three items.

1. __________________________

2. Estimate the difference sales of rings and necklaces.

2. __________________________

3. Which item sold the most?

3. __________________________

Write the ordered pair for each numbered point.

6 5 4 3 2 1 0 -6 -5 -4 -3 -2 -1-1 0 -2 -3 7 -4 -5 -6

T-178

4. __________________________ 4 1

5. __________________________ 2

6. __________________________

6 7. __________________________

Name: Instructor: Chapter 8 Test Form D (cont’d)

Date: Section:

Use the pie chart for problems 8–11.

Grades in a Statistics Class F 10% D ?%

A 23%

C 26% B 34%

8. What percent of the class made a B?

8. __________________________

9. What grade was earned by the largest group of students?

9. __________________________

10. What percent of the class make a D?

10. _________________________

11. If there were 70 students in the class, how many made a C? Round the to nearest whole number.

11. _________________________

Graph the ordered pairs on the same axis. Label the points 12-14. 12. (3, -2) 13. (-1, -4) 14. (4, 2)

12, 13, and 14.

T-179

Name: Instructor:

Date: Section:

Chapter 8 Test Form D (cont’d) Complete the ordered pair solution for the given equation. 15. y = - 1 x - 4 ; (-5, ), (10, ) 5

15. __________________________

16. 2x – 3y = 6; (3, ), (0, )

16. __________________________

17. y = 3x – 4; (1, ), (-1, )

17. __________________________

18. Is (10, 3) a solution to the equation x + 2y = 6?

18. __________________________

19. Graph x - y = 2

20. Graph y = -3

T-180

19.

20.

Name: Instructor: Chapter 8 Test Form D (cont’d)

Date: Section:

21. If a die is rolled, what is the probability that the number “6” 21. __________________________ will appear?

22. What is the probability of an “ace” being selected from a deck of cards (52 total)?

22. __________________________

23. What is the probability that a “Queen of Spades” will be selected from the deck (52 total)?

23. __________________________

If a bag contains 1 green, 3 yellow, 4 blue, 2 red, and 2 orange marbles: 24. What is the probability that a yellow marble will be selected?

24. _________________________

25. What is the probability of a green marble being selected?

25. __________________________

T-181

Name: Instructor:

Date: Section:

Chapter 8 Test Form E

Electoral Votes

60 50 40 30 20 10 0 California

Texas

Florida

Georgia

Indiana

Nebraska

State

1. Of the states listed, which has the most electoral votes?

1. __________________________

2. Of the states listed, which are almost equal in electoral votes?

2. __________________________

3. Approximate the number of electoral votes for Nebraska.

3. __________________________

Find the ordered pair for the following points (4–7). 6 5 4

4. __________________________

1 0 -6 -5 -4 -3 -2 -1 -1 0 -2 -3

5. __________________________ 5

6. __________________________

-4 -5

7. __________________________

-6

T-182

Name: Instructor: Chapter 8 Test Form E (cont’d)

Date: Section:

Use the pie chart for problems 8-12. A store kept track of the number of each item sold. Round the nearest whole number.

Sales

8. Which item sold the most?

8. __________________________

9. If there 800 items sold, how many DVDs were sold?

9. __________________________

10. If there 800 items sold, how many CDs were sold?

10. __________________________

11. If there 800 items sold, how many magazines were sold?

11. __________________________

12. How many more books were sold than video games?

12. __________________________

Graph the ordered pairs on the same axis. Label the points 13-15. 13. (3, -3) 14. (3, 0) 15. (-1, -2)

13, 14, and 15.

T-183

Name: Instructor: Chapter 8 Test Form E (cont’d)

Date: Section:

Complete the ordered pair solution for the given equation. 16. y =

1 x + 4 ; (–3, ), (3, ) 3

16. __________________________

17. x – 3y = 6; (6, ), (–3, )

17. __________________________

18. Is (–2, -1) a solution to the equation x + 2y = 0?

18. __________________________

19. Graph y = 3x

20. Graph x = 4

T-184

19.

20.

Name: Instructor: Chapter 8 Test Form E (cont’d)

Date: Section:

21. Graph 2x + 3y = 6

22.

22. If a die is rolled, what is the probability that a number Less than three will appear?

22. __________________________

23. What is the probability that a “Queen” will be selected from 23. __________________________ a deck of cards (52 total)?

24. What is the probability that a “Queen of Diamonds” will be 24. __________________________ selected from a deck of cards (52 total)?

25. If a bag contains 4 green marbles, 2 blue marbles, 1 yellow, 25. __________________________ and 3 red marbles, what is the probability of selecting a red marble?

T-185

Name: Instructor:

Date: Section:

Chapter 8 Test Form F The bar graph shows the relationships of a number times itself (x2).

Squared Numbers 28 26 24 22 20 18 16 14 12 10 8 6 4 2 0 1

The next bar would reach to what number? a. 30

b. 36

c. 6

d. 7

The bar shows an increase. a. True

b. False

Find the ordered pair for the following points (3–7). 6 5 4

3 2 1

0 -6 -5 -4 -3 -2 -1 -1 0

4 2

-2 -3 -4 -5 -6

a. (3, -3)

b. (-2, 3)

c. (3, 2)

d. (3, -2)

a. (2, -1)

b. (2, 1)

c. (0, 2)

d. (2, 0)

a. (3, 2)

b. (2, -3)

c. (-2, 3)

d. (2, 3)

a. (-3, 2)

b. (3, -3)

c. (-3, -3)

d. (-3, 3)

a. (0, 0)

b. 0

c. (-1, 0)

d. (0, -1)

T-186

Name: Instructor: Chapter 8 Test Form F (cont’d)

Date: Section:

Complete the ordered pair for the given equation. 8. y = 2x + 2

(1, )

a. (1, 0) 9. 3x + 2y = 6

c. (1, 4)

d. (1, 3)

b. (2, 4)

c. (2, 0)

d. (2, 2)

b. (4, 2)

c. (4, 6)

d. (4, 0)

c. (-2, -4)

d. (-2, 8)

(2, )

a. (2, 6) 10. y =

b. (1, 2)

1 x+2 4

(4, )

a. (4, 3) 11. y = 2x + 4

(-2, )

a. (-2, –8)

b. (-2, 0)

12. Is (–4, –1) a solution to the equation y =

1 x - 2? 4

a. yes

b. no

Use the following pie chart for problems 13-17. The chart shows the income at a pet store. Income

Aquariums 12% Collars, leashes, clothes 20%

Animals 40%

Pet Food ?%

13. What category brought in the most income? a. Animals

b. Pet Food

c. Collars, leashes, clothes

d. Aquariums

14. What percent of the income came from pet food? a. 28%

b. 40%

c. 12%

d. 18%

15. What category brought in the least income? a. Animals

b. Pet Food

c. Collars, leashes, clothes

d. Aquariums

T-187

Name: Instructor: Chapter 8 Test Form F (cont’d)

Date: Section:

16. If the total income for one month was $1250, how much was from animals? a. $250

b. $150

c. $500

d. $350

17. If the total income for one month was $1250, how much was from aquariums? a. $250

b. $150

c. $500

d. $350

18. Graph y = – 3. a.

b. y

d. y

19. Graph y = a.

1 x + 2. 2

b. y

x x

d. y

T-188

Name: Instructor: Chapter 8 Test Form F (cont’d)

Date: Section:

20. Find the probability of a “3” being picked from a deck of cards. a.

2 13

1 13

1 52

1 6

21. Find the probability of a “Heads” appearing when is coin is tossed. a.

1 2

2 1

22. Find the probability of a heart being picked from a deck of cards (52 total). a.

2 13

1 13

1 52

1 4

If a box contains 2 red marbles, 1 green marble, 3 yellow marbles, 2 blue marbles, 23. Find the probability of picking a yellow marble. a.

1 4

1 8

1 2

3 8

1 2

3 8

1 2

3 8

24. Find the probability of picking a red marble. a.

1 4

1 8

25. Find the probability of picking a green marble. a.

1 4

1 8

T-189

Name: Instructor:

Date: Section:

Chapter 8 Test Form G The bar graph shows the net income for a new manufacturing company.

Number of CD's

Types of CDs Sold in One Hour 60 50 40 30 20 10 0

13 2 Rock

Country

Classic

Instrumental

1. Which type of music sold the most CD’s? a. Rock

b. Country

c. Classic

d. Instrumental

c. 112

d. 46

c. 13

d. 2

2. The total number of CD’s sold was a. 54

b. 115

3. How many Rock CD’s were sold? a. 46

b. 54

4. How many more Country CD’s were sold than Rock? a. 41 b. 8 Find the ordered pair for each numbered point.

c. 54

d. 46

6 5

4 3 2 1

0 -6 -5 -4 -3 -2 -1 -1 0 -2 -3 -4 -5 -6

8 1

a. (5, 0)

b. (0, 5)

c. 5

d. (5, 1)

a. (3, 4)

b. (4, 3)

c. (-3, 4)

d. (-3, -4)

a. (4, 0)

b. (0, 4)

c. (-4, 0)

d. (0, -4)

a. (1, 0)

b. (0, 1)

c. (0, -1)

d. (1, 1)

a. (3, 3)

b. (-3, 3)

c. 3

d. (-3, -3)

T-190

Name: Instructor: Chapter 8 Test Form G (cont’d)

Date: Section:

Use the pie chart for problems 10–14.

Students Enrolled in Different Math Classes Trig 7%

Caluclus or above 9%

Introductory Algebra 19%

Pre-Calculus 5% Intermediate Algebra 27%

College Algebra ?%

10. Find the percent of students enrolled in Intermediate Algebra. a. 33%

b. 27%

c. 23%

d. 67%

11. What percent of the students were enrolled in Introductory Algebra? a. 5%

b. 9%

c. 7%

d. 19%

12. Find the percent of students enrolled in Intermediate Algebra. a. 33%

b. 27%

c. 23%

d. 67%

13. If there are 4800 students in all math classes, how many are enrolled in Pre-Calculus? a. 2400

b. 240

c. 24

d. 204

14. If there are 4800 students in all math classes, how many are enrolled in Intermediate Algebra? a. 1584

b. 1296

c. 1104

d. 3214

b. (–5, -4)

c. (–5, -2)

d. (–5, 2)

b. (3, 3)

c. (3, -3)

d. (3, 6)

Complete the ordered pair for the given equation. 15. y =

2 x–4 5

(–5, )

a. (–5, -6)

16. 3x – 2y = 9 a. (3, 0)

(3, )

T-191

Name: Instructor: Chapter 8 Test Form G (cont’d) 17. y = –x

Date: Section:

(2, )

a. (2, 5) 18. y = 4x + 3

c. (2, -2)

d. (2, 3)

b. (4, 7)

c. (4, 3)

d. (4, 24)

(4, )

a. (4, 19) 19. Graph. y =

b. (2, 2)

1 x+2 2

b. y

x x

d. y

20. Graph y = 3. a.

b. y

d. y

T-192

Name: Instructor:

Date: Section:

Chapter 8 Test Form G (cont’d) 21. Find the probability of tossing an odd number with one roll of the die. a.

1 6

1 2

3 6

d. 1

22. Find the probability that a “Tails” will appear on a coin toss. a.

2 1

1 2

23. What is the probability that an “ace” will be selected from a deck of cards? a.

4 13

1 13

1 52

2 52

24. What is the probability that a “6 of Hearts” will be selected from a deck of cards? a.

4 13

1 13

1 52

2 52

25. What is the probability that a number less than 4 will appear on a toss of a die? a.

2 6

1 6

1 3

1 2

T-193

Name: Instructor:

Date: Section:

Chapter 8 Test Form H

Games Won

Jones Murray High School Football Record

12 10 8 6 4 2 0 1

Years

1. Jones Murray’s football team won 8 games in what year? a. 1st

b. 2nd

c. 3rd

d. 4th

c. 3rd

d. 2nd

2. In what year did the team win the least games? a. 5th

b. 4th

3. How many games did the team win in the five years? a. 28

b. 32

c. 30

d. 26

4. Find the team’s average number of wins for the 5 years. a. 5 b. 6 Find the ordered pair for each numbered point. 6 5 4 3 2 9 1 0 -6 -5 -4 -3 -2 -1-1 0 -2 -3 8 -4 -5 -6

c. 8

d. 10

5 1

a. (4, 2)

b. (4, -2)

c. (2, 4)

d. (2, -4)

a. 2

b. 0

c. (0, 2)

d. (2, 0)

a. (3, -2)

b. (3, 2)

c. (-2, 3)

d. (-2, -3)

a. (4, -3)

b. (4, 3)

c. (-4, 3)

d. (-4, -3)

a. (2, -5)

b. (-5, 2)

c. (2, 5)

d. (-2, -5)

T-194

Name: Instructor: Chapter 8 Test Form H (cont’d)

Date: Section:

Use the pictograph for problems 10-14. A company looked at their office expenses for one month. Office Expenses

= $100

Supplies Furniture Equipment

10. How much was spent on supplies? a. $600

b. $1100

c. $400

d. $2100

c. $400

d. $2100

c. $400

d. $2100

11. How much was spent on furniture? a. $600

b. $1100

12. What was the total spent on office expenses? a. $600

b. $1100

13. How much more was spent on equipment than on furniture? a. $600

b. $500

c. $200

d. $2100

14. How much more was spent on equipment than on supplies? a. $700

b. $500

c. $200

d. $2100

15. Find the ordered pair solution for the equation - 2 x + 4 y = 8 . a. (0, 2)

b. (-4, 0)

c. (2, -1)

d. (0, 4)

16. Complete the ordered pair (3, ) for the equation y = –x + 4. a. (3, 1)

b. (3, 2)

c. (3, –1)

d. (3, –2)

17. Complete the ordered pair (3, ) for the equation y = 2x – 1. a. (3, 1)

b. (3, -1)

c. (3, 5)

d. (3, –5)

T-195

Name: Instructor: Chapter 8 Test Form H (cont’d)

Date: Section:

18. Graph. y = 3x + 2 a.

b. y y

d. y

19. Graph. y = 2 x - 3 3 a.

x x

d. y y

x x

T-196

Name: Instructor: Chapter 8 Test Form H (cont’d)

Date: Section:

20. Find the probability of the “ten of clubs” being selected from a deck of cards. 1 1 1 2 a. b. c. d. 6 13 52 52

21. Find the probability of a “king” begin selected from a deck of cards (52 total). a.

1 6

1 13

1 52

2 52

1 2

22. Find the probability of a “3” being rolled on a die toss. a.

1 6

1 3

3 6

23. What is the probability of an “odd” appearing on a roll of a die? a.

1 6

1 3

1 2

5 6

2 5

3 10

A bag holds 10 marbles, 2 blue, 3 red, 4 green and 1 yellow. 24. Find the probability of a blue marble being chosen. a.

1 2

b. 2

1 5

25. Find the probability of a red marble being chosen. a.

1 5

2 5

1 2

T-197

Name: Instructor:

Date: Section:

Cumulative Test Form A Chapters 7 and 8 1. Write 312.5% as a decimal and as a fraction in simplest terms.

1. __________________________

2. Write 0.012 as a percent.

2. __________________________

3. Write 86.5% as a fraction or mixed number in simplest terms. 4 4. Write 1 as a percent. 5

3. __________________________

5. From a freshman English class, 68% passed with A’s or B’s. Write as a fraction in simplest terms.

5. __________________________

6. From a high school band, percentage.

1 4

were seniors. Write

1 4

4. __________________________

as a 6. __________________________

Solve. (Round to the nearest tenth if needed.) 7. What number is 18% of 200?

7. __________________________

8. 42 is what percent of 252?

8. __________________________

9. 33 is 30% of what number?

9. __________________________

10. A set of 4 tires for John’s truck is on sale for 15% off the normal price of $280. Find the cost of the tires.

10. __________________________

11. Randi gets 3.5% commission on her jewelry sales. What is her commission if her total sales are $318. (Round to the nearest cent.)

11. __________________________

12. A couple borrowed $750 from a bank at 8% interest for 6 months. Find the total amount owed with simple interest. (Round to the nearest cent.)

12. __________________________

13. $1000 is compounded annually at 5% for 3 years. Find the total amount in the account. (Round to the nearest cent.) 14. Find the simple interest earned on $2500 saved for 1 12

13. __________________________

14. __________________________

years at an interest rate of 5.5%. (Round to the nearest cent.) 15. A prom dress is on sale for 20% off the regular price of $140. Find the sale price.

T-198

15. __________________________

Name: Instructor: Cumulative Test Form A Chapters 7 and 8 (cont’d) 16. Find the coordinates for A, B, and C.

Date: Section: 16. __________________________

B A x

Complete the ordered pair solutions for the equations. 2 17. y = x - 3; (3, ), (0, ), and (-3, ) 3 18. y = x – 3 ; (1, ), (0, ), and (-1, ) 19. Graph y = 2x – 3.

20. Graph x – 2y = 6.

17. __________________________ 18. __________________________ 19.

20.

T-199

Name: Instructor: Cumulative Test Form A Chapters 7 and 8 (cont’d) 21. Graph. x = -4

Date: Section:

21.

22. Find the probability of the “King of Diamonds” being selected from a deck of cards.

22. ________________________

23. If a die is rolled, what is the probability of a “6” appearing?

23. ________________________

Use the spinner for problems 24 and 25. 4

24. From the spinner, what is the probability of selecting the “3”?

24. ________________________

25. What is the probability of an even number being selected?

25. ________________________

T-200

Name: Instructor:

Date: Section:

Cumulative Test Form B Chapters 7 and 8 1. Write 26% as a decimal. a. 2.6

b. 260

c. 0.026

d. 0.26

c. 0.31%

d. 3.1%

2. Write 0.031 as a percent. a. 0.031%

b. 31%

3. Write 68.2% as a fraction or mixed number in simplest form.

341 500

4. Write

2 as a percent. 25

a. 45% 5.

340 50

b. 0.8%

34 5

d. 341

c. 8%

d. 80%

1 1 of the boxes at a charity drive were full. Write as a percent. 8 8

a. 1.25%

b. 125%

c. 12.5%

d. 0.125%

c. 528

d. 5.3

c. 0.16%

d. 16.0%

c. 18.8

d. 300

Solve. (Round to the nearest tenth if needed.) 6. What number is 18% of 95? a. 17.1

b. 0.171

7. 23 is what percent of 144? a. 6.3%

b. 1.6%

8. 75 is 25% of what number? a. 1875 9.

b. 33

All summer clothes are on sale for 75% off. Find the sale price of shorts that normally sell for $34. (Round to the nearest cent.)

a. $26

b. $28

c. $25.50

d. $8.50

10. Jack gets 4% commission on homes he sells. Find this commission on a $120,000 home. a. $1200

b. $480,000

c. $4,800

d. $480

11. Sandy borrowed $350 to buy her college text books. She will repay the loan in 6 months. How much are her monthly payments if the interest rate is 3%. (Round to the nearest cent.) a. $68.00 b. $59.21 c. $68.08 d. $60 12. $700 is compounded semiannually at 5.5%. Find the total amount after 5 years. a. $918.16 b. $91.82 c. $981.16 d. $981.61 13. Find the simple interest earned on $1000 invested for 4 years at 9% interest rate. a. $900 b. $100 c. $360 d. $740

T-201

Name: Instructor:

Date: Section:

Cumulative Test Form B Chapters 7 and 8 (cont’d) 14. Shoes which normally sell for $80 are on sale for $45. Find the discount rate. a. 47.35%

b. 43.75%

c. 47.50%

d. 43.00%

1 15. Is (-5, -4) a solution to the equation y = - x - 3? 5 a. yes b. no 16. Complete the ordered pair solution for the equation y = a.

5 3

1 3

b. -

1 2 x + ; (3, ) 3 3

c. -3

d. 0

17. Complete the ordered pair solution for the equation y = -2x - 4; (1, ) a. -4

b. 4

d. -2

c. 2

18. Graph. x - y = 4 a.

b. y

x x

d. y y

T-202

Name: Instructor: Cumulative Test Form B Chapters 7 and 8 (cont’d) 19. Graph. y = a.

2 x-3 3

Date: Section:

b. y

d. y

20. Graph. x = 3 a.

b. y

d. y

T-203

Name: Instructor: Cumulative Test Form B Chapters 7 and 8 (cont’d)

Date: Section:

21. Find the probability that a “10” would be selected from a deck of cards. a.

2 52

1 13

1 52

1 6

22. Find the probability that a “Jack of Diamonds” would be selected from a deck of cards. a.

2 52

1 13

1 6

1 52

Population (in millions)

Use the following bar graph for problems 23 - 25.

16.3

15.5

New York

Los Angeles

15 10 5 0

Source: Dept. of Economics, Social Information and Policy Analysis. 23. Find the ratio of New York to Los Angeles population. a.

16.3 15.5

15.5 16.3

155 163

163 155

24. Find the ratio of Los Angeles to New York population. a.

16.3 15.5

15.5 16.3

155 163

25. Find the difference in population between the 2 cities. a. 8,000

T-204

b. 80,000

c. 800,000

d. 8,000,000

Name: Instructor:

Date: Section:

Chapter 9 Pretest Form A 1. Find the complement of a 47° angle.

1. __________________________

2. Find the supplement of a 127° angle.

2. __________________________

3. Find the measure of ∠x

3. __________________________

x 35°

4. __________________________

Find the measure for x, y, and z. x y

40° z

Find the perimeter (or circumference) and area of each figure. For the circle, give the exact value and then use π ≈ 3.14 for an approximation. Round answers to the nearest tenth. 5.

5.6 m

5. __________________________ 2.5 m

6. __________________________ 3 feet 3 feet 7. __________________________

7. 10 inches 6 inches 8inches 8.

8. __________________________ 2m

T-205

Name: Instructor:

Date: Section:

Chapter 9 Pretest Form A (cont’d) 9. Find the volume of a rectangular box which has a length of 3 ft., a width of 2 12 feet, and a height of 1 12 feet.

9. __________________________

Convert. (Round to the nearest tenth if needed.) 10. 480 inches to feet

10. __________________________

11. 2 12 gallons to quarts

11. __________________________

12. 3.6 kilograms to grams

12. __________________________

Perform each indicated operation. 13. 5 gallons 1 quart + 3 gallons 3 quarts

13. __________________________

14. 9 gallons 3 quarts ÷ 3

14. __________________________

Convert. (Round to the nearest tenth if needed.)

15. 6 meters to kilometers

15. __________________________

16. 4 gallons to pints

16. __________________________

17. 94°F to degrees Celsius

17. __________________________

18. 26°C to degrees Fahrenheit

18. __________________________

19. If 4 feet 8 inches is cut from a 6 foot board, what is the length of the piece that is left?

19. __________________________

20. How much soil is needed to fill a rectangular hole 2 feet by 3 feet by 2 feet?

20. __________________________

T-206

Name: Instructor:

Date: Section:

Chapter 9 Pretest Form B 1. Find the complement of a 68° angle.

1. __________________________

2. Find the supplement of a 100° angle.

2. __________________________

3. Find the measure of ∠x

3. __________________________

x 20°

Find the measure for x, y, and z.

4. __________________________

x y

30° z

Find the perimeter (or circumference) and area of each figure. For the circle, give the exact value and then use π ≈ 3.14 for an approximation. Round answers to the nearest tenth. 5.

7.5 yards

5. __________________________ 2.5 yards

6. __________________________ 1m 1m 7. __________________________

7. 5 cm 3 cm 4 cm 8.

8. __________________________ 16 ft.

T-207

Name: Instructor:

Date: Section:

Chapter 9 Pretest Form B (cont’d) 9. Find the volume of a rectangular box which has length 12 inches, width 8 inches, and height 5 inches.

9. __________________________

Convert. (Round to the nearest tenth if needed.) 10. 324 inches to feet

10. __________________________

11. 24 pints to gallons

11. __________________________

12. 8.25 kilograms to grams

12. __________________________

Perform each indicated operation. 13. 8 meters 16 centimeters ÷ 2

13. __________________________

14. 4 quarts – 2 quarts 1 pint

14. __________________________

Convert.

15. 24 quarts to gallons

15. __________________________

16. 135 grams to kilograms

16. __________________________

Convert. (Round to the nearest tenth if needed.) 17. 85°F to degrees Celsius

17. __________________________

18. 16°C to degrees Fahrenheit

18. __________________________

19. If bows take 2 feet of ribbon to make, how many bows can be make from 12 yards of ribbon?

19. __________________________

20. How much baseboard is needed to go around a rectangular room that measures 12 feet by 8 feet?

20. __________________________

T-208

Name: Instructor:

Date: Section:

Chapter 9 Test Form A 1. Find the complement of a 79° angle.

1. __________________________

2. Find the supplement of a 35° angle.

2. __________________________

3. Find the measure of ∠x

3. __________________________

x 52° 4. Find the measure for x, y, and z.

x y

4. __________________________ 50°

z 5. __________________________

5. Find the measure of ∠x. 91° 63°

x 6. __________________________

6. Find the perimeter and area for the rectangle shown. 2.4 m 7.2 m 7. Find the circumference. (Give an exact value and then use π ≈ 3.14 for an approximation.)

7. __________________________

3m 4 in. 8. Find the perimeter and area.

8. __________________________ 9 in. 6 in. 15 in.

9. Convert 125 grams to milligrams.

9. __________________________

10. Convert 1 12 gallons to quarts.

10. __________________________

11. Convert 86 liters to milliliters

11. __________________________

12. Convert 200 centimeters to meters.

12. __________________________

T-209

Name: Instructor:

Date: Section:

Chapter 9 Test Form A (cont’d) 13. Find the area and perimeter of a rectangle that is 11 inches long and 8 inches wide.

13. __________________________

14. Simplify 2 feet 8 inches + 3 feet 6 inches.

14. __________________________

15. Find the area of the triangle shown. (Round to the nearest tenth.)

15. __________________________

7.5 ft. 6 ft. 16. Find the area of the circle shown. (Give an exact value and then use π ≈ 3.14 for an approximation.)

16. __________________________

10 m

17. Find the volume of a rectangular box whose length is 3 feet, width is 2 feet, and height is 1.5 feet.

17. __________________________

18. Find the volume of a cylinder shaped storage bin with diameter 3 yards and height 8 feet. (Round to the nearest tenth.)

18. __________________________

19. Convert 53°C to degrees Fahrenheit. (Round to the nearest tenth.)

19. __________________________

20. Convert 38°F to degrees Celsius. Round to the nearest tenth.

20. __________________________

21. Find the perimeter of a 20 in. by 24 in. poster.

21. __________________________

22. Four friends are going on a road trip and will share the driving equally. The trip is 220 kilometers each way. How far will each person drive on the entire road trip?

22. __________________________

23. The local 4-H group is making blankets for their baby

23. __________________________

lambs. Each blanket requires 1

1 2

yards of material and

they want to make 12 blankets. How many yards of material will they need to purchase? 24. An angle that measures less than 90° is _____________.

24. __________________________

25. An angle that measures more than 90° is ____________.

25. __________________________

T-210

Name: Instructor:

Date: Section:

Chapter 9 Test Form B 1. Find the complement of a 73° angle.

1. __________________________

2. Find the supplement of an 89° angle.

2. __________________________

3. Find the measure of x.

3. __________________________

94° x

115°

4. Find the measure of x and y.

30°

4. __________________________

5. Find the measure of x.

5. __________________________ 30° 105° x

6. Find the measure for x, y, and z.

6. __________________________

x y

52° z

Find the perimeter and area of the square.

7. __________________________

4.6 cm

8. Find the perimeter and area.

4 ft.

8. __________________________

3 ft. 2 ft. 6 ft. 9. Find the circumference. (Give the exact value and then use π ≈ 3.14 for an approximation.)

9. __________________________

10 m

10. Convert 8.25 grams to milligrams.

10. __________________________

11. Convert 16 kilometers to meters.

11. __________________________

12. Convert 400 milliliters to liters.

12. __________________________

T-211

Name: Instructor: Chapter 9 Test Form B (cont’d)

Date: Section:

13. Convert 39 feet to yards.

13. __________________________

14. Convert 4 meters to kilometers.

14. __________________________

15. Find the area of the triangle shown.

15. __________________________

3 cm 7.2 cm 16. Find the radius of a circle that has a diameter of 5.8 centimeters.

16. __________________________

17. Find the volume of a cube with sides of length 1.8 m. (Round to the nearest tenth.)

17. __________________________

18. Find the volume of a cylinder shaped storage bin if the base has a diameter of 12 feet, and the height is 15 feet. (Round to the nearest tenth.)

18. __________________________

19. Convert 36° C to degrees Fahrenheit. (Round to the nearest tenth.)

19. __________________________

20. Convert 94° F to degrees Celsius. (Round to the nearest tenth.)

20. __________________________

21. Find the area of a 4 in. by 6 in. photo.

21. __________________________

22. A ten gallon bucket of oil has 2 gal and 1 qt removed. How much oil is left?

22. __________________________

23. If 2

2 3

yards of material are needed to make 1 lap blanket,

23. __________________________

how much material will be needed to make 12 blankes for the residents of an assisted living center? True or False? 24. An acute angle is greater than 90°.

24. __________________________

25. An obtuse angle is greater than 90°.

25. __________________________

T-212

Name: Instructor:

Date: Section:

Chapter 9 Test Form C 1. Find the complement of a 75° angle.

1. __________________________

2. Find the supplement of a 125° angle.

2. __________________________

3. Find the measure of x. 125°

3. __________________________ x

4. Find the measure of x and y.

28°

4. __________________________

5. Find the measure of x.

5. __________________________ x 55° 55°

6. Find the measure of x, y, and z.

6. __________________________

120° z

x y

7. Find the perimeter and area of the rectangle.

7. __________________________

3 ft. 8 ft. 2m 8.

Find the perimeter and area.

8. __________________________ 2m

2.2 m 3m

9. Find the circumference of a circle that has a radius of 2.5 feet. (Give the exact value and then use π ≈ 3.14 for an approximation.)

9. __________________________

10. Convert 2.1 grams to milligrams.

10. __________________________

11. Convert 0.4 kiloliters to liters.

11. __________________________

T-213

Name: Instructor:

Date: Section:

Chapter 9 Test Form C (cont’d) 12. Convert 20 quarts to gallons.

12. __________________________

13. Convert 20 cm to meters.

13. __________________________

14. Convert 2 meters to millimeters.

14. __________________________

15. Find the area of the triangle shown.

15. __________________________

4 cm 8 cm 16. Find the area of the circle shown. (Give an exact value and then use π ≈ 3.14 for an approximation.) Round to the nearest tenth.

16. __________________________

4 ft.

17. Find the volume of a box whose length is 0.5 m, width is 0.3 m and height is 0.3 m.

17. __________________________

18. Find the volume of a soup can that has a diameter of 2.6 in. and is 4 in. tall. Give an exact value and then use π≈3.14 for an approximation and round to the nearest tenth.

18. __________________________

19. Convert 28°C to degrees Fahrenheit. (Round to the nearest tenth.)

19. __________________________

20. Convert 74°F to degrees Celsius. (Round to the nearest tenth.)

20. __________________________

21. A poster is 18 inches by 30 inches. Find the area of the poster.

21. __________________________

22. Steven and his roommate are driving back to college 288 km away. If they share the driving, how many kilmeteres will each drive?

22. __________________________

23. A nursing club at Capital High School is making lap blankets for a nursing home. Each blanket requires 1 yard and 1 foot of material. How many yards of material will be required to make 15 blankets?

23. __________________________

24. What kind of angle has a measurement of 90°?

24. __________________________

25. What kind of angle has a measurement that is between 90° and 180°?

25. __________________________

T-214

Name: Instructor:

Date: Section:

Chapter 9 Test Form D 1. Find the supplement of a 23° angle.

1. __________________________

2. Find the complement of an 80° angle.

2. __________________________

3. Find the measure of x.

3. __________________________

x 35° 4. Find the measure of x and y.

38°

4. __________________________

5. Find the measure of x.

5. __________________________ x 120° 30°

6. Find the measure of x, y, and z.

x y

6. __________________________ 42°

z 7. Find the perimeter and area. (Round to the nearest tenth.)

7. __________________________

7.5 cm 8. Find the perimeter and area. Round to the nearest tenth if needed.

1.4 m

8. __________________________

8m 9.

Find the circumference of the circle shown. (Give an exact value and then use π ≈ 3.14 for an approximation.) Round to the nearest tenth.

9. __________________________

10. Convert 8.02 grams to kilograms.

10. __________________________

11.

11. __________________________

Convert 4.9 kilometers to meters.

12. Convert 3.5 gallons to quarts.

12. __________________________

T-215

Name: Instructor: Chapter 9 Test Form D (cont’d)

Date: Section:

13. Convert 9 yards to feet. 14. Convert 5

1 2

13. __________________________ 14. __________________________

tons to pounds.

15. Find the area of the triangle shown. (Round to the nearest tenth.)

15. __________________________

9 ft. 7 ft. 16. Find the area of the circle shown. (Give an exact value and then use π ≈ 3.14 for an approximation.) Round to the nearest tenth.

16. __________________________

1.5 m

17. Find the volume of a box whose length is 1.5 cm, width is 0.3 cm and height is 0.3 cm.

17. __________________________

18. Find the volume of a grain silo that is built in a cylindrical shape. The diameter is 10 feet and the height is 4 feet. (Round to the nearest tenth.)

18. __________________________

19. Convert 91° C to degrees Fahrenheit. (Round to the nearest tenth.)

19. __________________________

20. Convert 74° F to degrees Celsius. (Round to the nearest tenth.)

20. __________________________

21. A poster, 36 in. by 24 in. will be framed. Find the perimeter of the poster.

21. __________________________

22. If 6 student share equally the driving of a 1350 km trip, how many kilometers will each drive?

22. __________________________

23. A Girl Scout group is making tote bags for a senior citizen center. Find the number of bags that can be made out of 1 yard. 12 yards of material if each tote bag requires 2

23. __________________________

True or False? 24. An acute angle is less than 90°.

24. __________________________

25. An obtuse angle is less than 90°.

25. __________________________

T-216

Name: Instructor:

Date: Section:

Chapter 9 Test Form E 1. Find the complement of a 23° angle.

1. __________________________

2. Find the supplement of an 87° angle.

2. __________________________

3. Find the measure of x.

3. __________________________

70° x 4.

Find the measure of x and y.

4. __________________________ 117° x y 5. __________________________

5. Find the measure of ∠x. x 65°

59°

6. Find the perimeter and area of the square. (Round to the nearest tenth if needed.)

6. __________________________

3.6 cm 3.6 cm 7. Find the circumference of the circle shown. (Give an exact value and then use π ≈ 3.14 for an approximation.) Round to the nearest tenth.

7. __________________________

3m 8.

Find the perimeter and area.

8. __________________________ 5 ft.

3 ft. 1 ft. 2 ft 9.

2 ft.

Find the area and perimeter. (Round to the nearest tenth.)

9. __________________________

12.2 m 15 m

T-217

Name: Instructor:

Date: Section:

Chapter 9 Test Form E (cont’d) 10. Convert 1.06 liters to milliliters.

10. __________________________

11. Convert 2 gallons to pints.

11. __________________________

12. Convert 8.65 kiloliters to liters.

12. __________________________

13. Convert 1.6 meters to centimeters.

13. __________________________

14. Add: 2 yards 1 foot + 3 yards 7 feet

14. __________________________

15. Add: 3 meters 50 cm + 8 meters 75 cm

15. __________________________

16. Find the area of the triangle shown.

16. __________________________

5m 8m 17. Find the area of the circle shown. (Give an exact value and then use π ≈ 3.14 for an approximation.) Round to the nearest tenth.

17. __________________________

6 cm

18. Find the volume of a cube whose sides are 3.8 cm. (Round to the nearest tenth.)

18. __________________________

19. Find the volume of a cylinder shaped vase with diameter 6 inches and height 8 inches. (Round to the nearest tenth.)

19. __________________________

20. Convert 70 °F to degrees Celsius. (Round to the nearest tenth.)

20. __________________________

21. Convert 32° C to degrees Fahrenheit. (Round to the nearest tenth.)

21. __________________________

22. Sarah has 12 yards of ribbon. She is making bows which 3 yard each. How many bows can she make? require 4

22. __________________________

23. A right angle is one that measures _____________.

23. __________________________

24. An acute angle is one that measures ____________.

24. __________________________

25. Find the area of 2 walls, each 8 × 14 feet in size.

25. __________________________

T-218

Name: Instructor:

Date: Section:

Chapter 9 Test Form F 1.

Find the supplement of a 27° angle.

a. 153°

c. 73°

d. 63°

c. 145°

d. 45°

Find the complement of a 55° angle.

a. 35°

b. 117°

b. 125°

Find the measure of ∠x x

a. 90°

b. 25°

c. 65°

d. 15°

Find the measure of ∠x and ∠y.

x y

a. x = 135°, y = 45°

115°

b. x = 45°, y = 90°

45°

c. x = 105°, y = 45°

d. x = 45°, y = 135°

c. 50°

d. 52°

Find the measure of ∠x. x 63°

a. 63°

65°

b. 65°

Find the area. 2.4 yd. 6.8 yd.

a. 16.32 yd 2

b. 18.84 yd 2

c. 9.2 yd 2

d. 18.4 yd 2

T-219

Name: Instructor:

Date: Section:

Chapter 9 Test Form F (cont’d) 7.

Find the perimeter. 2.4 yd. 5.6 yd. a. 16 yd2

b. 8 yd

d. 13.44 yd2

c. 16 yd

Find an approximation for the circumference. Use π ≈ 3.14. 8 cm

a. 200.96 cm

b. 25 cm

c. 50.24 cm

2 in

9. Find the perimeter for:

d. 25.12 cm

1 in 1 in

2 in. 6 in. a. 20 in.

b. 18 in.

c. 14 in.

d. 12 in.

c. 650 cm

d. 0.065 cm

c. 9 quarts

d. 10 quarts

b. 120 cm

c. 0.012 cm

d. 1.20 cm

b. 2 kg

c. 0.2 kg

d. 0.02 kg

10. Convert 6.5 meters to centimeters. a. 6.5 cm

b. 65 cm

11. Convert 2 12 gallons to quarts. a. 3 quarts

b. 8 quarts

12. Convert 12 meters to centimeters. a. 1200 cm

13. Convert 20 g to kg. a. 200 kg

T-220

Name: Instructor:

Date: Section:

Chapter 9 Test Form F (cont’d) 14. Simplify 3 yards 2 feet + 8 yards 4 feet. a. 12 yards

b. 13 yards

c. 12 yards 1 feet

d. 11 yards 3 feet

15. Find the area of the rectangle. 3m 8m a. 24 m2

b. 22m2

c. 11 m2

d. 1482

c. 16 cm2

d. 32 cm2

16. Find the area of the triangle. 6 cm 10 cm a. 60 cm2

b. 30 cm2

17. Find the area of the circle. (Use π ≈ 3.14 for an approximation.) 6m

a. 18.84 m2

b. 113.04 m2

c. 28.26 m2

d. 9.42 m2

18. Find the volume of a cube whose sides have lengths of 6 inches. a. 1296 in.3

b. 108 in.3

c. 18 in.3

d. 216 in.3

19. Convert 32°C to degrees Fahrenheit. (Round to the nearest tenth.) a. 89.0°

b. 86.0°

c. 86.9°

d. 89.6°

T-221

Name: Instructor:

Date: Section:

Chapter 9 Test Form F (cont’d) 20. Convert 92°F to degrees Celsius. (Round to the nearest tenth.) a. 33.3°

b. 32°

c. 34°

d. 35.1°

21. Find the perimeter of a photo which is 8 inches long and 3 inches wide. a. 11 inches

b. 22 inches

c. 16 inches

d. 24 inches

22. Luca is making big bows for decorations. Each bow requires 1 14 yards. She needs 12 bows. How much ribbon will she need? a. 14 yards

b. 16 yards

c. 18 yards

d. 15 yards

True or False? 23. An acute angle is an angle less than 90°. a. True

b. False

24. An obtuse angle is an angle less than 90°. a. True

b. False

25. Jamar is taking a trip that is 443 kilometers long. If he drives 118.7 kilometers on the first day and 156.6 kilometers on the second day, how many miles are left to drive on the third day? a. 169.3 km

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b. 168.3 km

c. 167.7 km

d. 169.1 km

Name: Instructor:

Date: Section:

Chapter 9 Test Form G 1. Find the complement of a 13° angle. a. 167°

b. 87°

c. 103°

d. 77°

c. 155°

d. 65°

2. Find the supplement of a 25° angle. a. 45°

b. 115°

3. Find the measure of ∠x

125° x

a. 90°

b. 45°

c. 25°

4. Find the measure of ∠x.

d. 35°

109° x

a. 70°

b. 71°

c. 89°

d. 90°

c. 90°

d. 30°

5. Find the measure of ∠x. x 60° a. 60°

b. 120°

60°

6. Find the area of a rectangle that is 7.2 meters long and 3.4 meters wide. a. 48.96 m 2

b. 21.2 m 2

c. 24.48 m 2

d. 10.6 m 2

T-223

Name: Instructor:

Date: Section:

Chapter 9 Test Form G (cont’d) 7. Find the perimeter of the rectangle. . 3.5 yd. 1 yd. a. 9 yards

b. 3.5 yards

c. 4.5 yards

d. 12.25 yards

8. Find an approximation for the circumference. Use π ≈ 3.14. 2m

a. 3.24 m

b. 6.14 m

c. 12.56 m

d. 6.28 m

9. Find the perimeter of the figure. 5m 2.5 m 1m 2m a. 15.5 m

b. 20 m

c. 10.5 m

d. 21 m

c. 56.200 mm

d. 0.56 mm

c. 6 quarts

d. 4 quarts

c. 500 mm

d. 50 mm

c. 0.002 kg

d. 200 kg

10. Convert 56.2 meters to millimeters. a. 0.562 mm

b. 56,200 mm

11. Convert 1 14 gallons to quarts. a. 7 quarts

b. 5 quarts

12. Convert 5 meters to centimeters. a. 0.005 mm

b. 5.0 mm

13. Convert 2 grams to kg. a. 20 kg

T-224

b. 2000 kg

Name: Instructor:

Date: Section:

Chapter 9 Test Form G (cont’d) 14. Convert 2

1 miles to feet. 2

a. 13,200 feet

b. 10,560 feet

c. 2640 feet

d. 2112 feet

c. 7 feet 4 inches

d. 8 feet 2 inches

15. Simplify 3 feet 8 inches + 4 feet 6 inches a. 8 feet 4 inches

b. 7 feet 2 inches

16. Find the area of a square that has sides measuring 4.5 centimeters. a. 18 cm2

b. 9 cm2

c. 20.25 cm2

d. 16 cm2

c. 20 ft.2

d. 40 ft.2

17. Find the area of 8 ft. 5 ft. a. 24 ft.2

b. 80 ft.2

18. Find the area of the circle to the nearest tenth. (Use π ≈ 3.14 for an approximation.) 1.3 cm

a. 4.1 cm2

b. 5.3 cm2

c. 8.2 cm2

d. 40.8 cm2

19. Find the volume of a cylinder shaped vase with diameter 12 in. and height 8 in. to the nearest tenth. a. 90.4 in3

b. 904.3 in3

c. 150.7 in3

d. 301.4 in3

T-225

Name: Instructor: Chapter 9 Test Form G (cont’d)

Date: Section:

20. Convert 17°C to degrees Fahrenheit. (Round to the nearest tenth.) a. 69°F

b. 62.6°F

c. –13.9 °F

d. 13.9°F

21. Convert 90°F to degrees Celsius. (Round to the nearest tenth.) a. 31.1°C

b. 32.2°C

c. 33°C

d. 33.3°C

22. Find the area of a road sign with length 14.8 feet and width 12 feet. a. 53.6 ft.2

b. 177.6 ft.2

c. 26.8 ft.2

d. 168 ft.2

c. less than 180°

d. greater than 180°

c. less than 180°

d. greater than 180°

23. An acute angle is ________________. a. less than 90°

b. greater than 90°

24. An obtuse angle is __________________. a. less than 90°

b. greater than 90°

25. Simplify 27.5 meters + 838 millimeters. Express the answer in meters. a. 2833.8 m

T-226

b. 283.38 m

c. 283.38 m

d. 28.338 m

Name: Instructor:

Date: Section:

Chapter 9 Test Form H 1. Find the supplement of an 8° angle. a. 8°

b. 96°

c. 82°

d. 172°

c. 33°

d. 123°

2. Find the complement of a 57° angle. a. 67°

b. 57°

3. Find the measure of ∠x 71° x a. 19°

b. 109°

c. 29°

4. Find the measure of ∠x.

d. 28°

x 39°

a. 51°

b. 141°

5. Find the measure of ∠x.

c. 140°

x 111°

a. 50°

d. 50°

b. 52°

17°

c. 120°

d. 128°

6. Find the perimeter of the square. 10.3 feet

a. 53.4 ft.

b. 20.6 ft.

c. 41.2 ft.

d. 106.1 ft.

T-227

Name: Instructor:

Date: Section:

Chapter 9 Test Form H (cont’d) 7. Find the area. 3.6 cm 1 cm a. 9.2 cm2

b. 3.6 cm2

c. 4.6 cm2

d. 7.2 cm2

8. Find an approximation for the circumference. Use π ≈ 3.14. Round to the nearest tenth. 6m

a. 18.8 m

b. 9.4 m

c. 28.3 m

d. 113.0 m

4m 9. Find the perimeter for: 2m 4m

1m 1m 1m

a. 16 m

b. 13 m

c. 18 m

d. 20 m

c. 20,000 m

d. 2000 m

10. Convert 200 centimeters to meters. a. 2 m

b. 20 m

11. Convert 18 quarts to gallons. a. 4 gallons

b. 72 gallons

c. 4 gallons 2

d. 4.4 gallons

12. Convert 150 g to kilograms. a. 0.15 kg

b. 150,000 kg

c. 1.5 kg

d. 15 kg

c. 0.182 km

d. 0.0182 km

13. Convert 18.2 meters to kilometers. a. 182 km

T-228

b. 1.82 km

Name: Instructor:

Date: Section:

Chapter 9 Test Form H (cont’d) 14. Convert

1 miles to feet. 3

a. 5280 feet

b. 1760 feet

c. 1820 feet

d. 15,840 feet

c. 5 ft. 6 in.

d. 6 ft. 4 in.

c. 5 m 6 cm

d. 5 m 16 cm

15. Simplify 2 ft. 10 in. + 3 ft. 6 in. a. 6 ft. 4 in.

b. 5 ft. 4 in..

16. Simplify 2 meters 7 cm + 3 meters 9 cm a. 5 m 12 cm

b. 6 m 6 cm

17. Find the perimeter of a rectangle that is 7.6 centimeters long and 2.4 centimeters wide. a. 36.5 cm

b. 18.2 cm

c. 10 cm

d. 20 cm

c. 152 ft.2

d. 17.5 ft.2

18. Find the area of 9.5 ft. 8 ft. a. 38 ft.2

b. 76 ft.2

19. Find the area of the circle. (Use π ≈ 3.14 for an approximation.) Round to the nearest tenth. 4.2 m

a. 13.8 m2

b. 13.2 m2

c. 6.6 m2

d. 13.1 m2

T-229

Name: Instructor:

Date: Section:

Chapter 9 Test Form H (cont’d) 20. Convert 24°C to degrees Fahrenheit. (Round to the nearest tenth if needed.) a. 4.5°F

b. 69°F

c. 75.2°F

d. 6°F

21. Convert 81°F to degrees Celsius. (Round to the nearest tenth if needed.) a. 27.2°C

b. 29°C

c. 28.7°C

d. 22.7°C

22. A new bulletin board is placed at the local college for student announcements. The bulletin board is 10 feet long and 8 feet wide. Find the area of the new board. a. 80 ft.2

b. 160 ft.2

c. 18 ft.2

d. 36 ft.2

c. less than 180°

d. greater than 180°

c. less than 180°

d. greater than 180°

23. An acute angle is ________________. a. less than 90°

b. greater than 90°

24. An obtuse angle is __________________. a. less than 90°

b. greater than 90°

25. Find the volume of a cylindrical storage bin with diameter 8 feet and height 10 feet. a. 250 ft.3

T-230

b. 502.4 ft.3

c. 50.2 ft.3

d. 251.2 ft.3

Name: Instructor:

Date: Section:

Chapter 10 Pretest Form A Perform each indicated operation. 1. (11y2 – 3y + 7) + (3y2 – 7y − 7)

1. _______________________

2. (9b + 2) – (3b2 – 4b + 6)

2. _______________________

3. Subtract (−z3 + 2z + 5) from (3z3 – 6z + 12).

3. _______________________

4. Find the value of −x2 – 3x + 5 if x = −2.

4. _______________________

Multiply and simplify. 5. a3 ⋅ a ⋅ a7

5. _______________________

6. 2x ⋅ 3x2 ⋅ 4x3

6. _______________________

7. (z9)3

7. _______________________

8. (3x2)4

8. _______________________

9. (2x2y)2(x3y2)3

9. _______________________

10. −3x(2x4 – 2x2 + 5)

10. _______________________

11. (x + 8)(x – 3)

11. _______________________

12. (x + 4)2

12. _______________________

13. (z + 6)(z – 6)

13. _______________________

14. (a – 2)(a2 + 2a – 3)

14. _______________________

15. 2(3x2 – 4x − 2)

15. _______________________

16. (−3x)3(2x)2

16. _______________________

Find the greatest common factor for each list of numbers. 17. 12, 36, 60

17. _______________________

18. 6m3, 2m2, 12m5

18. _______________________

Factor. 19. 7y2 + 14y – 28

19. _______________________

20. 3x4 – 9x3 + 6x2 – 3x

20. _______________________

T-231

Name: Instructor:

Date: Section:

Chapter 10 Pretest Form B Perform each indicated operation. 1. (5x2 – 3x + 2) + (−2x2 – 5x − 7)

1. _______________________

2. (y2 – 3y + 6) – (y2 − 3y – 6)

2. _______________________

3. (2x2 – 3x + 7) + (5x2 – 8x)

3. _______________________

4. Subtract (5x – 2) from (3x − 6).

4. _______________________

5. Find the value of 2x2 – 3x if x = −3.

5. _______________________

Multiply and simplify. 6. x3 ⋅ x ⋅ x5

6. _______________________

7. 2x(3x2)x5

7. _______________________

8. (x4)8

8. _______________________

9. (−3w2)3(2w)2

9. _______________________

10. (2x3)4

10. _______________________

11. (–2xy3z2)2

11. _______________________

12. (xy2)2(2x2y)3

12. _______________________

13. (2x – 5)(3x + 4)

13. _______________________

14. (x − 7)2

14. _______________________

15. −4(x2 − 4x + 3)

15. _______________________

16. (−2x)3(x2)4

16. _______________________

Find the greatest common factor for each list of numbers. 17. 5, 15, 35

17. _______________________

18. 3z2, 9z4, 12z3

18. _______________________

Factor. 19. 6x2 – 12x + 15

19. _______________________

20. 4y2 – 2y + 16

20. _______________________

T-232

Name: Instructor:

Date: Section:

Chapter 10 Test Form A Add or subtract as indicated. 1. (3y − 6) + (y2 – 6y − 5)

1. __________________________

2. (2x2 – 3x + 2) – (x2 + 4x − 3)

2. __________________________

3. (4.3x2 + 2.5x – 1.2) + (3.5x – 4)

3. __________________________

4. (5x + 6) – (−x2 – 3x + 2)

4. __________________________

5. Subtract (3x − 4) from (−6x + 8).

5. __________________________

6. Subtract (2x − 8) from (17x + 6).

6. __________________________

7. Find the value of 2x2 − x + 5 when x = 2.

7. __________________________

8. Find the value of 5x + 6 when x = −3.

8. __________________________

Multiply and simplify. 9. y5⋅ y

9. __________________________

10. (y7)3

10. __________________________

11. (3x)5

11. __________________________

12. (3xy3)3

12. __________________________

13. (2xy2)5

13. __________________________

14. x4 ⋅ x ⋅ x5

14. __________________________

T-233

Name: Instructor:

Date: Section:

Chapter 10 Test Form A (cont’d) 15. (3ab2)3(−2a2b)2

15. __________________________

16. (x3)2(x4)2

16. __________________________

17. −2z(z2 + 3z – 5)

17. __________________________

18. (3z – 4)2

18. __________________________

19. (2x + 1)(3x – 4)

19. __________________________

20. (x + 7)(x – 5)

20. __________________________

21. (a + 2)(a2 + a – 2)

21. __________________________

22. −3x(x − 2)

22. __________________________

23. Find the area of

23. __________________________ 2x x−5

Find the greatest common factor of the list of terms. 24. 15 and 45

24. __________________________

25. Factor out the GCF: 2x3 + 4x2 − 6x

25. __________________________

T-234

Name: Instructor:

Date: Section:

Chapter 10 Test Form B Add or subtract as indicated. 1. (2x − 6) + (3x2 − 7x + 4)

1. __________________________

2. (x2 – 7x + 5) – (2z2 − 4x + 3)

2. __________________________

3. (2.5x + 2.7) + (3x2 − 1.5x – 2.5)

3. __________________________

4. (4x − 6) – (−2x + 4)

4. __________________________

5. (3x2 + x – 2) + (x2 – 6)

5. __________________________

6. Subtract (−7x − 3) from (5x + 4).

6. __________________________

7. Find the value of 2z2 + 4z + 1 when z = −2.

7. __________________________

8. Find the value of 5z − 2 when z = 4.

8. __________________________

Multiply and simplify. 9. x3⋅ x2 ⋅ x

9. __________________________

10. (y7)5

10. __________________________

11. (3x2)3

11. __________________________

12. (4xy2z3)2

12. __________________________

13. (−3xy)3

13. __________________________

14. (2xy2z)2(−3x2yz2)

14. __________________________

T-235

Name: Instructor:

Date: Section:

Chapter 10 Test Form B (cont’d) 15. (y3)3(y2)5

15. __________________________

16. −3(x2 − 2x + 5)

16. __________________________

17. (x − 9)2

17. __________________________

18. (x + 5)(x – 5)

18. __________________________

19. (3x − 1)(2x + 5)

19. __________________________

20. (a + 1)(a2 + 2a + 1)

20. __________________________

21. −5x(x2 −2x + 3)

21. __________________________

22. __________________________

22. Find the area of 3x 2x − 3 Find the greatest common factor of the list of terms. 23. 15, 12, 24

23. __________________________

Factor out the greatest common factor. 24. 3x2 − 6x + 9

24. __________________________

25. 4x4 − 8x3 + 12x2

25. __________________________

T-236

Name: Instructor:

Date: Section:

Chapter 10 Test Form C Add or subtract as indicated. 1. (2x + 5) + (3x − 7)

1. __________________________

2. (4x2 − 2x + 6) – (4x2 − 2x + 3)

2. __________________________

3. (2.0z2 − 1.2z + 3.2) + (z – 4.5)

3. __________________________

4. (2x2 + x − 2) – (–3x2 − 2x + 6)

4. __________________________

5. (x2 − 2x + 5) − (−x2 + 5x − 4)

5. __________________________

6. Subtract (3x + 8) from (2x − 5).

6. __________________________

7. Find the value of y2 − y + 2 if y = −4.

7. __________________________

8. Find the value of x2 + 2x + 1 when x = 2.

8. __________________________

Multiply and simplify. 9. x ⋅ x2 ⋅ x5

9. __________________________

10. (x8)6

10. __________________________

11. (4x)3

11. __________________________

12. (−2x4y)2

12. __________________________

13. (2x5)4

13. __________________________

14. (2xy2)2(3xy2)

14. __________________________

T-237

Name: Instructor:

Date: Section:

Chapter 10 Test Form C (cont’d) 15. (y2)3(y3)5

15. __________________________

16. −3(x2 + 3x − 6)

16. __________________________

17. (x – 5)2

17. __________________________

18. (x + 4)(x – 5)

18. __________________________

19. (2x − 4)(3x + 1)

19. __________________________

20. (x − 2)(x2 + 2x − 5)

20. __________________________

21. −(x2 −5x − 8)

21. __________________________

22. Find the area of

22. __________________________ 1.5x 2.3x + 2

Find the greatest common factor of the list of terms. 23. 2y, 6y2, 12y3

23. __________________________

Factor out the greatest common factor. 24. 3x 2 − 12 x + 18

24. __________________________

25. 5x4 + 10x3 – 15x2

25. __________________________

T-238

Name: Instructor:

Date: Section:

Chapter 10 Test Form D Add or subtract as indicated. 1. (3x – 7) + (x2 – 6x + 4)

1. __________________________

2. (x2 – 7x − 2) + (5x2 –6x − 2)

2. __________________________

3. (6x – 7) – (−2x + 8)

3. __________________________

4. (x2 + 2x − 1) – (–4x − 7)

4. __________________________

5. (−3x2 + 12x − 2) − (x2 − 3x − 5)

5. __________________________

6. Subtract (x2 + x − 4) from (2x2 + 2x + 1).

6. __________________________

7. Find the value of 5x – 3 if x = −4.

7. __________________________

8. Find the value of y2 + y + 3 if y = −3.

8. __________________________

Multiply and simplify. 9. z5 ⋅ z ⋅ z4

9. __________________________

10. (y7)3

10. __________________________

11. (–2x4)6

11. __________________________

12. (−3x4)3

12. __________________________

13. (xy2)(xy)3

13. __________________________

14. 3x(x2y3)2

14. __________________________

T-239

Name: Instructor:

Date: Section:

Chapter 10 Test Form D (cont’d) 15. (2x)3(3x2)4

15. __________________________

16. −2(x2 + 6x + 9)

16. __________________________

17. (x + 2)2

17. __________________________

18. (4x – 3)(2x + 5)

18. __________________________

19. (x + 10)(x − 10)

19. __________________________

20. (y + 1)(2y2 − 5y − 8)

20. __________________________

21. −6(10x − 5)

21. __________________________

22. Find the area of

22. __________________________ 4x (x + 2)

Find the greatest common factor of the list of terms. 23. 3z, 39z3, 6z2

23. __________________________

Factor out the greatest common factor.

24. 8x6 − 4x4 − 16x2

24. __________________________

25. 2x2y − 4xy2

25. __________________________

T-240

Name: Instructor:

Date: Section:

Chapter 10 Test Form E Add or subtract as indicated. 1. (5x − 3) + (2x − 9)

1. __________________________

2. (3z2 − 4z) – (−3z2 − 2z + 5)

2. __________________________

3. (12y + 3) – (3y − 7)

3. __________________________

4. (x2 − 5x + 12) – (x2 + 2x − 5)

4. __________________________

5. (−8y2 + 5y − 12) − (y2 +8y + 5)

5. __________________________

6. Subtract (3x − 4) from (−2x − 5).

6. __________________________

7. Find the value of 12x – 7 if x = −3.

7. __________________________

8. Find the value of 3y2 − 18y + 9 if y = −1.

8. __________________________

Multiply and simplify. 9. a ⋅ a6 ⋅ a7

9. __________________________

10. (a3b)5

10. __________________________

11. (b8)5

11. __________________________

12. (xy2)2(x2y3)3

12. __________________________

13. (p2)3(p3)5

13. __________________________

14. 3x4(2x4)

14. __________________________

T-241

Name: Instructor:

Date: Section:

Chapter 10 Test Form E (cont’d) 15. (2x)3(−2x)3

15. __________________________

16. −3(x2 + 2x − 9)

16. __________________________

17. −5x(x + 7)

17. __________________________

18. (5x − 3)(7x + 2)

18. __________________________

19. (3x + 2)(5x − 1)

10. __________________________

20. (z − 3)2

20. __________________________

21. (a + 2)(a2 + 2a – 5)

21. __________________________

22. Find the area of

22. __________________________ 3x + 1 3x + 1

Find the greatest common factor of the list of terms. 23. 5x, 15x2, 20x5

23. __________________________

Factor out the greatest common factor. 24. 15x4− 5x2 + 10x

24. __________________________

25. 3x5y2 + 9x4y3 − 12x3y5

25. __________________________

T-242

Name: Instructor:

Date: Section:

Chapter 10 Test Form F Add or subtract as indicated. 1. (7x – 2) + (3x − 2) a. 10x – 4

b. 10x + 2

c. 10x – 2

d. 4x + 2

b. 5x2 + 2x + 6

c. 5x2 – 12x − 6

d. 5x2 – 2x − 6

b. – 1

c. – 11

d. 6y – 1

c. 10x2 – 3x + 3

d. 7x2 + 15

c. 5y2 + 8y – 1

d. 17y2 + 2y – 5

c. x + 3

d. −x – 13

c. 2

d. −2

b. 39

c. 43

d. −43

9. x ⋅ x5 ⋅ x6 a. x6

b. x7

c. x11

d. x12

10. (x5)6 a. x8

b. x11

c. x15

d. x30

11. (x2y3)2 a. x4y5

b. x4y8

c. x4y9

d. x4y6

12. (p3)3(p2)4 a. p15

b. p17

c. p6p6

d. p12

13. (2x3)2(−2x2)3 a. −32x6

b. −32x12

c. 32x12

d. −24x12

2. (x2 – 7x + 2) + (4x2 + 5x − 8) a. 5x2 + 12x − 6 3. (3y – 6) – (−3y + 5) a. 6y – 11

4. (7x2 – 3x + 8) – (−3x + 7) a. 7x2 – 6x + 1

b. 7x2 – 6x + 15

5. (11y2 + 5y – 3) – (6y2 – 3y – 2) a. 5y2 + 2y – 5

b. 5y2 + 2y – 1

6. Subtract (−2x – 2) from (3x – 5) a. 5x – 3

b. –x – 3

7. Find the value of 2x2 – x – 12 a. 6

if x = −2.

b. −14

8. Find the value of z – 12z if z = 3 a. −33 Multiply and simplify:

T-243

Name: Instructor:

Date: Section:

Chapter 10 Test Form F (cont’d) 14. (−2x2)(−2x2)2 a. 8x6

b. −4x6

c. 4x6

d. −8x6

b. 12x2y3

c. 12x4y3

d. 12x2y4

b. −5x3 + 15x2 + 40x

c. −5x3 − 15x2 – 40x

d. −5x3 + 15x – 8

b. −4x − 12

c. −4x – 3

d. 4x − 3

b. x2 − 16

c. x2 + 16

d. x2 − 8x + 16

b. 5x2 – x + 2

c. 6x2 – x – 35

d. 6x2 – x + 2

b. 13x

c. −4x2 + 12x

d. −4x2 – 12x

b. y3 – 3y2 + 5y – 3

c. y3 – 3y2 + y – 3

d. y3 – 3y2 + y + 3

15. 2xy3(3x2y)(4x) a. 24x4y4 16. −5x(x2 + 3x + 8) a. −5x3 + 15x + 40x 17. −4(x + 3) a. −4x + 12 18. (x − 4)2 a. x2 − 8x − 16 19. (3x + 7)(2x – 5) a. 5x2 – x – 35 20. (x + 3)(−4x) a. −4x2 – 3 21. (y – 1)(y2 – 2y + 3) a. y3 – y2 + 6y – 3 22. Find the area of 2x a. Area = 4x2 – 10x

2x – 5 b. Area = 4x – 5

c. Area = 4x2 – 5

d. Area = 2x2 – 10x

c. 3xy

d. 3x

b. 7x2(x2 − 2x + 4)

c. 7x2(x2 − 7x + 2)

d. 7x(x3 − 2x + 4)

b. 2x2y(xy2 + 2)

c. 2x(x2y3 + 2xy)

d. 2(x3y3 + 2x2y)

Find the greatest common factor of the list of terms. 23. 3xy, 21x2, 24x2y2 a. 3x2y

b. 3xy2

Factor out the greatest common factor. 24. 7x4 − 14x2 + 28x a. 7(x4 – 2x2 + 4) 25. 2x3y3 + 4x2y a. 2xy(x2y2 + 2x) T-244

Name: Instructor:

Date: Section:

Chapter 10 Test Form G Add or subtract as indicated. 1. (−4x – 2) + (3x − 7) a. x − 5

b. −x – 9

c. −7x – 9

d. −x – 5

b. 5x2 – 2x − 3

c. 4x2 – 6x – 13

d. 5x2 − 6x – 13

b. –z – 13

c. −z − 1

d. −z + 1

c. 14x2 − 2x – 4

d. 8x2 – 2x – 4

c. −x2 + 8x + 7

d. −x2 + 8x + 3

c. 5x2 – 4x – 11

d. 5x2 + 4x – 11

c. 57

d. −57

b. 3.9

c. 3.6

d. 2.8

9. x4 ⋅ x ⋅ x6 a. x11

b. x10

c. x21

d. x8

10. (y2)3(y2) a. y10

b. y7

c. y12

d. y8

11. (y3)4 a. y7

b. y12

c. y81

d. y34

12. (x4y)2 a. x10y

b. x6y3

c. x8y

d. x8y2

13. (y2)3(2y3)5 a. 2y21

b. 32y21

c. 2y13

d. 32y13

2. (4x2 − 2x − 5) + (x2 – 4x − 8) a. 4x2 – 6x – 3 3. (3z – 6) – (−4z + 7) a. 7z – 13

4. (11x2 − 7x + 3) – (−3x2 + 5x – 7) a. 8x2 − 12x − 4

b. 14x2 − 12x + 10

5. (2x2 + 3x + 5) – (3x2 + 5x – 2) a. −x2 − 2x + 7

b. 5x2 + 8x + 3

6. Subtract (2x2 + x + 3) from (7x2 − 3x − 8) a. 9x2 + 4x + 11

b. 9x2 – 4x – 11 if y = −3.

7. Find the value of −y – 18y a. 51

b. −51

8. Find the value of 2.5x2 – x + 1.5 a. 2.4

if x = 1.2

Multiply and simplify:

T-245

Name: Instructor:

Date: Section:

Chapter 10 Test Form G (cont’d) 14. (2x2)3(−3x)3 a. −216x9

b. −18x8

c. −18x9

d. −216x8

b. 6y4z6

c. 36y6z8

d. 6y3z4

b. 27x7

c. 27x12

d. 9x12

b. −3x3 + 8x + 12

c. −3x3 − 24x + 12

d. −3x3 + 8x + 12

b. 10x2 – 6x – 3

c. 5x2 – 6x – 3

d. 10x2 – 30x – 15

b. 9y2 + 25

c. 9y2 − 30y + 25

d. 9y2 + 30y + 25

b. 8x2 – 6x + 35

c. 18x2 – 14x + 12

d. 8x2 – 6x − 35

b. x2 – 14x + 49

c. x2 – 49

d. x2 – 14

c. 3x2 + 11x + 18x

d. 3x2 – 11x − 18x + 16

c. Area = 9x

d. Area = 18x

c. 6

d. 24

15. (3y2z)2(2yz3)2 a. 6y4z8 16. (3x)3(x4) a. 9x7 17. −3x(x2 + 8x + 12) a. −3x3 − 24x2 – 36x 18. 5(2x2 – 6x – 3) a. 2x2 – 30x – 15 19. (3y − 5)(3y + 5) a. 9y2 – 25 20. (2x − 5)(4x − 7) a. 8x2 – 34x + 35 21. (x – 7)2 a. x2 + 49 22. (x + 2)(3x2 + 5x + 8) a. 3x3 + 11x2 + 18x + 16

b. 3x3 + 11x2 + 16

23. Find the area of 6x 3x a. Area = 18x2

b. Area = 9x2

Find the greatest common factor of the list of terms. 24. 6, 18, 24 a. 3

b. 18

25. Factor out the greatest common factor. 8xy2 + 24x2y2 + 16xy3 a. 8xy2(3x + 2y) b. 8xy2(1 + 3x + 2y) c. 8x(y2 + 3xy2 + 2y2)

T-246

d. 8xy2(5xy)

Name: Instructor:

Date: Section:

Chapter 10 Test Form H Add or subtract as indicated. 1. (2.1x − 5.2) + (1.2x2 + x − 3.2) a. 1.2x2 + 2x – 8

b. 1.2x2 + 3.1x – 8.4

c. 1.2x2 + 3.1x + 8

d. 1.2x2 + 3.1 + 8.4

c. 3y2 − y + 9

d. –y2 – 13y – 1

c. x2 – 2x + 12

d. 2x2 − 3x + 12

c. 4x2 + 2x – 11

d. 2x2 + 12x – 11

c. x + 1

d. –x + 1

c. −2x2 + 3x − 10

d. −2x2 − x

c. 8

d. −8

b. 25

c. 21

d. −21

9. x ⋅ x6 ⋅ x4 a. x15

b. x24

c. x10

d. x11

10. (a3)4 a. a7

b. a81

c. a12

d. a

11. (4x3)3 a. 12x27

b. 64x9

c. 12x9

d. 64x6

12. (2x2)3(−3x2)3 a. 2x6 – 3x6

b. −6x12

c. −216x6

d. −216x12

13. (y2)2(y3)2 a. y9

b. y10

c. y5

d. y13

2. (y2 – 7y + 4) − (2y2 + 6y + 5) a. 3y2 – 13y – 1

b. 3y2 − y – 1

3. (−x + 5) + (x2 – 3x + 7) a. x2 − 4x + 12

b. x2 − 3x + 12

4. (3x2 + 7x − 9) – (−x2 − 5x – 2) a. 2x2 + 2x − 11

b. 4x2 + 12x − 7

5. Subtract (3x + 4) from (2x + 5) a. x − 9

b. –x + 9

6. Subtract (x − 5) from (2x2 − 2x + 5) a. 2x2 − x

b. 2x2 − 3x + 10

7. Find the value of x2 + 5x − 6 a. 0

if x = 2.

b. −12

8. Find the value of 18x2 + 2x + 5 a. 11

if x = 1

Multiply and simplify:

T-247

Name: Instructor:

Date: Section:

Chapter 10 Test Form H (cont’d) 14. x2 ⋅ x ⋅ x3 a. x4

b. x5

c. x6

d. x7

b. 8x10

c. 16x10

d. 8x13

b. −2x2 + x – 11

c. −2x2 + 2x – 11

d. −2x2 – 2x – 11

b. 12x3 − 24x2 + 40x

c. 12x3 − 6x + 10

d. 12x3 − 6x + 40

b. 9x2 − 16

c. 9x2 − 24x + 16

d. 9x2 − 24x − 16

b. 2x3 – 5x2 – 11x + 24

c. 2x3 – 6x2 – 11x – 8

d. −4x5 – 11x – 8

b. 10x2 − 6x − 15

c. 10x2 + 19x − 15

d. 16x2 + 24x + 9

b. 16x2 – 24x + 9

c. 16x2 – 9

d. 16x2 – 6x + 9

b. x2 − 14x + 11

c. −2x3 − 14x − 22

d. −2x3 – 14x + 11

c. Area = 7x2 – 3.5x

d. Area = 12.25x2 − 3.5x

c. 18

d. 3

15. (4x2)2(2x3)2 a. 64x10 16. −2(x2 + x – 11) a. −2x2 – 2x + 22 17. 4x(3x2 − 6x + 10) a. 12x3 − 24x + 10 18. (3x + 4)(3x − 4) a. 3x2 − 16 19. (x – 3)(2x2 + x − 8) a. 2x3 – 5x2 – 11x – 8 20. (2x − 5)(5x − 3) a. 10x2 − 31x + 15 21. (4x − 3)2 a. 16x2 − 6 22. (x2 + 7x − 11)(−2x) a. −2x3 − 14x2 + 22x 23. Find the area of (3.5x – 1) a. Area = 3.5x2 – 3.5x

3.5x b. Area = 7x2 – 3.5

Find the greatest common factor of the list of terms. 24. 18, 36, 54 a. 6

b. 2

25. Factor out the greatest common factor. 7xy2 + 14x2y + 28x2y2 a. 7xy(2x + 4y)

T-248

b. 7xy(y + 2x + 4xy)

c. 7x(y2 + 2xy + 4y2)

d. 7xy(y + 2xy + 4x)

Name: Instructor:

Date: Section:

Final Exam Test Form A Perform the indicated operations. 1. 732 – 485

1. __________________________

362 × 16

2. __________________________

3. 31,521 ÷ 3

3. __________________________

Estimate the sum by rounding each number to the nearest hundred. 4. __________________________

4. 5219 + 3852 + 1936 Simplify: 5. −3 + (–8)

5. __________________________

6. 16 – 42

6. __________________________

7. −3(4)

7. __________________________

8. (−56) ÷ −8

8. __________________________

9. 3 − |−3|

9. __________________________

10. Evaluate 3 − x( y + 2)

when x = 2 and y = −1.

11. __________________________

11. Find the perimeter for the triangle shown. (2x + 3)

10. __________________________

(2x + 3)

3x feet Solve the following equations: 12. 2x − 8 = 12

12. __________________________

13. 3x + 8 = 4x – 2

13. __________________________

Translate the following into a mathematical expression. Use x to represent “a number”. 14. The product of 3 and a number decreased by fifteen. Perform the indicated operations. Write answers in simplest form. 2 3 15. ⋅ 5 4 3 1 + 16. 5 2 17. −

16 12 ÷ 5 5

18. __________________________

15. __________________________ 16. __________________________ 17. __________________________

T-249

Name: Instructor: Final Exam Test Form A (cont’d)

Date: Section:

7x 18. Simplify 8 21x 16

19. Solve

18. __________________________

4 1 +x=− 5 3

19. __________________________

20. Evaluate −4x − 5 when x = −

1 . 4

20. __________________________

21. A piece 2 13 feet is cut from an 8 foot board. How much

21. __________________________

is left after the piece is cut? 22. Find the area for

22. __________________________ 2 yd. 3 1 13 yd.

Perform the indicated operations. Round to the nearest hundredth if needed. 23. −39.92 – (−61.58)

23. __________________________

24. 10.5 ⋅ 2.34

24. __________________________

25. 0.0896 ÷ (−0.32)

25. __________________________

26. Simplify 2.3x – 1.6 – 0.9x

26. __________________________

100

27. Find the square root

27. __________________________

28. Find the unit rate for 197 miles for 10 gallons of gas. Solve the proportions: 39 13 = 29. 6 x

28. __________________________

29. __________________________

30. Given that the triangles are similar, find the missing length. 8 16

30. __________________________

5 x

31. Write 3.6% as a decimal.

39. __________________________

32. Write 25% as a fraction in simplest form.

32. __________________________

T-250

Name: Instructor: Final Exam Test Form A (cont’d)

Date: Section:

3 as a percent. 8 Solve. (Round to the nearest tenth if needed.) 34. 132 is what percent of 264?

34. __________________________

35. 6% of what number is 420?

35. __________________________

36. A new spring outfit is on sale for 20% off the regular price of $125. Find the new price.

36. __________________________

37. Find the simple interest earned on $1500 saved for 2 12

37. __________________________

33. Write

33. __________________________

years at an interest rate of 5.5%. 38. Graph x + 2y = 4.

38.

6 5 4 3 2 1 x

0 -6

-5

-4

-3

-2

-1

-2 -3 -4 -5 -6 y

39. Find the mean, median and mode for 75, 36, 82, 75, 92, 84

39. __________________________

40. What is the probability of picking an even numbered disk out of a bag with disks numbered 1 – 20?

40. __________________________

41. Find the complement of a 73° angle.

41. __________________________

42.

42. __________________________

Find the cirumference of a circle with diameter 8 feet. Use π ≈ 3.14 for an approximation.

43. Convert 0.27 grams to milligrams. 44. Convert 2 12 gallons to quarts.

43. __________________________

45. Add: (3x – 7) + (14x – 2)

45. __________________________

46. Find the value of 5x – 2 when x = −3. Multiply and simplify:

44. __________________________

46. __________________________

47. y6 ⋅ y5

47. __________________________

48. (−3x2)4

48. __________________________

49. –2x(2x2 − x + 1)

49. __________________________

50. Find the greatest common factor of the list of terms. 6y2, 9y, 24y3, 12y4

50. __________________________

T-251

Name: Instructor:

Date: Section:

Final Exam Test Form B Perform the indicated operations. 1. 461 – 238

1. __________________________

324 × 37

2. __________________________

3. 3705 ÷ 15

3. __________________________

4. Evaluate 42 ⋅ 33

4. __________________________

Simplify. 5. − 7 – 11

5. __________________________

6. 21 – 36

6. __________________________ 7. __________________________

7. (−9)( 5) 8.

−33 − 11

8. __________________________

9. 18 – (−7) + (−9)

9. __________________________

10. Evaluate 3x – y when x = −1 and y = −3.

10. __________________________

11. Simplify 3x − 2 + 4x − 7 by combining like terms.

11. __________________________

12. Multiply: 2(3x – 7)

12. __________________________

13. Find the perimeter for the rectangle shown. 2x yd.

13. __________________________

4x + 3 yd. Solve the following equations. 14. 8x – 19 = 7x + 12

14. __________________________

15. 12x – 17 = 27

15. __________________________

Translate the following into a mathematical expression. Use x to represent “a number”. 16. Seven less than twice a number. Perform the indicated operations. Write answers in simplest form. 3 8 ⋅ 17. 4 9 7 3 18. − 12 4 T-252

16. __________________________

17. __________________________ 18. __________________________

Name: Instructor:

Date: Section:

Final Exam Test Form B (cont’d) 19.

11 4 ÷ 20 5

19. __________________________

20. Simplify:

21. Solve:

4 5x 8 15 x

20. __________________________

1 2 +x= 7 3

22. Evaluate 5xy when x =

21. __________________________ 4 1 and y = . 5 5

22. __________________________

23. A 12-foot board is cut in three pieces to make shelves. Find the length of the shelves.

23. __________________________

24. Find the area of

24. __________________________ 4 ft. 2 14 ft.

Perform the indicated operations. Round the result to the nearest hundredth if needed. 25. 42.3 – 3.78

25. __________________________

26. 1.4 × 3.8

26. __________________________

27. Simplify: 11.5x – 0.9x + 32.5

27. __________________________

28. Find the square root and simplify.

49 64

Solve the proportions. 13 1 = 29. 39 y

28. __________________________

29. __________________________

30. Given that the triangles are similar, find the missing length. 5

30. __________________________

x 3 12

31. Write 129.4% as a decimal.

31. __________________________

32. Write 0.859 as a percent.

32. __________________________

33. Write 48% as a fraction in simplest form. 3 34. Write as a percent. 8 Solve (Round to the nearest tenth if needed.) 35. What number is 12% of 500?

33. __________________________ 34. __________________________ 35. __________________________

T-253

Name: Instructor: Final Exam Test Form B (cont’d)

Date: Section:

36. 118 is what percent of 354? Round to the nearest tenth.

36. __________________________

37. 20% of what number is 60?

37. __________________________

38. Graph 2x + y = 6.

38.

6 5 4 3 2 1 x

0 -6

-5

-4

-3

-2

-1

-2 -3 -4 -5 -6 y

39. Find the mean, median, and mode for: 32, 54, 27, 44, 86.

39. __________________________

40. What is the probability of selecting a 10 of Diamonds from a deck of cards?

40. __________________________

41. Find the supplement of a 75° angle.

41. __________________________

42. Find the circumference of a circular flower bed with 22 diameter of 2 12 yards. Use π ≈ for an approximation. 7 (Round to the nearest tenth.)

42. __________________________

43.

43. __________________________

Convert 32.4 meters to millimeters.

44. Convert 3 liters to centiliters.

44. __________________________

45.

Convert 7 quarts to gallons.

45. __________________________

46. Subtract: (2x – 5) – (5x + 2).

46. __________________________

Multiply and simplify: 47.

47. __________________________

y4 ⋅ y6 ⋅ y

48. (x7)3

48. __________________________

49. −2x(3x2 – 5x + 7)

49. __________________________

50.

50. __________________________

T-254

Find the greatest common factor of the list of terms. 15x3, 25x2, 70x3y2

Name: Instructor:

Date: Section:

Final Exam Test Form C Perform the indicated operations. 1. 294 + 371

1. __________________________

2. 704 – 695

2. __________________________

361 × 29

3. __________________________

4. 1190 ÷ 14

4. __________________________

5. Evaluate 3 2 ⋅ 2 4

4. __________________________

Simplify. 6. __________________________

6. − 9 + (– 3)

7. __________________________

7. −27 + 8 8. (−10)(−2)

8. __________________________

64 −8

9. __________________________

10. Evaluate 2xy when x = −2 and y = 3.

10. __________________________

11. Multiply: −3(11x – 4)

11. __________________________

Solve the following equations: 12. 9x – 6x = 12

12. __________________________

13. 7 – (x + 5) = 32

13. __________________________

Translate the following into a mathematical expression. Use x to represent “a number.” 14. Twelve less than a number.

14. __________________________

Perform the indicated operations. Write answers in simplest form. 15.

5 8 ⋅ 6 15

15. __________________________

16.

2 1 + 3 5

16. __________________________

3 17. Simplify: 4 x 9 16 x

17. __________________________

T-255

Name: Instructor: Final Exam Test Form C (cont’d) 18. Solve

Date: Section:

1 5 + 2x = 3 12

18. __________________________ 19. __________________________

1 1 and y = . 2 3

19. Evaluate 4x – y when x =

20. __________________________ 3 20. 12 yards of ribbon are cut into pieces yd. each to make 4 bows. How many bows can be made from the 12 yards?

21. __________________________

21. Find the area of the rectangle shown: 2 12

yd.

5 yd. Perform the indicated operations. Round the result to the nearest tenth if needed. 22. 4.29 – 0.361

22. __________________________

23. 1.93 ∙ 36.2

23. __________________________

24. 0.6 ÷ 1.5

24. __________________________

25. Simplify: 7.3x – 5.8 + 2.5x

25. __________________________ 25 49

26. Find the square root and simplify. 27.

Solve the proportion:

26. __________________________ 27. __________________________

x 5 = 12 48

28. Given that the triangles are similar, find the missing length. 6

28. __________________________

x 15

29. Write 0.32 as a percent.

29. __________________________

30. Write 20% as a fraction or mixed number in simplest form. 3 31. Write as a percent. (Round to the nearest tenth if 7 needed.)

30. __________________________ 31. __________________________

Solve. Round to the nearest tenth if needed. 32. What number is 17% of 720?

32. __________________________

33. 13% of what number is 39?

33. __________________________

34. Find the simple interest earned on a $1500 investment if the interest rate is 6% for 2 12 years.

34. __________________________

T-256

Name: Instructor: Final Exam Test Form C (cont’d) 35. Graph y =

Date: Section:

1 x −4. 2

35.

5 4 3 2 1 x

0 -6

-5

-4

-3

-2

-1

-2 -3 -4 -5 -6 y

36. Find the mean, median, and mode for: 12, 18, 85, 14, 12, 69, 73 Round to the nearest tenth if needed.

36. __________________________

37. What is the probability of an even number appearing if a single die is rolled?

37. __________________________

38. Find the complement of a 83° angle.

38. __________________________

39. Find the perimeter of the rectangle shown.

39. __________________________

3.2 m 6.8 m 40. Find the volume of a box with length 4 feet, width 2 feet and height 1 12 feet.

40. __________________________

41.

41. __________________________

Find the circumference of the flower base of a fountain if the diameter is 5 feet. Use π ≈ 3.14 for an approximation.

42. Convert 2.3 meters to millimeters.

42. __________________________

42.

43. __________________________

Convert 382.5 meters to kilometers.

44. Convert 8 quarts to gallons.

44. __________________________

45. Add: (3x + 2) + (4x2 – 7x + 8)

45. __________________________

46. Find the value of x2 + xy when x = −2 and y = 3.

46. __________________________

Multiply and simplify. 47. x ⋅ x6 ⋅ x6

47. __________________________

48. (3x2)4

48. __________________________

49. −x(x2 – 4x + 5)

49. __________________________

50. Factor out the greatest common factor. 33x2y + 11xy + 66xy2

50. __________________________

T-257

Name: Instructor:

Date: Section:

Final Exam Test Form D Perform the indicated operations. 1. 691 + 432

1. __________________________

2. 896 – 798

2. __________________________

19 × 34

3. __________________________

4. 2380 ÷ 14

4. __________________________

Simplify. 5. −9 + 10

5. __________________________

6. −45 + 6

6. __________________________ 7. __________________________

7. (−2)(8) 8. (−40) ÷ (−4) 9. Evaluate x – y

8. __________________________ 9. __________________________

when x = −1 and y = −3.

10. Multiply: 3(2x – 5)

10. __________________________

11. Find the perimeter of the rectangle shown.

11. __________________________ 2x ft.

4x ft. Solve the following equations. 12. 12 + 3x = –3 – 2x

12. __________________________

13. −2(4 – x) = 16

13. __________________________

Translate the following into a mathematical expression. Use x to represent “a number.” 14. Seventeen less than twice a number. Perform the indicated operations. Write answers in simplest form. 3 14 15. ⋅ 7 15 2 1 − 16. 3 4 3 9 17. ÷ 15 10

T-258

14. __________________________

15. __________________________ 16. __________________________ 17. __________________________

Name: Instructor: Final Exam Test Form D (cont’d)

Date: Section:

2 18. Simplify 5 x 4 15 x 2 x=6 19. Solve 3

20. Evaluate

1 x+ y 2

18. __________________________

19. __________________________ when x =

1 1 , y=− . 2 3

1 of a yard of ribbon, how many bows can 3 be made from 20 yards of ribbon?

21. If a bow takes

20. __________________________

21. __________________________

Perform the indicated operations. Round the result to the nearest hundredth if needed. 22. 37.49 – 28.371

22. __________________________

23. 17.2 × 3.6

23. __________________________

24. 0.46 ÷ 0.002

24. __________________________

25. Simplify 2.7x – 3.8 + 1.2x

25. __________________________

26. Find the square root and simplify. 27. 28.

36 49

x 2 = 5 15 Given that the triangles are similar, find the missing length.

Solve the proportion:

26. __________________________ 27. __________________________ 28. __________________________

3 25

5 3 of an inch on a road 29. If 125 miles is represented by 4 map, find the distance between 2 cities which are 3 34

29. __________________________

inches apart. 30. Write 0.163 as a percent. 31. Write 30% as a fraction in simplest form. 3 32. Write as a percent. Round to the nearest tenth. 7 Solve. (Round to the nearest tenth if needed.)

30. __________________________ 31. __________________________ 32. __________________________

33. 17 is what percent of 340?

33. __________________________

34. 11% of what number is 66?

34. __________________________

35. Find the simple interest rate earned on $500 for 2 years at 5.5%

35. __________________________

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Name: Instructor: Final Exam Test Form D (cont’d) 36. Graph y = –3x + 6.

Date: Section: 36.

6 5 4 3 2 1 x

0 -6

-5

-4

-3

-2

-1

-2 -3 -4 -5 -6 y

37. Find the mean, median, and mode for: 3, 6, 2, 8, 11, 5 Round to the nearest tenth if needed.

37. __________________________

38. What is the probability of a 6 being rolled if a single die is tossed?

38. __________________________

39. Find the complement of a 65° angle.

39. __________________________

40. Find the area of the triangle shown.

40. __________________________

2.4 m 6.2 m 41. Find the volume of a cube with sides of length 2.3 cm (Round to the nearest tenth.)

41. __________________________

42.

42. __________________________

Find the circumference of a water tower if the radius is 10 feet. Use π ≈ 3.14 for an approximation.

43. Convert 93.4 meters to centimeters.

43. __________________________

44.

44. __________________________

Convert 387.5 grams to kilograms.

45. Convert 2 12 gallons to quarts.

45. __________________________

46. Add: (3x + 7) + (x2 – 5x – 2)

46. __________________________

47. Find the value of 3x – 5y when x = −1 and y = −3.

47. __________________________

Multiply and simplify.

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48. x6 ⋅ x8 ⋅ x

48. __________________________

49. (–2x)3(3x2)2

49. __________________________

50.

50. __________________________

Factor the greatest common factor. 5xy2 + 15x + 25x2y2

Name: Instructor:

Date: Section:

Final Exam Test Form E Perform the indicated operations. 1. 432 + 791 a. 1233

b. 1213

c. 1113

d. 1223

b. 178

c. 182

d. 183

b. 50

c. 91

d. 99

2. 495 – 317 a. 188 3. 495 ÷ 5 a. 10

4. Estimate the sum 3560 + 2250 + 1725 by rounding each number to the nearest hundred. a. 4035

b. 6500

c. 7600

d. 8400

b. 2

c. 48

d. –2

b. −7

c. −1

d. 15

b. 4

c. –4

d. −12

c. 3

d. 1

Simplify: 5. −6(8) a. −48 6. (−3) + (−8) + 4 a. 7 7.

16 −4

a. 12

8. Evaluate (x + y)2 when x = −2 and y = 3. a. −1

b. 4

9. Simplify 7x+ 5x – 3 + 3x – 7 by combining like terms. a. 5x – 15

b. 15x – 10

c. 20x – 10

d. 5x – 5

b. −8x – 5

c. 8x + 2

d. −8x − 20

10. Multiply 4(−2x – 5) a. −8x + 5

11. Find the perimeter of the rectangle shown. 2m 5x m a. 10x + 4

b. 14x + 2

Solve the following equations: 12. x + 7 = −2x – 5 a. 6 b. 4

c. 10x + 2

d. 20x + 4

c. −4

d. −3

T-261

Name: Instructor: Final Exam Test Form E (cont’d)

Date: Section:

13. x + 12 = −8 a. 4

b. −20

c. –4

d. 20

Perform the indicated operations. Write answers in simplest form. 14.

3 12 ⋅ 9 15

1 5

15.

2 1 − 3 4

1 12

16.

4 15

3 5

1 5

1 2

5 12

7 12

1 25

c. 25

1 4

1 3

2 3

3 2

3 8

11 15

1 15

2 1 ÷ 5 10

a. 4

17.

Simplify:

1 2

18. Solve x − a.

5 2x 15 4x

1 2 = 3 5

1 2

19. How many bows can be made from 12 yards of ribbon if each bow takes a. 48

b. 36

c. 24

1 of a yard? 4 d. 32

20. Find the area of the rectangle shown. 2 12 yards 4 12 yards

a. 11 yd.2

b. 11 14 yd.2

c. 12 14 yd.2

d. 9 yd.2

b. −36.414

c. −36.42

d. 36.5

21. −22.56 + (−13.854) a. 36.4 T-262

Name: Instructor: Final Exam Test Form E (cont’d)

Date: Section:

22. 16.53 ÷ 3 a. 5.51

b. 5.52

c. 5.6

d. 5.55

c. 1.077x

d. 1.77x

c. 6

d. 18

23. Simplify 1.7x + 0.07x a. 1.177x

b. 1.14x

24. Find the square root: a. 3

b. 9

25. Write as a unit rate: 585 km in 10 hours. (Round to the nearest tenth.) a. 5850

b. 5.9

d. 585

c. 12

d. 9

x 3 . = 13 39

26. Solve the proportion a. 7

c. 58.5

b. 13

27. Given that the triangles are similar, find the missing length. 10

5 7 x

a. 13

b. 7.14

28. If 35 miles is represented by a. 485

c. 14

d. 7

1 on a scale drawing, how many miles is represented by 3 12 ? 4

b. 480

29. Write 4.9% as a decimal. a. 490 b. 49

c. 490

d. 470

c. 0.049

d. 0.49

30. Write 0.023 as a percent. (Round to the nearest tenth if needed.) a. 0.23% b. 2.3% c. 0.023%

3 as a percent. 8 a. 375% b. 37.5%

d. 23%

31. Write

c. 3.75%

d. 0.375%

32. What number is 12% of 150? a. 12.5 b. 1250

c. 18

d. 1800

33. 18 is what percent of 195? a. 9.2% b. 9.3%

c. 9.7%

d. 9%

Solve. (Round to the nearest tenth if needed.)

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Name: Instructor: Final Exam Test Form E (cont’d)

Date: Section:

34. Find the sales tax on a total bill of $250 if the tax rate is 6%. a. $21.50

b. $15

c. $13

35. Graph y = 2x + 2. a.

d. $17

b. 6

6 5 4 3 2 1 0 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -2 -3 -4 -5 -6

5 4 3 2 1 0 -6

-5

-4

-3

-2

-1

-1 -2 -3 -4 -5 -6

1 0 -6 -5 -4 -3 -2 -1 -1 0

0 -6

-5

-4

-3

-2

-1

-2

-3

-4

-5

-6

36. Find the mean for: a. 19

12, 18, 27, 25, 33. Round to the nearest tenth if necessary. b. 15

c. 20

d. 23

37. What is the probability of drawing an Ace of hearts out of a deck of cards? 1 1 1 a. b. c. d. 26 3 13 38. Find the supplement of a 35° angle. a. 45° b. 145°

c. 55°

1 52

d. 125°

39. Find the perimeter of the triangle shown. 4.7 yd. a. 15 yd.

b. 14.6 yd.

4.7 yd.

5.2 yd. c. 13.8 yd.

d. 17 yd.

40. Find the area of the rectangle shown. 3.2 m a. 23 yd.2

T-264

b. 24 yd.2

7.2 m c. 23.04 yd.2

d. 14.6 yd.2

Name: Instructor: Final Exam Test Form E (cont’d)

Date: Section:

41. Find the volume of a cube with sides of length 5 cm. a. 150 cm3

b. 125 cm3

c. 75 cm3

d. 25 cm3

42. Find the circumference of a circular flower bed with diameter 10 ft. Use π ≈ 3.14 for an approximation. a. 31.6 ft.

b. 31.41 ft.

c. 32 ft.

d. 31.4 ft.

c. 1350

d. 0.135

c. 4

d. 0.04

c. 12.8

d. 12.833

43. Convert 1.35 meters to millimeters. a. 0.00135

b. 135

44. Convert 0.4 grams to milligrams. a. 0.0004

b. 400

45. Convert 385 feet to yards. a. 128 13

b. 128

46. Add: (2x2 + 7x – 12) + (x2 – 3x + 6) a. 3x2 + 10x – 6

x2 + 4x – 6

c. 3x2 + 4x – 6

d. 3x2 + 10x + 18

y12

c. y6

x24

c. x10

d. 6x4

b. 36x6

c. 36x12

d. 6x12

Multiply and Simplify: 47. y6y3 a. y18

48. (x4)6 a.

49. (2x)2(3x2)2 a. 6x6

50. Find the greatest common factor of the following terms: 3xy, 9x2y, 12xy2 a. 3x2y

b. 3xy

c. x2y2

T-265

Name: Instructor:

Date: Section:

Final Exam Test Form F Perform the indicated operations. 1. 372 + 49 a. 421

b. 411

c. 311

d. 321

b. 585

c. 475

d. 484

b. 8892

c. 8782

d. 6892

b. 3

c. −11

d. 11

b. −7

c. 23

d. −23

b. −12

c. 1

d. –1

b. 6

c. −9

d. −6

8. Evaluate x – 2y when x = −2 and y = 3. b. 8 a. −4

c. 6

d. −8

9. Multiply −2(3x + 8) b. –6x + 8 a. −6x – 8

c. −6x + 16

d. −6x − 16

2. 769 – 285 a. 575 3. 342 × 26 a. 8692 Simplify: 4. −7 + 4 a. −3 5. −15 − 8 a. 7 6. 3(−4) a. 12 7. 63 ÷ −7 a. 9

10. Find the perimeter of the rectangle shown. 2x ft. (3x + 4) ft. a. (10x + 8) ft.

(5x + 4) ft.

Solve the following equations: 11. −x + 8 = −5 b. 3 a. −3 12. 5x – 7 = −47 59 a. − 5

T-266

b. 8

c. 18x ft.

d. 19x ft.

c. −13

d. 13

c. –8

54 5

Name: Instructor: Final Exam Test Form F (cont’d)

Date: Section:

13. 11x – 12 = 12x – 8 a. –4

b. 4

c. 20

d. −20

Perform the indicated operations. Write answers in simplest form. 14.

3 10 ⋅ 5 11

30 11

15.

7 12

b. −

17. Simplify:

6 5

2 3

1 4

2 3

c. −

a. −

1 2

b. 1

6 5

d. −

6 5

2 3

3 2

11 15

1 15

d. −

c. −24

d. −48

1 2x 3 4x

1 x

18. Solve x +

3 8x

2 1 = . 5 3

19. Evaluate 2x2y when x = 2 and y = −3 a. 24 b. 48 20.

13 16

−3 5 ÷ 13 26

a. −1

6 11

1 3 − 12 4

a. −

16.

1 15

1 a yard? 2 d. 8 bows

How many bows can be made from 12 yards of ribbon, if each bow takes

a. 6 bows

b. 24 bows

c. 30 bows

21. Find the area of the rectangle shown. 3 13 yards 6 14 yards

a. 20 56 yd.2

b. 9 56 yd.2

c. 20 yd.2

d. 12 165 yd.2

T-267

Name: Instructor: Final Exam Test Form F (cont’d)

Date: Section:

22. Add: −40.25 + (−16.387) Round to the nearest tenth. a. 65.6

b. −56.6

c. −65.6

d. 56.6

c. 163.4

d. 16.34

c. 2x – 2

d. 2x – 2.01

c. 6

d. 18

c. y = 17

d. y =

23. Multiply: 17.2 × 9.5 a. 14.34

b. 143.4

24. Simplify 5.2x – 2.01 – 3.2x a. 2x

b. 4x

25. Find the square root: a. 2

36 .

b. 3

26. Solve the proportion:

a. y = 15

y 1 = 3 51

b. y = 153

3 51

27. Given that the triangles are similar, find the missing length. 2 7 6 x a. 14

b. 28

c. 12

d. 21

c. 0.45

d. 0.045

28. Write 4.5% as a decimal. a. 45

b. 450

29. Write 125% as a mixed number or fraction in simplest form. 5 7 a. 1 20 b. 1 14 c. 1 20

17 20

Solve. (Round to the nearest tenth if needed.) 30. What number is 6% of 120? a. 72 b. 720

c. 0.72

d. 7.2

31. 5% of what number is 150? a. 3 b. 300

c. 30

d. 3000

32. 37 is what percent of 185? a. 27 b. 2

c. 20

d. 200

33. Find the sales tax on a total bill of $138 if the tax rate is 5%. a. $6.89 b. $0.69 c. $69

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d. $6.90

Name: Instructor: Final Exam Test Form F (cont’d)

Date: Section:

34. Graph y = 2x + 1. a.

1 0 -6 -5 -4 -3 -2 -1-1 0

1 1

0 -6 -5 -4 -3 -2 -1-1 0

-2

-3

-4

-5

-6

d. 6 5 4 3 2 1 0 -6 -5 -4 -3 -2 -1-1 0 -2 -3 -4 -5 -6

6 5 4 3 2 1 1

0 -6 -5 -4 -3 -2 -1-1 0

-2 -3 -4 -5 -6

35. Find the mean for: a. 19.9

5, 12, 28, 17, 24, 17, 33. Round to the nearest tenth if necessary. b. 194

c. 19.4

d. 19

c. 17

d. 28

36. Find the mode for: 5, 12, 28, 17, 24, 17, 33 a. 24

b. 12

37. Find the median for: a. 17

5, 12, 28, 17, 24, 17, 33 b. 12

c. 28

d. 24

38. What is the probability of an even number appearing on a toss of a single die? a.

1 3

6 3

1 2

1 6

39. Find the complement of a 49° angle. a. 141°

b. 131°

c. 51°

d. 41°

40. Find the volume of a cube whose sides are 1.6 m. Round to the nearest tenth. b. 4.1 m3 c. 2.6 m3 d. 4.8 m3 a. 9.6 m3 41. Find the circumference of a circular flower bed with diameter 8 ft. Use π ≈ 3.14 for an approximation. Round to the nearest foot. a. 27 ft. b. 28 ft. c. 24 ft. d. 25 ft. 42. Convert 90 meters to millimeters. a. 90 mm

b. 90,000 mm

c. 0.09 mm

d. 9000 mm T-269

Name: Instructor: Final Exam Test Form F (cont’d)

Date: Section:

43. Convert 3.5 grams to milligrams. a. 35 mg

b. 3500 mg

c. 350 mg

d. 0.035 mg

44. Convert 2640 feet to miles. a.

1 mi. 4

b. 2 mi.

1 mi. 2

d. 1 mi.

45. Add: (3x2 + 7x – 2) + (x2 – 6x + 4) a. 4x2 + x – 6

b. 4x2 + x + 2

c. 4x2 − 13x – 6

d. 4x2 + x − 4

46. Find the value of (2x)2y when x = 2 and y = −2. a. −16

b. 16

c. 32

d. −32

b. x32

c. x12

d. x13

b. x4

c. x15

d. x8

b. −27x9

c. 3x9

d. −3x9

Multiply and Simplify: 47. x8 ⋅ x ⋅ x4 a. x33 48. (x3)5 a. x10 49. (−3x3)3 a. −27x6

50. Find the greatest common factor of the following terms: 2x2, 12xy, 16x2y2 a. 4x2y2

T-270

b. 2x

c. 2x2

d. 2xy

Name: Instructor:

Date: Section:

Final Exam Test Form G Perform the indicated operations. 1. 28 + 352 a. 379

b. 380

c. 370

d. 360

b. 376

c. 274

d. 264

b. 1072

c. 107

d. 1027

b. 3702

c. 3170

d. 3720

b. −15

c. −7

d. 7

b. –2

c. −8

d. 8

b. 15

c. −2

d. 2

b. −4

c. 18

d. −18

b. 8

c. 7

d. −7

b. −18

c. −7

d. 18

b. 12

c. 10

d. 16

b. 30 – 10x

c. 30 – 2x

d. 30 + 10x

2. 362 – 98 a. 336 3. 3216 ÷ 3 a. 172 4. 310(12) a. 370 Simplify: 5. −11 – 4 a. 15 6. −3 − (−5) a. 2 7. 3(−5) a. –15 8. –24 ÷ (−6) a. 4 9. −3 −8 + 4 a. −15

Evaluate when x = −2 and y = 3. 10. 3xy a. 7 11. x2 + 2(y) a. 18 12. Multiply: 5(6 – 2x) a. 10x – 30

T-271

Name: Instructor: Final Exam Test Form G (cont’d)

Date: Section:

13. Find the perimeter of the rectangle shown. x ft. (x + 9.2) ft. a. 13.2x ft.

b. (4x + 9.2) ft.

c. (2x. + 9.2) ft.

d. (4x + 18.4) ft.

14. 21 – x = 2x + 6 a. 28

b. 27

c. 5

d. 15

15. x – 9 = 12 a. 21

b. –21

c. 3

d. −3

Solve the following equations:

Perform the indicated operations. 3 2 16. ⋅ 5 9 6 2 b. a. 5 15 17.

3 1 − 2 3

1 3

18.

−

a. − 19.

21.

2 45

4 5

2 5

2 9

b. −

1 9

1 3

c. −

d. −

1 2

6 3 ÷ 7 14

1 4

Solve x + 2 3

b. 4

18 98

6 49

5 4

4 x−2

9 x−2

7 3

c. −

1 3

2 3

4 x−2 5 x−2

4 5

a. 3

T-272

5 14

1 1 +− 3 6

20. Simplify:

7 6

2 =3 3

Name: Instructor: Final Exam Test Form G (cont’d)

Date: Section:

22. Evaluate (xy)2 when x = 1 and y = −2. a. −4

c. −2

b. 4

23. How many bows can be made from 18 yards of ribbon if each bow takes a. 18 bows

b. 3.5 bows

c. 72 bows

d. 2 1 of a yard? 4

d. 22 bows

24. Add: −37.82 + (−12.151) Round to the nearest hundredth. a. 49.44

b. 49.7

c. −49.45

d. −49.97

25. Multiply: 3.8(27.5) Round to the nearest tenth. a. 105

b. 104.82

c. 104.5

d. 105.4

c. 14.9x

d. 8.9x + 6

c. 8

d. 16

c. x = 28.9

d. x = 25

26. Simplify 3.7x + 5.2x – 6 a. 14x

b. 8.9x – 6

27. Find the square root: a. 4

b. 2 x 5 = 17 85

28. Solve the proportion: a. x = 289

b. x = 28

29. Given that the triangles are similar, find the missing length. 3 x a. 14

b. 28

c. 12

30. If 70 miles is represented by

d. 21

1 4 inch on a scale drawing, how many miles are represented by 3 3

inches? a. 320 miles

b. 160 miles

c. 280 miles

d. 380 miles

c. 0.023

d. 230

31. Write 2.3% as a decimal. a. 2.3 32. a.

b. 0.23

Write 85% as a fraction in simplest form. 3 10

13 20

17 20

17 100

T-273

Name: Instructor: Final Exam Test Form G (cont’d) 33. Write

Date: Section:

2 as a percent. (Round to the nearest tenth if needed.) 5

a. 40%

b. 4%

c. 0.4%

d. 0.40%

c. 4860

d. 48.6

34. What number is 18% of 270? a. 1500

b. 15

35. Graph y = −2x + 1. a.

0 -6 -5 -4 -3 -2 -1-1 0

-2

-3

-4

-5

-6

d. 6 5 4 3 2 1 0 -6 -5 -4 -3 -2 -1 -1 0 -2 -3 -4 -5 -6

6 5 4 3 2 1

36. Find the mean for: a. 16

0 -6 -5 -4 -3 -2 -1-1 0 -2 -3 -4 -5 -6

9, 3, 12, 18, 16, 9, 10. Round to the nearest tenth if necessary. b. 10

c. 11

d. 9

c. 11

d. 9

c. 11

d. 9

c. 127°

d. 37°

37. Find the mode for: 9, 3, 12, 18, 16, 9, 10. a. 16 38. Find the median for: a. 16

b. 10 9, 3, 12, 18, 16, 9, 10 b. 10

39. Find the supplement of a 53° angle. a. 62° b. 53°

40. Find the volume of a cube whose sides are 3.5 cm. Round to the nearest tenth. a. 49.2 cm3 b. 42.9 cm3 c. 10.5 cm3 d. 28.5 cm3 41. Find the circumference of a circular flower bed with diameter 18 ft. Use π ≈ 3.14 for an approximation. Round to the nearest tenth of a foot. a. 56.5 ft. b. 55.6 ft. c. 57 ft. d. 57.6 ft.

T-274

Name: Instructor: Final Exam Test Form G (cont’d)

Date: Section:

42. Convert 9.2 meters to millimeters. a. 920 mm

b. 0.0092 mm

c. 0.092 mm

d. 9200 mm

c. 120 mg

d. 0.0012 mg

c. 162 yd.

d. 160 yd.

c. 7x + 29

d. 7x – 5

b. 8

c. 4

d. 16

b. x8

c. x24

d. x9

b. x7

c. x21

d. x10

b. −2x4

c. –4x4

d. 4x4

43. Convert 1.2 grams to milligrams. a. 12 mg

b. 1200 mg

44. Convert 486 feet to yards. a. 1458 yd.

b. 168 yd.

45. Add: (x + 17) + (6x − 12) a. 7x + 5

b. 7x − 12

46. Find the value of x2y2 when x = 2 and y = −2. a. 32 Multiply and Simplify: 47. (x4)(x3)(x2) a. x12 48. (x7)3 a. x12 49. −(2x2)2 a. −4x

50. Find the greatest common factor of the following terms: 3xyz, xyz, 6x a. x

b. 3x

c. 3xy

d. yx

T-275

Name: Instructor:

Date: Section:

Final Exam Test Form H Perform the indicated operations. 1. 931 + 799 a. 1720

b. 1730

c. 1629

d. 11620

b. 423

c. 333

d. 324

b. 307

c. 357

d. 370

b. −1900

c. −190

d. 1900

b. −3

c. 15

d. –15

b. −39

c. –3

d. 39

b. 12

c. −7

d. 7

b. −5

c. 76

d. −76

2. 795 – 362 a. 433 3. 1850 ÷ 5 a. 375 4. 380(−5) a. 190 Simplify: 5. −9 + 6 a. 3 6. −21 − 18 a. 3 7. (−3)(4) a. −12 8. –95 ÷ (−19) a. 5

9. Evaluate x ÷ y when x = −2 and y = 3 a. −

3 2

2 3

c. −

2 3

3 2

10. Multiply: −2(3x – 2) a. −6x + 2

b. −6x + 4

c. –6x – 4

d. −6x – 2

11 Find the perimeter of the rectangle shown. 3x yd. 6x – 5 yd. a. 14x yd.

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b. 28x yd.

c. (18x – 10) yd.

d. (9x – 5) yd.

Name: Instructor: Final Exam Test Form H (cont’d)

Date: Section:

Solve the following equations: 12. 3x – 2x = 13 a.

13 3

b. 13

c. −13

d. −

b. 9

c. –4

d. 4

13 3

13. 3x – 15 = 3 a. 6

Perform the indicated operations. Write answers in simplest form. 3 4 ⋅ 28 15

14.

a. 35

1 9

a. 17.

3 11

1 21

d. 21

3 20

1 5

1 20

4 33

7 22

4 22

1 6

1 2

3 13

4 11 3 2 ÷ 39 13

7 2 Simplify: − x 7 2x

a. –1 19.

a. 2

18.

1 35

2 1 − 5 4

15.

16.

b. 0

c. 2x

d. 7

7 5

c. 2

d. –2

16 3

2 3

d. −

Solve 7x = 5 5 7

20. Solve 3x – 7 = 9. a.

3 16

2 3

T-277

Name: Instructor: Final Exam Test Form H (cont’d)

Date: Section:

21. How many bows can be made from 30 yards of ribbon if each bow takes a. 45 bows

b. 60 bows

c. 15 bows

1 a yard? 2

d. 30 bows

Perform the indicated operations. Round the results to the nearest hundredth if needed. 22. 13.059 + 16.128 a. 29.91

b. 29.19

c. 29

d. 29.99

b. 40

c. 0.04

d. 4.0

b. 3.12

c. 31.2

d. 312

b. 3

c. 1

c. 45

d. 7

c. x = 18

d. x = 45

23. 0.8(0.05) a. 0.4 24. 9.36 ÷ 0.3 a. 0.312 3

1 25. Simplify   ⋅ 27  3

a. 0

26. Find the square root. a. 14

b. 45.5

27. Solve the proportion a. x = 27

1 3

2 x . = 9 81

b. x = 36

28. Given that the triangles are similar, find the missing length. 3 6 9 x a. 6

b. 15

c. 12

29. If 50 miles is represented by a. 2000 miles

d. 18

1 on a scale drawing, how many miles is represented by 2 12 ? 4

b. 200 miles

c. 500 miles

d. 5000 miles

c. 0.03

d. 0.003

30. Write 0.3% as a decimal. a. 3

T-278

b. 30

Name: Instructor: Final Exam Test Form H (cont’d) 31. a.

Date: Section:

Write 12% as a fraction in simplest form. 6 10

32. Write

6 5

3 25

4 25

3 as a percent. (Round to the nearest tenth if needed.) 7

a. 3%

b. 37%

c. 42.8%

d. 42.9%

c. 2.94

d. 1.5

33. 14% of what number is 21? a. 150

b. 294

34. Find the sales tax on a total bill of $118 if the tax rate is 6.5% a. $7.76

b. $7.56

c. $7.67

d. $7.70

35. Find simple interest on a loan of $1500 for 2 12 years at 5% a. $187.90 36. Graph y =

b. $188.50

c. $187.50

d. $185.70

1 x −1 3

b. 6 5 4 3 2 1 0 -6 -5 -4 -3 -2 -1 -1 0 -2 -3 -4 -5 -6

6 5 4 3 2 1 0 -6 -5 -4 -3 -2 -1 -1 0 -2 -3 -4 -5 -6

d. 6 5 4 3 2 1 0 -6 -5 -4 -3 -2 -1 -1 0 -2 -3 -4 -5 -6

37. Find the mean for: a. 96

6 5 4 3 2 1 0 -6 -5 -4 -3 -2 -1 -1 0 -2 -3 -4 -5 -6

96, 31, 85, 74, 88. Round to the nearest tenth if necessary. b. 85 c. 74.8 d. 55.6

38. Find the mode for: 96, 31, 85, 74, 88 a. 96 b. 85

c. 74.8

d. none

39. Find the median for: 96, 31, 85, 74, 88 a. 96 b. 85

c. 74.8

d. none

T-279

Name: Instructor: Final Exam Test Form H (cont’d)

Date: Section:

40. Find the complement of a 172° angle. a. 19°

b. 82°

c. 18°

d. 8°

41. Find the volume of a cube whose sides are 0.5 cm. a. 15 cm3

b. 125 cm3

c. 0.125 cm3

d. 12.5 cm3

42. Find the circumference of a circular flower bed with diameter 7 ft. Use π ≈ 3.14 for an approximation. Round to the nearest tenth of a foot. a. 21.9 ft.

b. 22.0 ft.

c. 22.9 ft.

d. 21.8 ft.

c. 230 mm

d. 0.00023 mm

c. 380 mg

d. 0.0038 mg

c. 0.5 in.

d. 18 in.

b. −x2 + 6x – 12

c. x2 + 6x + 12

d. −x2 + 6x – 2

b. x28y9

c. x10y6

d. x11y9

b. x7

c. x27

d. x3

b. 16x12

c. 16x7

d. −16x7

43. Convert 0.23 meters to millimeters. a. 0.23 mm

b. 23 mm

44. Convert 3.8 grams to milligrams. a. 0.038 mg

b. 3800 mg

45. Convert 6 feet to inches. a. 2 in.

b. 72 in.

46. Subtract: (3x –7) − (−x2 –3x + 5) a. x2 + 6x – 12 Multiply and Simplify: 47. x4y3(x7y3) a. x11y6 48. (x4)3 a. x12 49. (−2x3)4 a. −16x12

50. Find the greatest common factor of the following terms: x2y, xy2, x2y2 a. x2y2

T-280

b. xy2

c. xy

d. x2y

Chapter 1

Tests

Answers

Pretest Form A 1. tens 2. 2602 3. 135 4. 6647 5. 22 cm 6. 241 7. 3000 8. 1400 9. 24 feet 10. 36 sq cm 11. 27 12. 8064 R 8 13. 36 14. 32 ⋅ 24 15. 33 16. 15 17. 10 18. 1 19. 14 20. 2x − 7

Pretest Form B 1. thousands 2. 10,532 3. 205 4. 12.792 5. 38 feet 6. 295 7. 27,000 8. 1900 9. 11 10. 144 sq m 11. 368 12. 526 R 2 13. 144 14. 2 ⋅ 63 ⋅ 82 15. 18 16. 35 17. 25 18. $27 19. 35 20. 6x

Test Form A 1. twenty-five thousand, four hundred eighty-nine 2. 3604 3. 1127 4. 455 5. 14,282 6. 85 7. 0 8. 1 9. 11,832 10. 26 11. 5 12. 33,000 13. 2300 14. 300 15. 4 16. 8 17. P = 12 cm, A = 9 cm2 18. P = 12 yds, A = 8 yds2 19.

x 7

20. 2x + 17 21. $59.50 22. $1495 23. $26 24. 35 25. yes

T-281

Chapter 1

Tests

Answers (cont’d)

Test Form B 1. ninety-six thousand, three hundred sixty-two 2. 6408 3. 118 4. 437 5. 11,501 6. 258 7. undefined 8. 1 9. 0 10. 576 11. 12 12. 14 13. 15,600 14. 3300 15. 1700 16. 2 17. 5 18. P = 16 in, A = 16 in2 19. P = 18 ft, A = 14 ft2 20. 3x + 5

Test Form C 1. ten thousand, three hundred seventy-one 2. 13,496 3. 181 4. 299 5. undefined 6. 41 7. 9342 8. 1 9. 0 10. 128 11. 3 12. 16 13. 4000 14. 1400 15. 2600 16. 21 17. 12 18. P = 20 cm, A = 21 cm2 19. P = 12 ft, A = 9 ft2 20. 2x − 5 21. 3x – 19 22. $72 23. $26 24. 87 25. no

Test Form D 1. forty-two thousand, forty-two 2. 61,205 3. 500 4. 588 5. 7776 6. 18 7. 0 8. 1 9. undefined 10. 675 11. 36 12. 0 13. 8000 14. 1500 15. 400 16. 24 17. 16 18. P = 12 m, A = 9 m2 19. P = 14 m, A = 10 m2 20. 2(x + 7) 21. (x + 6) − 4 22. $2596 23. $2495 24. $18 25. yes

21.

x 12

22. $297 23. 93° 24. $48 25. no

T-282

Chapter 1

Tests

Answers (cont’d)

Test Form E 1. sixteen thousand, two hundred ninety-five 2. 103,941 3. 128 4. 104 5. 5676 6. 358 7. 1 8. 0 9. 38 10. 29 11. 159 12. 30,000 13. 3500 14. 2500 15. 14 16. 7 17. P = 28 ft, A = 49 sq ft 18. P = 20 ft, A = 16 sq yd 19. x(x − 3) 20. x − 6 21. 966.4 calls 22. $71 23. $96 24. 94° 25. yes

Test Form F 1. b 2. c 3. a 4. a 5. c 6. a 7. b 8. a 9. c 10. c 11. a 12. a 13. a 14. b 15. c 16. d 17. b 18. b 19. b 20. c 21. b 22. a 23. d 24. b 25. d

Test Form G 1. c 2. b 3. a 4. b 5. d 6. b 7. d 8. b 9. a 10. b 11. a 12. c 13. c 14. d 15. d 16. a 17. a 18. d 19. c 20. d 21. a 22. c 23. a 24. b 25. b

T-283

Chapter 1

Tests

Test Form H 1. b 2. a 3. c 4. c 5. c 6. a 7. a 8. d 9. c 10. b 11. d 12. d 13. c 14. b 15. b 16. b 17. a 18. c 19. b 20. a 21. b 22. b 23. d 24. c 25. a

T-284

Answers (cont’d)

Chapter 2

Tests

Answers

Pretest Form A 1. −1400 2. 7 3. >

Pretest Form B 1. 8 2. −196 3. >

Test Form A 1. 5 2. −12 3. −18 4. −21 5. −7 6. 24 7. 9 8. 15 9. 52 10. 10 11. −30 12. −7 13. 4 14. −15 15. −2 16. 8 17. −2 18. −3 19. 7 20. 6 21. $256 22. +1580 23. −1 24. −7 25. 9

5. 5 6. −12 7. −8 8. −1 9. 13 10. −10 11. 5 12. 15 13. 0 14. –27 15. 36 16. −5 17. 12 18. −9 19. −2 20. 9

5. 5 6. 11 7. −17 8. −12 9. 8 10. −7 11. −24 12. −28 13. 24 14. −32 15. −308 16. 6 17. 28 18. −15 19. −4 20. −6

T-285

Chapter 2

Tests

Answers (cont’d)

Test Form B 1. 5 2. −21 3. 12 4. −11 5. –8 6. −7 7. 36 8. 5 9. −5 10. 12 11. 10 12. 5 13. −25 14. −2 15. −41 16. −1 17. –1 18. 11 19. −4 20. −9 21. $300 22. +150, −185, −$35 23. −3 24. 7 25. –9

Test Form C 1. 0 2. −16 3. 15 4. –6 5. 13 6. −2 7. 0 8. 81 9. 15 10. −8 11. −1 12. −9 13. −35 14. 23 15. −27 16. 4 17. −1 18. 25 19. 1 20. 1 21. 11° 22. −$13 23. 9 24. −6 25. –3

Test Form D 1. 8 2. −13 3. −80 4. –6 5. −25 6. 7 7. –88 8. 3 9. 23 10. 21 11. –8 12. −15 13. –72 14. 29 15. −9 16. −17 17. 1 18. 0 19. 9 20. −2 21. 12,290 + (−250), 12,030 ft 22. 1° 23. 5 24. 12 25. 6

T-286

Chapter 2

Tests

Answers (cont’d)

Test Form E 1. −6 2. −14 3. 45 4. −5 5. −8 6. 1 7. −19 8. 2 9. −16 10. 10 11. −43 12. −16 13. −33 14. −27 15. 0 16. −8 17. 0 18. −2 19. −4 20. −1 21. 3 22. $76 23. 49 24. 32 25. −1

Test Form F 1. b 2. a 3. c 4. b 5. b 6. c 7. a 8. b 9. a 10. c 11. a 12. b 13. b 14. b 15. a 16. c 17. c 18. b 19. b 20. b 21. d 22. a 23. c 24. a 25. b

Test Form G 1. b 2. b 3. b 4. a 5. d 6. d 7. a 8. a 9. c 10. c 11. b 12. b 13. b 14. a 15. b 16. c 17. a 18. b 19. b 20. d 21. d 22. d 23. d 24. a 25. a

T-287

Chapter 2

Answers

Answers (cont’d)

Test Form H 1. b 2. b 3. b 4. d 5. b 6. c 7. b 8. b 9. c 10. a 11. c 12. b 13. b 14. a 15. a 16. b 17. a 18. b 19. a 20. b 21. a 22. a 23. a 24. c 25. d

Cumulative Test Form A Chapters 1 and 2 1. 4 2. 5 3. 8 4. 5 5. −34 6. 36 7. −5 8. 108 9. 65 10. −3 11. 5 12. 7 13. −1 14. 10 15. 20 cm 16. 44 yd 17. 3x 18. 11 − 2x 19. 15 20. −9 21. 17 22. 0 23. −1 24. 120 25. $111

Cumulative Test Form B Chapters 1 and 2 1. c 2. a 3. d 4. b 5. d 6. a 7. c 8. a 9. c 10. d 11. a 12. b 13. d 14. a 15. c 16. d 17. c 18. d 19. a 20. b 21. a 22. b 23. b 24. a 25. b

T-288

Chapter 3

Tests

Answers

Pretest Form A 1. 10x 2. −7x −6 3. −15x 4. −32x + 28 5. (10x + 2) cm 6. no 7. −4 8. 18 9. 10 10. −6 11. −3 12. 4 13. 6 14. 8x − 7 15 15. 3x 16. 5 17. 5 18. −27 + 13 = −14 19. −7x = 21 20. 6

Pretest Form B 1. –3x 2. 9x − 23 3. −16y 4. −10x – 6 5. (46x − 15) ft 6. yes 7. 20 8. −5 9. 0 10. −24 11. −2 12. −6 13. –3 14. x + 12 15. 31 − 3x 16. 15 17. −15 18. 38 − (−5) = 43 19. 20 − 7 = 13 20. 3(x + 2) = 39

Test Form A 1. −x − 4 2. 4x − 2y − 2 3. 10x + 1 4. x + 10y 5. −6x + 12 6. −3x + 13 7. 9x + 4 8. 4x − 4 9. −22 10. −2 11. −17 12. 8 13. −4 14. 3 15. 3 16. 4 17. (5x + 17)cm 18. 6x in 19. 6 20. 41 women 21. 44 ft 22. (15 x − 10 ) sq. ft. 23. no 24. 5x +12 = 27 ; x = 3 25. 2x − 16 = x + 10 ; x = 26

T-289

Chapter 3

Tests

Answers (cont’d)

Test Form B 1. 12x 2. 14a + 6b 3. 5x + 5 4. 5a 5. 8x − 6 6. −x + 7 7. x + 1 8. 12x − 15 9. −1 10. −3 11. −2 12. 4 13. 2 14. 9 15. 9 16. 3 17. (8x − 6) inches 18. (12x + 8) cm. 19. no 20. yes 21. x – 6 = 22

Test Form C 1. 7a − 9 2. 5y + 14 3. 4y − 10 4. −10x + 2y 5. −2x − 8 6. 9a − 3b 7. 11x – 13 8. 2y + 1 9. −7 10. 4 11. 3 12. 9 13. 10 14. 7 15. −4 16. 13 17. −6 18. −15 19. yes 20. (6x + 8) feet 21. (8x + 20) meters 22. 20 23. 27 24. –20 25. $18

Test Form D 1. 4x 2. 7b − 5 3. 10a + 2b 4. 30x + 7 5. −12y + 32 6. 16x − 56 7. 5x + 8 8. 6y − 4 9. 4 10. 1 11. −17 12. −23 13. 7 14. 5 15. 4 16. 10 17. 22 18. 2 19. 24 20. 6 21. 1 = 36 inches 22. W = 12 inches 23. yes 24. 12x 25. 6x − 2

22.

x 4

= 16 no

23. 6 24. 20 25. 35 girls

T-290

Chapter 3

Tests

Answers (cont’d)

Test Form E 1. 5x − 13 2. 14y − 9 3. 5 x + 3 y 4. −3x + 3 5. −15x + 35 6. −6y − 16 7. 2x − 6 8. 4x + 16 9. 15 10. 5 11. −3 12. −35 13. −29 14. −4 15. −2 16. 11 17. 9 18. (7x + 14)cm 19. 48 20. 50 21. 6 22. 12 mph 23. 7 24. yes 25. no

Test Form F 1. b 2. a 3. c 4. a 5. c 6. d 7. c 8. d 9. c 10. a 11. c 12. a 13. b 14. b 15. b 16. c 17. c 18. b 19. c 20. d 21. a 22. b 23. b 24. a 25. d

Test Form G 1. d 2. d 3. c 4. c 5. a 6. b 7. c 8. b 9. c 10. b 11. c 12. d 13. c 14. b 15. c 16. b 17. a 18. b 19. c 20. b 21. a 22. a 23. c 24. a 25. b

T-291

Chapter 3

Tests

Test Form H 1. c 2. b 3. c 4. b 5. d 6. d 7. c 8. a 9. a 10. c 11. d 12. a 13. b 14. a 15. c 16. d 17. c 18. b 19. a 20. a 21. d 22. c 23. a 24. b 25. a

T-292

Answers (cont’d)

Chapter 4

Tests

Answers

Pretest Form A 1. undefined 2. −1 3. 25 ⋅ 3 19 4. 5 1 5. 4 6. 4 3y 7. 7 12 8. 35 35 9. 49 19 10. 3 20 9x + 4 y 11. 6 12. x 2 13. − 3 3 14. 8 1 15. 15 3 16. − 10 21 17. 2x 18. 15 19. 51 2 20. 3 ft 5

Pretest Form B 1. 0 2. 9 3. 24 ⋅ 3 ⋅ 5 41 4. 3 1 5. − 20 2 6. 2 15 2y 7. 3 11 8. 12 1 9. − 5 10. no 7x 11. 6 3 12. 10 5 13. − x 2 11 14. 5 15 1 15. −1 9 20 16. 48 x 17. 32 18. −28 1 19. 7 15 2 20. 3 ft 5

Test Form A 19 1. 5 9 2. 2 6 3. 10 7 1 4. 4 5 3 5. − 8 9 6. 37 2 7. 7 11x 8. 4 5 9. 16 1 10. 1 10 1 11. −1 2 2 12. 15 21 8 13. 9 1 14. 6 3 15. 2x 2 1 16. − 2 5 17. 16 3 18. 19 7 19. 8 1 20. 4 5 1 21. 1 6 7 22. 7 feet 15

T-293

Chapter 4

Tests

Answers (cont’d)

Test Form A (cont’d) 2 23. 4 cups 3 24. $80 7 25. feet 8

Test Form B 19 1. 7 41 2. 9 5 3. 11 8 1 4. 9 4 1 5. − 5 17 6. 26 47 7. 108 4 8. 1 15 1 9. 4 −6 + 2 x 10. 9 −6 − 3x 11. 10 4 12. 5 13 x 13. 35 4 14. 5 1 15. 1 2 1 16. −1 2 17. 13 18. 25 19. −192 1 20. 2 1 21. 5 7 22. 1 15 23. $52 24. 6 cups 36 25. square feet 49

Test Form C 41 1. 9 22 2. 7 3 3. 7 5 6 4. 12 7 14 5. −1 31 3 6. 4 9 7. 25 31 8. − 41 3 9. 1 5 10. −x 3 11. 1 7 5 12. 12 2 13. 9 5 14. 4 3 15. 10 9 xy 16. 2 17. 3 2 18. − 3 3 19. −9 5 7 20. − 30 21. −11 4 22. −1 5 2 23. −1 7 2 24. 4 feet 9

T-294

Chapter 4

Tests

Answers (cont’d)

Test Form C (cont’d) 25. $68

Test Form D 34 1. 7 19 2. − 8 2 3. 7 9 1 4. −10 5 6 5. 1 11 1 6. 4 2 7. 17 37 8. 80 3 + 2x 9. 5 1 10. x 1 11. 5 4 12. −2 5 x 13. 10 7 14. − 15 2 15. 5 7x 16. 16 17. 25 1 18. 11 19. −15 1 20. 1 4 4 21. −1 5 11 22. − 12 2 10 feet 23. 5

Test Form D (cont’d) 9 24. square cm 10 25. 3 cups

T-295

Chapter 4

Tests

Answers (cont’d)

Test Form E 49 1. 9 −68 2. 11 6 3. 4 7 5 4. 5 8 2 5. 10 9 7 6. 10 1 7. 5 1 8. − 87 9. 9x 4 10. 7 5 1 11. 3 1 12. 4 2 1 13. − 35 1 14. 12 2y 15. 3 16. 3 1 17. 1 2 18. −9 7 19. 12 3 20. 4 2 21. − 25 1 22. 81 1 23. 9 yards 2 24. $189 25. 405 miles

Test Form F 1. d 2. c 3. b 4. c 5. c 6. c 7. d 8. c 9. b 10. d 11. b 12. d 13. a 14. c 15. c 16. b 17. b 18. c 19. a 20. c 21. b 22. a 23. c 24. a 25. b

Test Form G 1. a 2. a 3. c 4. a 5. b 6. b 7. a 8. a 9. a 10. c 11. b 12. a 13. b 14. c 15. c 16. a 17. d 18. c 19. b 20. c 21. a 22. a 23. a 24. c 25. a

T-296

Chapter 4

Tests

Answers (cont’d)

Test Form H 1. a 2. d 3. d 4. a 5. c 6. a 7. b 8. b 9. c 10. a 11. d 12. b 13. d 14. a 15. c 16. a 17. c 18. d 19. b 20. a 21. c 22. c 23. c 24. d 25. a

Cumulative Test Form A Chapters 3 and 4 1. −x − 3 2. −20xy 3. 30x − 10 4. yes 5. −7 6. 2(x + 7) 5 7. 8 8. 23 ⋅ 32 ⋅ 5 5y 9. 6 1 10. 72 1 11. 9 12 12. 35 13. 2 11 14. 15 4 15. − 15 16. 30 1 17. − 13 7 18. − 33 1 19. 4 5 20. 6 3 21. 30 cm 5 22. 11 feet 7 23. 7 10 4 24. 7 5 25. $150

Cumulative Test Form B Chapter 3 and 4 1. b 2. b 3. a 4. b 5. a 6. d 7. c 8. a 9. b 10. c 11. b 12. d 13. c 14. a 15. c 16. c 17. b 18. d 19. c 20. a 21. a 22. b 23. a 24. b 25. d

T-297

Chapter 5

Tests

Answers

Pretest Form A 1. thousandths 2. > 12 3. 25 4. 283.3 5. 98.482 6. 244.4 7. 2.22x − 15 8. −10.287 9. 583 10. −7.35 11. 5.1 12. 0.2936 13. 300 14. 1.24 15. 42 ft 16. 3.924 17. 0.625 18. 45 sq. in. 19. 42; 40 20. 37.68

Pretest Form B 1. > 2. ten thousandths 4 3. 4 5 4. 38.35 5. 37.352 6. 43.08 7. 3.52x + 6.37y 8. 0.0012 9. 525.8 10. 0.05 11. 2.232 12. 0.06824 13. 44; 43; 22 14. 0.125 15. −7.11 16. −0.000027 17. 230 18. 0.875 19. 5.8 sq. in. 20. $13.51

Test Form A 1. thirty-seven and two hundred thirty-four thousandths 2. thirteen and one hundred twenty-eight ten-thousandths 3. 227.402 4. 0.714 5. 6.972 6. −8.331 7. 15.906 8. 0.34 9. 1.79 10. −2.6x 1 11. 3 4 1 12. 16 500 13. < 14. > 15. 0.6 16. 17 17. −11.3 18. −25.2 19. 0.95 20. 0.0625 21. −57.9 22. 39; 31; no mode 23. 86 24. 25.13 feet 25. 9.6 sq. cm.

T-298

Chapter 5

Tests

Answers (cont’d)

Test Form B 1. thirty-three and one thousandths 2. 300.009

Test Form C 1. thirty-six and fifty-two hundredths 2. 207.043 3. 0.6 11 4. 13 20 5. > 6. < 7. 273.5 8. 8.904 9. 1.89 10. −10.47 11. 0.0212 12. 208 13. −0.000008 14. 0.018; 0.08; 0.18; 0.8 15. −0.4 16. 40 17. 0.424 18. 3.152 19. 2.1 20. −0.0000096 21. −105 22. 33; 28; 24 23. 82 24. 17.92 square feet 25. 8.4π ≈ 26.376 cm

Test Form D 1. one hundred twenty and fiftysix thousandths 2. 2200.024 3. 0.778 18 4. 35 25 5. = 6. < 7. 87.9 8. 10.072 9. 48.562 10. −10.47 11. 0.0084 12. 46.525 13. 0.504 14. 0.0008 15. 0.1 16. 0.062; 0.206; 0.26; 0.6 17. 32 18. 6.3 19. −48.92 20. 0.125 21. 4.1 22. 48; 45; 34 23. $1.12 24. $17.50 25. 37.72 cm 2

6 25

4. 0.625 5. = 6. > 7. 139.5 8. 44.511 9. −5.19 10. 10.54 11. −1.92 12. −20.05 13. 2.197 14. −40.32 15. −6.3 16. 2.814 17. 2.1 18. 5 19. 0.09 20. −1.93 21. −4.4 22. 79; 85; 85 23. 16.25 sq ft 24. 31.4 ft 25. 0.072; 0.7; 0.702; 0.72

T-299

Chapter 5

Tests

Answers (cont’d)

Test Form E 1. one hundred twenty-four and thirty-six hundredths 2. fifteen and one hundred two ten thousandths 3. 3227.51 4. 0.714 1 5. 6 4 6. 285.370 7. < 8. < 9. 45.876 10. 3.85 11. 8.96 12. 0.284 13. 15 14. −0.792 15. 6.32 16. 38 17. 4.8 18. −0.08 19. 19.5 20. 1.728 21. 0.0625 22. 36.75; 37; no mode 23. 1.3π ≈ 4.082 yards 24. 59.52 in 2 25. $3079.85

Test Form F 1. a 2. c 3. c 4. b 5. d 6. b 7. c 8. a 9. b 10. d 11. c 12. a 13. a 14. c 15. a 16. b 17. d 18. c 19. b 20. c 21. a 22. d 23. a 24. a 25. d

Test Form G 1. c 2. b 3. c 4. b 5. a 6. b 7. d 8. d 9. a 10. b 11. a 12. d 13. c 14. b 15. a 16. c 17. d 18. b 19. d 20. b 21. c 22. b 23. b 24. b 25. a

T-300

Chapter 5

Tests

Answers (cont’d)

Test Form H 1. b 2. c 3. a 4. d 5. b 6. b 7. c 8. a 9. b 10. c 11. a 12. b 13. d 14. a 15. c 16. b 17. a 18. b 19. a 20. b 21. b 22. d 23. a 24. c 25. d

T-301

Chapter 6

Tests

Answers

Pretest Form A 13 1. 50 1 4 2 2. 8.75 7 3. 100 43 4. 7 4 5. 5 6. 25.8 miles/gal 7. $0.06/ounce 4 12 = 8. 12 36 1 3 2 = 4 9. 50 75 10. no 11. yes 12. 16.7 13. 3 14. 18 140 15. 33 16. 198 miles 17. 1 : 3 18. 4 19. 10 20. 14.318

Pretest Form B 28 1. 54 32 2. 424 102 3. 17 6 4. 5 4 5. 1 6. $5.75/hour 7. $0.13/ounce 1 1 1 = 2 8. 6 9 24 96 = 9. 1 4 10. 875 miles 11. no 12. 0.625 1 13. 64 14 14. 3 15. 68 16. 1 : 5 1 17. 1 2 18. 8.5 ft 19. 9 20. 17.2

Test Form A 25 1. 78 8 2. 37 2 3. 3 11 4. 18 1 5. 3 7 6. 18 5 computers 7. 11 students 1 teacher 8. 22 students 32 miles 9. 1 gallon 8 16 = 10. 10 20 2 4 11. = 6 12 12. false 13. true 14. 38 miles/gallon 15. 120 calories/serving 16. 4 17. 18 18. 6 19. 0.85 20. 12 21. 15 feet 22. 100 accepted 23. 7 10 24. 11 25. 12.247

T-302

Chapter 6

Tests

Answers (cont’d)

Test Form B 2 1. 7 16 2. 31 1 3. 5 6 4. 5 2 printers 5. 5 computers 4 computers 6. 35 students 25 students 7. 1 teacher 5 books 8. 1 student 1 1 1 4 2 = 2 9. 24 72 6 8 = 10. 3 4 11. false 12. true 13. 55 miles/hour 14. 90 calories/ounce 15. 14 16. 1 17. 1 18. 20 19. 10 20. 225 sq ft 21. 36 students 22. 12 completions 23. 5 2 24. 9 25. 10.583

Test Form C 39 1. 196 7 2. 5 1 3. 4 4 4. 1 1 printer 5. 5 students 1 gallon 6. 13 miles 1 hour 7. 60 miles 1 4 2 = 9 8. 36 72 6 36 9. = 1 6 10. true 11. false 12. 120 calories/glass 13. 66 miles/hour 14. 28 15. 7 16. 3 17. 96 18. 35 19. 16 20. 15 free throws 21. 250 miles 22. 24 ounce bag 23. 6 5 24. 7 25. 13.229

Test Form D 9 1. 7 158 2. 75 7 3. 1 73 4. 18 1 tutor 5. 6 students 1 gallon 6. 20 miles 1 person 7. 2 sandwiches 2 14 8. = 3 21 4 8 9. = 12 24 10. false 11. true 12. $1.29/soda 13. $1.39/gallon 14. 32 15. 6 16. 21 17. 66 18. 93 19. 10 20. 40 gallons 21. 175 miles 22. 120 minutes 23. 4 5 24. 7 25. 11.619

T-303

Chapter 6

Tests

Answers (cont’d)

Test Form E 13 1. 5 2 2. 5 3 3. 1 20 4. 31 1 computers 5. 3 students 1 teacher 6. 21 students 65 miles 7. 1 hours 7 ounces 8. 4 cents 28 miles 9. 1 gallon 1 1 3 10. 2 = 24 48 2 4 11. = 8 16 12. true 13. false 14. 25 miles/gallon 15. 35 calories/ounce 16. 30 17. 60 1 18. 2 19. 1.08 20. 4.5 21. 24 completions 22. 427.5 miles 23. 8 4 24. 7 25. 14.832

Test Form F 1. c 2. a 3. a 4. d 5. c 6. b 7. a 8. c 9. b 10. d 11. a 12. b 13. a 14. a 15. b 16. c 17. a 18. a 19. a 20. c 21. b 22. c 23. c 24. a 25. c

Test Form G 1. d 2. c 3. b 4. c 5. b 6. d 7. a 8. a 9. c 10. a 11. b 12. d 13. a 14. d 15. b 16. b 17. c 18. d 19. d 20. b 21. c 22. b 23. a 24. c 25. b

T-304

Chapter 6

Tests

Answers (cont’d)

Test Form H 1. d 2. a 3. a 4. c 5. d 6. d 7. b 8. d 9. b 10. a 11. b 12. a 13. d 14. a 15. b 16. a 17. d 18. c 19. b 20. c 21. c 22. b 23. a 24. b 25. b

Cumulative Test Form A Chapters 5 and 6 1. 65.74 2. −70.6 3. 1.89 4. 53.01 5. −0.6 6. 34.7 7. < 17 8. 500 9. 0.286 10. 12.46 11. −1.85x + 4.3 7 12. 8 13. −2.83 14. −47.91 5 15. 2 1 16. 9 17. 29 km/hr 18. 6 computers/printer 19. yes 20. 2 21. 16 22. 6 23. 27 ft 24. 43.9 miles 25. 5.75 cm

Cumulative Test Form B Chapter 5 and 6 1. c 2. a 3. d 4. c 5. b 6. d 7. b 8. b 9. c 10. d 11. a 12. d 13. a 14. d 15. d 16. b 17. c 18. b 19. b 20. b 21. c 22. d 23. b 24. c 25. a

T-305

Chapter 7

Tests

Answers

Pretest Form A 1 72% 2 0.35 3 123% 1 4 250 5 225% 6 0.12(95) = x 7 0.056x = 372 10.8 12 = 8 x 100 18 x 9 = 72 100 10 60 11 10% 12 112 13 438 seniors 14 $630 15 $2005.15 16 $320 17 $551.91 18 $187.50 19 $270 20 $14,500

Pretest Form B 1. 5002 females 2. 0.87 3. 0.3%

Test Form A 1. 3.29 2. 0.053 3. 0.001 4. 32.1% 5. 390% 6. 73% 18 7. 1 25 29 8. 50 3 9. 1000 10. 37.5% 11. 20% 12. 240% 13. 24 14. 640 15. 12.5% 16. 6.3 17. 400 18. 10% 19. $125.00 20. $560 21. 5% 22. $510 23. $588.61 24. $90 25. $5850

T-306

3 125

5. 6. 7.

62.5% 0.128(160) = x 0.17x = 156 34 17 = 8. x 100 30 x 9. = 90 100 10. 50 11. 3% 12. 8.6 13. $24,300 14. 2100 15. $816.00 16. $200 17. $798.60 18. 29.4% 19. $144 20. $2196

Chapter 7

Tests

Answers (cont’d)

Test Form B 1. 1.93 2. 0.035 3. 0.87 4. 40% 5. 39.7% 6. 320% 19 7. 20

Test Form C 1. 0.38 2. 0.035 3. 0.0009 4. 332% 5. 4% 6. 6.3%

Test Form D 1. 0.12 2. 0.024 3. 0.08 4. 60% 5. 97% 6. 530% 43 7. 50 7 8. 500 7 9. 1 20 10. 60% 11. 20% 12. 31.25% 13. 4.8 14. 2800 15. 12.5% 16. 20 17. 775 18. 6.5% 19. $63 20. $487.50 21. 6% 22. $303.95 23. $2339.72 24. $2380 25. $5700

8. 9.

1 5

1 8

10. 37.5% 11. 375% 12. 55.6% 13. 20.25 14. 1150 15. 37.5% 16. 0.35 17. 1000 18. 30% 19. $52.50 20. $70 21. 6.5% 22. $1620 23. $1559.66 24. $175 25. $3400

7 20

1 400

10. 6.25% 11. 62.5% 12. 125% 13. 16.8 14. 400 15. 15% 16. 6 17. 500 18. 200% 19. $81.25 20. $4.96 21. 4.6% 22. $465.19 23. $1040.82 24. $297.50 25. $1085

T-307

Chapter 7

Tests

Answers (cont’d)

Test Form E 1. 3.9 2. 0.05 3. 0.007 4. 40% 5. 0.3% 6. 73% 1 7. 1 2

Test Form F 1. b 2. a 3. c 4. b 5. c 6. d 7. c 8. d 9. c 10. c 11. c 12. a 13. b 14. d 15. c 16. a 17. d 18. c 19. a 20. c 21. b 22. b 23. a 24. c 25. b

Test Form G 1. c 2. c 3. b 4. c 5. a 6. c 7. d 8. b 9. a 10. b 11. d 12. a 13. d 14. b 15. a 16. c 17. c 18. b 19. a 20. c 21. a 22. c 23. c 24. d 25. b

61 250

1 200 10. 50% 11. 95% 12. 187.5% 13. 3.45 14. 13,700 15. 34% 16. 75.6 17. 15,000 18. 25% 19. $6.64 20. 576 21. 6% 22. $604.44 23. $2662 24. $21,375 25. $23.98

T-308

Chapter 7

Tests

Answers (cont’d)

Test Form H 1. b 2. d 3. d 4. d 5. a 6. b 7. a 8. b 9. b 10. c 11. b 12. d 13. a 14. c 15. c 16. b 17. a 18. b 19. d 20. c 21. a 22. b 23. a 24. b 25. a

T-309

Chapter 8

Tests

Answers

Pretest Form A 1. Kentucky 2. Utah 3. same number of smokers 4. 4 5. May 6. 19 7. Apr and June 8. Pirates of the Caribbean: Dead Man’s Chest 9. $250 million 10. $225 million 11. $300 million 12. yes 13. (5, −5), (−5, −3), (0, −4) 14.

Pretest Form A (cont’d) 1 20. 4

Pretest Form B 1. 2002 2. increased 3. freshmen 4. 8000 5. 10,000 6. yes 7. Jill's Daily Schedule Other 13%

Class time 24%

Driving 4% Sleep 21% Work 17%

Study 21%

8. $200 9. 4 10. $1200 11. yes 12.

15.

13. y

16. y

14. y

1 2 1 18. 6 1 19. 52

17.

T-310

15. (4, 1), (–4, 3), (0, 2) 16. (1, −1), (−1, −3), (0, −2) 1 17. 6

Chapter 8

Tests

Answers (cont’d)

Pretest Form B (cont’d) 1 18. 2 1 19. 13 1 20. 4

Test Form A 1. $3 million 2. year 1 3. $3.5 million 4. (−4, 3) 5. (−3, 0) 6. (1, 1) 7. (3, −2) 8. (0, −3) 9. white 10. 36 11. 13% 12. 30 13. 1250 14 and 15

Test Form A (cont’d) 21. y

1 2 2 23. 3 1 24. 13 1 25. 52

22.

14.

15. x

16. (2, −1), (0, −5) 17. (3, 2), (0, 1) 18. no 19. y

20. y

T-311

Chapter 8

Tests

Answers (cont’d)

Test Form B 1. June 2. $29,000 3. (4, 2) 4. (2, −1) 5. (0, −3) 6. (−2, 0) 7. (−4, 1) 8. horses 9. cats 10. 156 11. 42 12. 114 13 and 14.

Test Form B (cont’d) 21.

Test Form C 1. $800 2. $600 3. $2500 4. (4, 0) 5. (5, −3) 6. (0, −1) 7. (−3, −3) 8. (−4, 0) 9. (1, 3) 10. 175 11. 150 12. 50 13. 100 14 and 15.

13.

14.

1 2 1 23. 3 1 24. 52 1 25. 13

22.

15. 14.

15. (2, −3), (−2, −7) 16. (3, −7), (1, −1) 17. (4, 1), (0, −5) 18. yes 19.

16. (1, −3), (−1, −5) 17. (2, −2), (−2, −10) 18. (3, 1), (−3, −1) 19. no 20.

20. y

21. y

T-312

Chapter 8

Tests

Answers (cont’d)

Test Form C (cont’d) 22.

Test Form D 1. $500 2. $50 3. bracelets 4. (0, 2) 5. (5, 0) 6. (5, −3) 7. (−1, −3) 8. 34% 9. B 10. 7% 11. 18 12, 13, and 14.

Test Form D (cont’d) 1 21. 6 1 22. 13 1 23. 52 1 24. 4 1 25. 12

1 2 1 24. 4 1 25. 13

23.

14.

12.

13.

15. (−5, −3), (10, −6) 16. (3, 0), (0, −2) 17. (1, −1), (−1, −7) 18. yes 19. y

20. y

T-313

Chapter 8

Tests

Answers (cont’d)

Test Form E 1. California 2. Georgia and Indiana 3. 5 4. (3, 0) 5. (0, 2) 6. (−4, 2) 7. (1, −2) 8. books 9. 168 10. 120 11. 72 12. 152 13, 14, and 15

Test Form E (cont’d) 21.

Test Form F 1. b 2. a 3. d 4. d 5. d 6. d 7. a 8. a 9. c 10. a 11. b 12. b 13. a 14. a 15. d 16. c 17. b 18. c 19. c 20. b 21. a 22. d 23. d 24. a 25. b

1 3 1 23. 13 1 24. 52 3 25. 10

22.

14. x

15. 13.

16. (−3, 3), (3,5) 17. (6, 0), (−3, −3) 18. no 19. y

20. y

T-314

Chapter 8

Tests

Answers (cont’d)

Test Form G 1. b 2. b 3. a 4. b 5. a 6. a 7. d 8. b 9. b 10. b 11. a 12. d 13. b 14. b 15. a 16. a 17. c 18. a 19. c 20. a 21. b 22. b 23. b 24. c 25. d

Test Form H 1. d 2. c 3. c 4. b 5. a 6. d 7. a 8. d 9. b 10. c 11. a 12. d 13. b 14. a 15. b 16. a 17. c 18. a 19. a 20. c 21. b 22. a 23. c 24. c 25. d

Cumulative Test Form A Chapters 7 and 8 1 1. 3.125, 3 8 2. 1.2% 173 3. 200 4. 180% 17 5. 25 6. 25% 7. 36 8. 16.7% 9. 110 10. $238 11. $11.13 12. $780 13. $1157.63 14. $206.25 15. $112 16. A(2, 0), B(−2, 1), C(−3, −2) 17. (3, −1), (0, −3), (−3, −5) 18. (1, −2), (0, −3), (−1, −4) 19. y

20. y

21. y

T-315

Chapter 8

Tests

Cumulative Test Form A Chapter 7 and 8 (cont’d) 1 22. 52 1 23. 6 1 24. 4 1 25. 2

Cumulative Test Form B Chapter 7 and 8 1. d 2. d 3. a 4. c 5. c 6. a 7. d 8. d 9. d 10. c 11. b 12. a 13. c 14. b 15. b 16. a 17. d 18. b 19. b 20. a 21. b 22. d 23. d 24. c 25. c

T-316

Answers (cont’d)

Chapter 9

Tests

Answers

Pretest Form A 1. 43° 2. 53° 3. 55° 4. x = 140°, y = 40°, z = 140° 5. P = 16.2 m, A = 14 sq m 6. P = 12 ft, A = 9 sq ft 7. P = 24 in., A = 24 in 2 8. C = 41 ÷ 12.6 m, A = 41 ÷ 12.6 sq m

Pretest Form B 1. 22° 2. 80° 3. 150° 4. x = 150°, y = 30°, z = 150° 5. P = 20 yd, A = 18.8. sq yd 6. P = 4 m, A = 1 sq m 7. P = 12 cm, A = 6 cm 2 8. C = 16 1 ÷ 50.3 ft, A = 641 ÷ 201.0 sq ft 9. 480 cu in 10. 27 feet 11. 3 gal 12. 8250 g 13. 4 m 8 cm 14. 1 qt 1 pt 15. 6 gallons 16. 0.135 kg 17. 29.4° C 18. 60.8° F 19. 18 bows 20. 40 ft

Test Form A 1. 11° 2. 145° 3. 38° 4. x = 130°m, y = 50°, z = 130° 5. 26° 6. P = 19.2 m, A = 17.28 sq m 7. C = 6π ≈ 18.84 sq. in. 8. P = 48 in., A = 102 sq. in. 9. 125,000 mg 10. 6 qt 11. 86,000 ml 12. 2 m 13. P = 38 in, A = 86 sq in 14. 6 ft 2 in 15. 22. 5 sq ft 16. 251 ÷ 78.5 sq m 17. 9 cu ft 18. 56.5 cu ft 19. 127.4° F 20. 3.3° C 21. 136 in 22. 110 kilometers 23. 20 yd 24. acute 25. obtuse

1 4

cu. ft.

10. 40 feet 11. 10 qt 12. 3600 g 13. 9 gal 14. 3 gal 1 qt 15. 0.006 km 16. 32 pints 17. 34.4° C 18. 78.8° F 19. 1 ft 4 in 20. 12 sq ft

T-317

Chapter 9

Tests

Answers (cont’d)

Test Form B 1. 17° 2. 91° 3. 21° 4. x = 150°, y = 30° 5. 45° 6. x = 128°, y = 52°, z = 128° 7. P = 18.4 cm, A = 21.16 sq. cm. 8. P = 17 ft, A = 14 sq ft 9. C = 10π ≈ 31.4 m 10. 8250 mg 11. 16,000 meters 12. 0.4 l 13. 13 yd 14. 0.004 km 15. 10.8 cm 16. 2.9 17. 5.8 cu m 18. 1695.6 cu ft 19. 96.8° F 20. 34.4° C 21. 24 sq. in. 22. 7 gal 3 qt 23. 32 yd 24. false 25. true

Test Form C 1. 15° 2. 55° 3. 35° 4. x = 152°, y = 28° 5. 70° 6. x = 60 °, y = 120°, z = 60° 7. P = 22 ft, A = 24 sq ft 8. P = 10.2 m, A = 5 sq m 9. C = 5π ≈ 15.7 ft. 10. 2100 mg 11. 400 l 12. 5 gallons 13. 0.2 m 14. 2000 mm 15. 16 sq cm 16. 41 ÷ 12.6 sq ft 17. 0.045 cu m 18. 6.76π ≈ 21.2 cu. in. 19. 82.4° 20. 23.3° 21. 540 sq. in. 22. 144 km 23. 20 yd 24. right angle 25. obtuse

Test Form D 1. 157° 2. 10° 3. 55° 4. x = 142°, y = 38° 5. 30° 6. x = 138°, y = 42°, z = 138° 7. P = 30 cm, A = 56.3 sq cm 8. P = 18.8 m, A = 11.2 sq m 9. 6π ≈ 18.8 sq m 10. 0.00802 kg 11. 4900 m 12. 14 qt 13. 27 ft 14. 11,000 lbs. 15. 31.5 sq ft 16. 2.251 ÷ 7.1 sq m 17. 0.135 cu cm 18. 314 ft 19. 195.8° 20. 23.3° 21. 120 in. 22. 225 km 23. 24 bags 24. true 25. false

T-318

Chapter 9

Tests

Answers (cont’d)

Test Form E 1. 67° 2. 93° 3. 20° 4. x = 63°, y = 117° 5. 56° 6. P = 14.4 cm, A = 13.0 sq cm 7. 3π ≈ 9.4 m 8. P = 18 ft, A = 14 sq ft 9. P = 54.4 m, A = 183 sq. m 10. 1060 ml 11. 16 pints 12. 8650 l 13. 160 cm 14. 7 yd 2 ft 15. 12 m 25 cm 16. 20 sq m 17. 36π ≈ 113 sq. cm 18. 54.9 cu cm 19. 226.1 cu in 20. 21.1° 21. 89.6° 22. 16 bows 23. 90° 24. less than 90° 25. 224 sq ft

Test Form F 1. a 2. a 3. b 4. a 5. d 6. a 7. c 8. d 9. b 10. c 11. d 12. a 13. d 14. b 15. a 16. b 17. c 18. d 19. d 20. a 21. b 22. d 23. a 24. b 25. c

Test Form G 1. d 2. c 3. d 4. b 5. a 6. c 7. a 8. c 9. d 10. b 11. b 12. c 13. c 14. a 15. d 16. c 17. c 18. b 19. b 20. b 21. b 22. b 23. a 24. b 25. d

T-319

Chapter 9

Tests

Test Form H 1. d 2. c 3. a 4. b 5. b 6. c 7. b 8. a 9. d 10. a 11. c 12. a 13. d 14. b 15. a 16. d 17. d 18. a 19. a 20. c 21. a 22. a 23. a 24. b 25. b

T-320

Answers (cont’d)

Chapter 10

Tests

Answers

Pretest Form A 1. 14 y 2 − 10 y

Pretest Form B 1. 3x 2 + 2 x − 5 2. 12 3. 7 x 2 − 11x + 7 4. −2x − 4 5. 27 6. x9 7. 6x8 8. x32 9. −104w8 10. 16x12

Test Form A 1. y 2 − 3 y − 11

2. 3. 4. 5. 6. 7. 8. 9.

−3b 2 + 13b − 4 4 z3 − 8z + 7 7 a11 24x 6 z 27 81x8

4x13 y8

10. −6 x5 + 6 x3 − 15 x 11. x 2 + 5 x − 24 12. x 2 + 8 x + 16 13. z 2 − 36 14. a3 − 7a + 6 15. 6 x 2 − 8 x − 4 16. −108x5 17. 12 18. 2m 2

(

19. 7 y 2 + 2 y − 4

(

)

2 6 4

11. 4x y z

4.3x 2 + 6 x − 5.2 x2 + 8x + 4 −9x + 12 15x + 14 11 −9 y6

10. y 21 11. 243x5 12. 32x 2 y10

13. −27x6 y 3

13. 6 x + 7 x − 20 14. x 2 − 14 x + 49 15. −4 x 2 + 16 x − 12 16. −8x11 17. 5 18. 3z 2

( ) 20. 2 ( 2 y − y + 8 ) 19. 3 2 x 2 − 4 x + 5

)

x2 − 7 x + 5

8 7

12. 8x y

20. 3x x3 − 3x 2 + 2 x − 1

2. 3. 4. 5. 6. 7. 8. 9.

14. x10 15. 108a 7 b8 16. x14 17. −2 z 3 − 6 z 2 + 10 z 18. 6 x 2 − 5 x − 4 19. x 2 − 16 20. x 2 + 2 x − 35 21. a3 + 3a 2 − 4 22. −3x 2 + 6 x 23. 2 x 2 − 10 x 24. 15

(

25. 2 x x 2 + 2 x − 3

)

T-321

Chapter 10

Tests

Answers (cont’d)

Test Form B 1. 3x 2 − 5 x − 2 2. − x 2 − 3x + 2 3. 3x 2 + x + 0.2 4. 6 x − 10 5. 2 x 2 + x + 4 6. 12x + 7 7. 1 8. 18 9. x6

Test Form C 1. 5x − 2 2. 3 3. 2 z 2 − 0.2 z − 1.3 4. 5 x 2 + 3 x − 8 5. 2 x 2 − 7 x + 9 6. −x − 13 7. 22 8. 9 9. x8 10. x 48 11. 64x 3

Test Form D 1. x 2 − 3x − 3 2. 6 x 2 − 13 x − 4 3. 8x − 15 4. x 2 + 6 x + 6 5. −4 x 2 + 15 x + 3 6. x 2 + x + 5 7. −23 8. 9 9. z10 10. y 21

10. y 35 11. 27x 6

12. 4x8 y 2

2 4 6

12. 16x y z

15. y

16. −3 x 2 + 6 x − 15 17. x 2 + 18 x + 81 18. x 2 − 25 19. 6 x 2 + 13x − 5 20. a3 + 3a 2 + 3a + 1 21. −5 x3 + 10 x 2 − 15 x 22. 6 x 2 − 9 x 23. 3 25. 4 x

T-322

)

( x − 2 x + 3) 2

14. 3x5 y 6

15. y 21

(

13. x 4 y 5

14. 12x3 y 6

14. −12x 4 y5 z 4

24. 3 x 2 − 2 x + 3

12. 16x2 y8

13. 16x 20

13. −27x3 y 3

16. −3x 2 − 9 x + 18 17. x 2 − 10 x + 25 18. x 2 − x − 20 19. 6 x 2 − 10 x − 4 20. x3 − 9 x + 10 21. − x 2 + 5 x + 8 22. 3.45 x 2 + 3x 23. 2y

(

24. 3 x 2 − 4 x + 6 25. 5 x

11. −27x12

)

( x + 2 x − 3) 2

15. 648x11 16. −2 x 2 − 12 x − 18 17. x 2 + 4 x + 4 18. 8 x 2 + 14 x − 15 19. x 2 − 100 20. 2 y 3 − 3 y 2 − 13 y − 8 21. −60x + 30 22. 2 x 2 + 4 x 23. 3z

(

24. 4 x 2 2 x 4 − x 2 − 4 25. 2 xy ( x − 2 y )

)

Chapter 10

Tests

Answers (cont’d)

Test Form E 1. 7x − 12 2. 6 z 2 − 2 z − 5 3. 9y + 10 4. −7x + 17 5. −9 y 2 − 3 y − 17

Test Form F 1. a 2. d 3. a 4. d 5. c 6. a 7. d 8. a 9. d 10. d 11. d 12. b 13. b 14. d 15. a 16. c 17. a 18. d 19. c 20. d 21. b 22. a 23. d 24. d 25. b

Test Form G 1. b 2. d 3. a 4. b 5. a 6. b 7. c 8. b 9. a 10. d 11. b 12. d 13. b 14. a 15. c 16. b 17. a 18. d 19. a 20. a 21. b 22. a 23. b 24. c 25. b

6. −5x − 1 7. −43 8. 30 9. a14 10. a15b5 11. b40 12. x8 y13 13. p 21 14. 6x8 15. −64x6 16. −3x 2 − 6 x + 27 17. −5 x 2 − 35 x 18. 35 x 2 − 11x − 6 19. 15 x 2 + 7 x − 2 20. z 2 − 6 z + 9 21. a3 + 4a 2 − a − 10 22. 9 x 2 + 6 x + 1 23. 5x

(

24. 5 x 3x 2 − x + 2

(

)

25. 3 x3 y 2 x 2 + 3xy − 4 y 3

)

T-323

Chapter 10

Tests

Test Form H 1. b 2. d 3. a 4. b 5. d 6. b 7. c 8. b 9. d 10. c 11. b 12. d 13. b 14. c 15. a 16. a 17. b 18. b 19. b 20. a 21. b 22. a 23. d 24. c 25. b

T-324

Answers (cont’d)

Final

Exams

Test Form A 1. 247 2. 9545 3. 10,507 4. 11,000 5. −11 6. −33 7. −12 8. 7 9. 0 10. 1 11. (7x + 6) feet 12. 3 13. 10 14. 5x − 17

Test Form A (cont’d) 37. $206.25 38.

3 15. 10 1 16. 1 10 1 17. −1 3 2 18. 3 17 19. − 15

Answers

39. 74, 78.5, 75 40.

1 2

41. 39° 42. 31.4 feet 43. 270 44. 10 45. −4x − 3 46. −17 47. y14 48. 81x8 49. −2 x3 + 2 x 2 − 2 x 50. 3y

20. −4 21. 5 22.

2 ft 3

8 sq yd 9

23. 21.66 24. 39.308 25. − 0.28 26. 1.4x − 1.6 27. 9 28. 19.7 mpg 29. 2 30. 0.057 31. 0.036 32.

Test Form B 1. 223 2. 23,782 3. 247 4. 576 5. −18 6. −33 7. −45 8. 3 9. 16 10. −11 11. 7x − 9 12. −10x + 20 13. (12x + 6) yd 14. 31 15. 6 16. 2x − 7 17.

2 3

1 6 11 19. 16 3 20. 2 11 21. 21 4 22. 5

18. −

23. 4 feet 24. 4

1 feet 2

25. 31.21 26. 11.75 27. 10.6x + 32.5 28.

6 7

29. 3 30. 20 31. 1.294 32. 10.7%

1 4

33. 87.5% 34. 50% 35. 14,000 36. $100

33.

12 25

34. 37.5%

T-325

Final

Exams

Answers (cont’d)

Test Form B (cont’d) 35. 60 36. 33.3% 37. 300 38.

Test Form C 1. 665 2. 9 3. 19,448 4. 85 5. 108 6. −12 7. −33 8. 20 9. −8 10. 24 11. −20x + 35 12. 4 13. −30 14. x − 12

Test Form C (cont’d) 35.

39. 48.6, 44, no mode 1 40. 52

41. 99° 42. 7.9 yd 43. 32,400 44. 300 45. 1

3 gal 4

46. −2x − 1 47. y6 48. x 21 49. −6 x3 + 10 x 2 − 14 x 50. 5x 2

4 9 13 16. 15 4 17. 3 1 18. 24 2 19. 1 3

15.

20. 16 bows 21. 12

1 sq yd 2

22. 3.98 23. 69.9 24. 0.4 25. 9.8x − 5.8 26.

7 10

27. 20 28. 2 29. 57% 30.

7 20

31. 42.9% 32. 122.4 33. 300 34. $225

T-326

36. 40.4, 18, 12 37.

2 3

38. 76° 39. 20 m 40. 12 cu ft 41. 15.7 ft 42. 2300 43. 0.3825 44. 2 45. 4 x 2 − 11x + 14 46. −2 47. x9 48. 32x 20 49. − x3 + 3x 2 − 8 x 50. 11xy (3x + 1 + 6 y )

Final

Exams

Answers (cont’d)

Test Form D 1. 1123 2. 98 3. 1344 4. 170 5. 3 6. −39 7. −15 8. 10 9. 2 10. 15x − 20 11. 12x ft 12. −3 13. 12 14. 2x − 17

Test Form D (cont’d) 36.

Test Form E 1. d 2. b 3. d 4. c 5. a 6. c 7. c 8. d 9. a 10. c 11. a 12. b 13. a 14. b 15. c 16. a 17. c 18. c 19. a 20. b 21. b 22. a 23. d 24. a 25. c 26. c 27. c 28. c 29. c 30. d 31. b 32. c 33. a 34. b 35. b 36. d 37. c 38. c 39. b 40. c 41. b 42. d 43. c 44. b

2 5 5 16. 12 2 17. 9 3 18. 2

15.

19. 9 20. −

1 12

37. 5.8, 5.5, no mode 38.

1 2

39. 115° 40. 7.44 sq m 41. 12.2 cu cm 42. 62.8 ft 43. 9340 44. 0.3875 45. 10 46. x 2 − 2 47. −21 48. x9 49. −72x7 50. 5 x ( y 2 + 3 + 5 xy 2 )

21. 60 bows 22. 9.12 23. 87.78 24. 230 25. 3.9x − 3.8 26.

7 9

27. 6 28. 15 29. 625 mi 30. 13.5% 31.

3 10

32. 28.6% 33. 5% 34. 600 35. $55

T-327

Final

Exams

Answers (cont’d)

Test Form E (cont’d) 45. a 46. a 47. a 48. c 49. b 50. b

Test Form F 1. a 2. d 3. b 4. a 5. b 6. b 7. a 8. b 9. c 10. a 11. a 12. c 13. d 14. b 15. b 16. d 17. b 18. d 19. c 20. b 21. a 22. b 23. c 24. d 25. b 26. a 27. d 28. d 29. b 30. d 31. b 32. c 33. d 34. c 35. c 36. c 37. a 38. d 39. b 40. b 41. d 42. b 43. b 44. c

Test Form F (cont’d) 45. c 46. d 47. d 48. a 49. b 50. b

T-328

Final

Exams

Answers (cont’d)

Test Form G 1. b 2. d 3. b 4. d 5. c 6. c 7. b 8. b 9. a 10. d 11. b 12. a 13. d 14. c 15. c 16. b 17. b 18. d 19. b 20. a 21. b 22. b 23. c 24. d 25. c 26. b 27. a 28. d 29. d 30. c 31. c 32. c 33. a 34. d 35. a 36. c 37. d 38. b 39. d 40. b 41. a 42. d 43. b 44. c

Test Form G (cont’d) 45. a 46. a 47. d 48. a 49. c 50. a

Test Form H 1. b 2. a 3. d 4. d 5. a 6. b 7. a 8. a 9. a 10. c 11. c 12. c 13. d 14. b 15. b 16. c 17. c 18. a 19. b 20. c 21. b 22. b 23. a 24. c 25. c 26. d 27. b 28. d 29. c 30. d 31. c 32. d 33. a 34. c 35. c 36. a 37. c 38. d 39. b 40. c 41. c 42. b 43. c 44. b

T-329

Final

Exams

Test Form H (cont’d) 45. b 46. b 47. c 48. a 49. b 50. c

T-330

Answers (cont’d)