Chapter 1 Fractions As part of your California Mathematics Review, you must understand how to add, subtract, multiply, and divide fractions. Answers for fraction problems will be given in a simplified form. In this chapter, you will review and practice all of the skills needed to add, subtract, multiply, and divide fractions as well as to simplify them.
1.1
Simplifying Improper Fractions
28 = 28 ÷ 3 = 9 remainder 1 3 The quotient, 9, becomes the whole number portion of the mixed number. & 1 28 = 9 Á The remainder, 1, becomes the top number of the fraction. 3 3 % % The bottom number of the fraction always remains the same. 7 Simplify . 5 7 is the same as 7 ÷ 5. 7 ÷ 5 = 1 with a remainder of 2. 5 Rewrite as a whole number with a fraction. 1 25 Simplify
Example 1:
Example 2: Step 1: Step 2:
Simplify the following improper fractions. 8 = ___ 6 2 2. = ___ 1 30 3. = ___ 4
13 = ___ 5 18 5. = ___ 7 12 6. = ___ 11
1.
4.
21 = ___ 3 10 8. = ___ 9 17 9. = ___ 3
12 = ___ 7 27 11. = ___ 6 17 12. = ___ 2
7.
10.
15 = ___ 4 41 14. = ___ 9 11 15. = ___ 4
8 = ___ 5 3 17. = ___ 2 7 18. = ___ 2
13.
16.
Fractions that have the same denominator (bottom number) can be added quickly. Add the numerators (top numbers) and keep the bottom number the same. Simplify the answer. The first one is done for you. 19.
3 7
+ 37 +
5 7
20.
3 5
+ 25 +
1 5
21.
6 10
+
9 10
+
=
7 10
11 7
= 1 47
22.
4 9
+ 79 +
23.
6 12
24.
1 8
+
8 12
3 9
+
+ 58 +
7 8
1 12
25.
2 3
26.
2 15
+
9 15
+
11 15
27.
4 11
+
3 11
+
6 11
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+ 13 +
1 3
15
Chapter 1 Fractions
1.2 Example 3: Step 1: Step 2: Step 3:
Changing Mixed Numbers to Improper Fractions
1 Change 3 to an improper fraction. 3 Multiply the whole number (3) by the bottom number of the fraction (3). 3Ă—3=9 Add the top number to the product from Step 1. 9 + 1 = 10 Put the answer over the bottom number (3).
2. Add this number. 3. Put the answer here. & . 1 10 3 = 3 3 % % -4. This number stays the same. 1. Multiply these two numbers. Change the following mixed numbers to improper fractions. 2 1. 4 =_______ 7
2. 1
3 5. 8 =_______ 5
1 1 3 9. 2 =_______ 13. 10 =_______ 17. 6 =_______ 9 2 5
9 =_______ 6. 2 2 =_______ 10. 3 1 =_______ 14. 4 7 =_______ 18. 7 4 =_______ 13 3 6 8 5
4 3. 5 =_______ 5
1 3 3 1 7. 1 =_______ 11. 5 =_______ 15. 1 =_______ 19. 2 =_______ 4 7 4 5
1 4. 9 =_______ 3
7 1 2 3 8. 10 =_______ 12. 1 =_______ 16. 3 =_______ 20. 5 =_______ 8 3 8 8
Whole numbers become improper fractions when you put them over 1. The first one is done for you. 21. 5 =
5 1
22. 7 =_______
16
23. 2 =_______
25. 9 =_______
24. 17 =_______ 26. 4 =_______
27. 10 =_______ 29. 12 =_______ 28. 6 =_______
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30. 3 =_______
1.3 Greatest Common Factor
1.3
Greatest Common Factor
To reduce fractions to their simplest form, you must be able to find the greatest common factor.
Example 4:
Find the greatest common factor (GCF) of 35 and 28.
To find the greatest common factor (GCF) of two numbers, first list the factors of each number. The factors of 35 are: 1, 5, 7, and 35. The factors of 28 are: 1, 2, 4, 7, 14, and 28. What is the largest number they both have in common? 7 7 is the greatest (largest number) common factor.
Find all the factors and the greatest common factor (GCF) of each pair of numbers below.
1. 2. 3. 4. 5. 6. 7. 8. 9.
Pairs 12 27 36 30 12 9 35 28 50 10 16 24 54 6 16 12 28 36
Factors
GCF 10. 11. 12. 13. 14. 15. 16. 17. 18.
Pairs 27 39 32 12 15 30 14 16 22 55 15 25 32 16 18 24 68 51
Factors
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GCF
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Chapter 1 Fractions
1.4 Example 5:
Step 1:
Reduce
Reducing Proper Fractions
8 to lowest terms. 12
First you need to find the greatest common factor of 8 and 12. Think: What is the largest number that can be divided into 4 and 8 without a remainder? These must be the same number.
?8 ? 12 8 and 12 can both be divided by 4. Step 2:
Divide the top and bottom of the fraction by the same number. 2 8 2 8÷4 = Therefore, = . 12 ÷ 4 3 12 3
Reduce the following fraction to lowest terms.
1.
10 = 20
7.
12 = 16
13.
4 = 16
19.
3 = 21
25.
60 = 64
2.
9 = 27
8.
28 = 36
14.
42 = 63
20.
21 = 56
26.
4 = 18
3.
6 = 54
9.
20 = 35
15.
2 = 16
21.
34 = 42
27.
4.
12 = 24
10.
45 = 54
16.
4 = 6
22.
20 = 25
5.
12 = 36
11.
45 = 50
17.
3 = 12
23.
5 = 15
29.
30 = 39
6.
28 = 35
12.
21 = 28
18.
6 = 66
24.
7 = 63
30.
2 = 8
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51 = 105 3 28. = 9
1.5 Multiplying Fractions
1.5 Example 6:
Multiplying Fractions
8 3 Multiply 4 × 8 10 8 35 × 8 10
Step 1:
Change the mixed numbers in the problem to improper fractions.
Step 2:
When multiplying fractions, you can cancel and simplify terms that have a common factor. The 8 in the first fraction will cancel with the 8 in the second fraction. 35 8¢ × 8¢ 10 The terms 35 and 10 are both divisible by 5, so 35 simplifies to 7, and 10 simplifies to 2.
½ 1 7 ½ 35 × ½ 2 1 10 ½ 1 7 1 7 × = =3 1 2 2 2
Step 3:
Multiply the simplified fractions.
Step 4:
You cannot leave an improper fraction as the answer. You must change it to a mixed number.
Multiply and reduce your answers to lowest terms. 1 2 1. 4 × 2 2 3 2.
1 4 ×6 5 8
2 1 3. 3 × 2 9 7 4.
2 6 ×1 7 9
5. 3 × 2
1 4
3 1 6. 1 × 1 8 4 7. 4 ×
6 13
1 3 8. 5 × 3 5 7
7 1 9. 2 × 1 5 8 10.
3 4 ×6 7 4
1 17. 4 × 4 8 18.
2 1 ×3 5 4
4 4 11. 2 × 2 5 7
1 1 19. 5 × 7 9 2
1 5 12. 3 × 3 6
1 4 20. 6 × 2 5 2
13. 1
3 1 ×7 14 5
1 1 21. 4 × 1 3 5
1 3 14. 2 × 1 4 2
2 1 22. 2 × 4 5 7
1 1 15. 2 × 3 2 4
7 3 23. 1 × 1 4 9
9 2 16. 6 × 3 10
1 2 24. 3 × 1 5 4
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Chapter 1 Fractions
1.6 Example 7:
Dividing Fractions
5 3 1 ÷2 4 8
Step 1:
Change the mixed numbers in the problem to improper fractions.
Step 2:
Invert (turn upside down) the second fraction and multiply.
Step 3:
Cancel where possible and multiply.
7 8 × 4 21
1 7¢ 8¢ 2 2 × = ½ 3 21 1 4¢ ½ 3
Divide and reduce answers to lowest terms. 1 3 1. 8 ÷ 4 8
9.
3 ÷2 4
1 3 17. 1 ÷ 2 4
1 7
5 1 10. 5 ÷ 1 9 3
1 2 18. 11 ÷ 4 3 6
1 5 3. 2 ÷ 2 8
1 1 11. 2 ÷ 7 4 4
2 3 19. 7 ÷ 5 5
4 5 4. 7 ÷ 7 9 9
3 1 12. 9 ÷ 4 5 5
5 1 20. 6 ÷ 3 8 8
2. 4 ÷ 1
5. 6 ÷
3 4
13. 10
1 1 ÷5 12 2
3 1 21. 2 ÷ 1 2 8
1 1 6. 7 ÷ 4 2 8
3 4 14. 1 ÷ 6 5 10
1 2 22. 6 ÷ 2 9 3
2 1 7. 2 ÷ 3 5 5
1 15. 4 ÷ 2 2
1 1 23. 3 ÷ 2 2 4
2 1 8. 5 ÷ 2 3 9
1 7 16. 3 ÷ 4 12
8 4 24. 4 ÷ 7 14
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7 21 ÷ . 4 8
1.7 Finding Numerators
1.7
REMEMBER:
Example 8:
Finding Numerators
Any fraction that has the same non-zero numerator (top numbers) and denominator (bottom number) equals 1. 7 =1 7
3 =1 3
15 =1 15
32 =1 32
4 =1 4
Any fraction multiplied by 1 in any fraction form remains equal. Example 9:
PROBLEM:
5 3 15 5 15 × = so = 8 3 24 8 24 3 = 7 21
Find the missing numerator (top number)
Step 1:
Ask yourself, "What was 7 multiplied by to get 21?"
3 is the answer.
Step 2:
The only way to keep the fraction equal is to multiply the top and bottom number by the same number. The bottom number was multiplied by 3. so multiply the top number by 3, as shown below. 9 3 3 × = 7 3 21
Find the missing numerators from the following equivalent fractions. 1.
1 = 3 27
6.
2 = 5 25
11.
7 = 8 32
16.
2 = 9 18
21.
3 = 10 50
26.
7 = 15 30
2.
9 = 11 77
7.
3 = 8 72
12.
2 = 7 35
17.
11 = 12 24
22.
3 = 5 45
27.
6 = 7 21
3.
6 = 15 45
8.
4 = 17 34
13.
4 = 7 35
18.
5 = 8 32
23.
4 = 13 26
28.
2 = 5 45
4.
9 = 10 40
9.
9 = 14 28
14.
1 = 3 9
19.
6 = 11 44
24.
6 = 7 28
29.
5 = 9 45
5.
3 = 11 88
10.
21 = 22 44
15.
10 = 13 65
20.
1 = 7 63
25.
4 = 9 81
30.
1 = 2 8
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Chapter 1 Fractions
1.8
Least Common Multiple
Find the least common multiple (LCM) of 12 and 14. To find the least common multiple (LCM), of two numbers, first list the multiples of each number. The multiples of a number are 1 times the number, 2 times the number, 3 times the number, and so on. The multiples of 12 are: 12> 24> 36> 48> 60> 72> 84=== The multiples of 14 are: 14> 28> 42> 56> 70> 84=== What is the smallest multiple they both have in common? 84 84 is the least (smallest number) common multiple of 12 and 14.
Find the least common multiple (LCM) of each pair of numbers below.
1. 2. 3. 4. 5. 6. 7. 8. 9.
22
Pairs Multiples LCM Pairs 5 5> 10> 15> 20> 25> 30> 35 35 10. 6 7 7> 14> 21> 28> 35 18 3 11. 9 4 6 15 12. 5 9 10 12 13. 6 9 8 14 14. 2 6 4 5 15. 3 12 9 7 16. 6 3 12 12 17. 5 8 8 9 18. 6 5 15
Multiples
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LCM
1.9 Adding Fractions
1.9 Example 10:
Adding Fractions
2 1 Add 3 + 2 2 3
Step 1:
Rewrite the problem vertically, and find a common denominator. Think: What is the smallest number I can divide 2 and 3 into without a remainder? 6, of course. 1 3 = 2 6 2 = +2 3 6
Step 2:
To find the numerator for the top fraction, think: What do I multiply 2 by to get 6? You must multiply the top and bottom numbers of the fraction by 3 to keep the fraction equal. For the bottom fraction, multiply the top and bottom number by 2. Add whole numbers and fractions, and simplify. 1 3 3 = 3 2 6 4 2 = 2 +2 3 6
Step 3:
= 5 Add and simplify the answers. 1 1 3 4. 3 7. 7 1. 2 4 7 8 1 3 1 +1 +2 +5 8 2 5 3 4 7 + 8
2. 2
3.
2 7 4 + 5
2 3 1 +1 2
5. 10
1 8 2 + 6
6. 9
8.
9.
7 8 1 +3 9 4
5 6 7 +3 10 8
5 8 3 + 4
10. 1
5 12 4 +1 9
11. 2
12.
2 7 2 + 5
13.
14.
15.
7 6
= 6
1 9 3 +3 4 2
7 8 3 +3 12 1
1 4 2 +3 3 7
1 6 4 5 3 +4 7
16. 4
17.
7 12 2 +4 3 7 8 1 +1 9
18. 5
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9 10 7 +2 10
19. 6
20.
21.
1 9 3 +7 4 1
1 2 1 +3 7 2
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Chapter 1 Fractions
1.10 Example 11:
Subtracting Mixed Numbers from Whole Numbers Subtract 15 3
3 4 15
Step 1: Step 2:
3 4 You cannot subtract three-fourths from nothing. You must borrow 1¶ from 15. 4 4 = 1 , you will You will need to put the 1 in the fraction form. If you use 4 4 be ready to subtract.
Rewrite the problem vertically.
3
4 4 4 3 3 4 1 11 4 15¢
Subtract. 1.
7 3 2 5
2.
24 1 12 2
3.
13 2 11 3
4.
28 5 21 8
5. 12 1 6 8
24
6.
8
11. 18
3 6 4 7. 22
12.
1 9 2 8. 14
13.
1 7 5 9.
15
35 7 22 9
17. 10
3 1 1 3
18.
5 1 2 8
21. 2 22.
4 8 5 6
19. 2
23.
37 2 24 7
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4 1 2
6 5
1 9
24. 12
3 4 20.
5 9
2
1 5 7
3 4 8 15.
15 15 1 16
40 11 36 13
14. 12
2 12 9 10.
16.
5 7 6
11 25.
1 9
9 7
1 8
1.11 Subtracting Mixed Numbers with Borrowing
1.11 Example 12: Step 1:
Step 2:
Step 3:
Subtracting Mixed Numbers with Borrowing
5 1 Subtract 7 5 4 6 Rewrite the problem and find a common denominator. 3 1 ×3 7 7 4 ×3 12 10 5 ×2 5 5 6 ×2 12 You cannot subtract 10 from 3. You must borrow 1 from the 7. The 1 will be in 3 12 which you must add to the you already have, making the fraction form 12 12 15 . 12 Subtract whole numbers and fractions, and simplify. 6 15 3¢ 7¢ 12 10 5 12 5 1 12
Subtract and simplify. 1.
2.
3.
4.
1 8 3 3 4 5
2 7 1 2 2 8
11 12 1 3 3 4
4 6 2 4 3 7
3 10 1 7 2
5. 15
6.
7.
8.
1 5 2 1 15 3
5 12 1 12 6 16
2 7 4 6 5 20
9.
10.
11.
12.
4 9 1 3 3 7
4 5 1 3 10 8
2 3 5 17 6 18
1 2 7 1 8 11
1 2 2 2 3
13. 14
5 6 1 3 3
14. 11
15.
16.
11 13 2 8 13 9
3 8 1 1 4 6
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17.
18.
19.
20.
2 5 9 7 10 8
1 6 4 1 5 9
2 5 3 5 5 7
3 4 1 11 3 15
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Chapter 1 Fractions
1.12
Deduction - Fractions Off
Sometimes sale prices are advertised as 14 off or multiply the original price by the fraction off. Example 13:
1 3
off. To find out how much you will save, just
CD players are on sale for 13 off. How much can you save on a $240 CD player? 80 © 240 1 © × = 80 You can save $80=00. 3¢ 1 1
Find the amount of savings in the problems below. J.P. Nichols is having a liquidation sale on all furniture. Sale prices are How much can you save on the following furniture items?
Liquidation Furniture Sale Item
1 4
off the regular price.
off all items in the store
Regular Price
Savings
1. Couch
$750
________
2. Loveseat
$499
________
3. Recliner
$682 2
________
4. Dining Room Set
$12799
________
5. Bedroom Set
$2544
________
Buy Rite Computer Store is having a 12 off sale on selected computer items in the store. How much can you save on the following items?
Buy Rite Computer Store SALE: off selected items in the store Item
Regular Price
Savings
6.
Midline Computer
$1482
________
7.
Notebook Computer
$3600
________
8.
Tape Backup Drive
$304
________
9.
Laser Printer
$755
________
Digital Camera
$624
________
10.
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1.13 Fraction Word Problems
1.13
Fraction Word Problems
Solve and reduce answers to lowest terms. 1. Sara works for a movie theater and sells candy by the pound. Her first customer bought 2 18 pounds of candy, the second bought 45 of a pound, and the third bought 12 pound. How many pounds did she sell to the first three customers?
7. In January, Jeff filled his car with 12 18 gallons of gas the first week, 10 34 gallons the second week, 11 12 gallons the third week, and 10 12 gallons the fourth week. How many gallons of gas did he buy in January?
2. Beth has a bread machine that makes a loaf of bread that weighs 2 12 pounds. If she makes a loaf of bread for each of her three sisters, how many pounds of bread will she make?
3. A farmer hauled in 81 bales of hay. Each of his cows ate 34 bales. How many cows did the farmer feed?
4. Juan was competing in a 1000 meter race. He had to pull out of the race after running of it. How many meters did he run?
6. A chemical plant takes in 6 13 million gallons of water from a local river and discharges 7 2 12 million back into the river. How much water does not go back into the river?
7 8
8. Li Tun makes sandwiches for his family. He has 9 34 ounces of sandwich meat. If he divides the meat equally to make 4 12 sandwiches, how much meat will each sandwich have?
9. The company water cooler started with 5 38 gallons of water. Employees drank 2 14 gallons. How many gallons were left in the cooler?
5. Tad needs to measure where the free throw line should be in front of his basketball goal. 10. Rita bought 13 pound hamburger patties for He knows his feet are 1 14 feet long and the her family reunion picnic. She bought 40 free-throw line should be 15 feet from the patties. How many pounds of hamburger did backboard. How many toe-to-toe steps does she buy? Tad need to take to mark off 15 feet?
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Chapter 1 Fractions
Chapter 1 Review Simplify. 1.
13 4
2.
9 2
3.
19 6
4.
7 3
Multiply and simplify. 21. 2 12 × 4 35 22.
2 9
× 3 16
23. 1 15 × 2 11 12 24. 1 45 × 4 29 Divide and simplify. 25. 5 59 ÷ 1 13
Reduce. 5.
3 12
26.
6.
8 24
27. 1 12 ÷
7.
14 35
28. 11 23 ÷ 4 16
8.
35 45
Change to an improper fraction. 9. 6 23
+
15. 6 12 + 2 5
+
35. 3 and 9 36. 12 and 8
5 6
3 5
Subtract and simplify. 17. 7 2 15 18. 2 13
1 6
20. 3 34
1 4
19. 10 12 7 23
28
Find the least common multiple for the following sets of numbers. 33. 3 and 8 34. 25 and 10
1 8
14. 8 34 + 1 29 16.
Find the greatest common factor for the following sets of numbers. 29. 16 and 14
32. 48 and 30
Perform the appropriate operation and simplify. 4 8
3 4
31. 28 and 36
5 19
12. 10 27
13.
÷2
30. 64 and 56
10. 4 11.
3 4
37. Which of the following can be used to compute 14 + 15 ? 1+1 (A) 4+5 1×5 1×4 + (B) 4×5 5×4 1 1 + (C) 4×5 5×4 1 1 + (D) 4×5 5×5 c American Book Company Copyright °
Chapter 1 Review
38. Which of the following can be used to 2 1 compute ? 3 2 2 1 (A) 3×2 2×3 1 1 (B) 3×2 2×3 2×2 1×3 (C) 3×2 2×3 2 1 (D) 3×2 39. Mrs. Tate brought 4 12 pounds of candy to divide among her 24 students. If the candy was divided equally, how many pounds of candy did each student receive?
42. The Vargas family is hiking a 24 23 mile trail. The first day, they hiked 7 12 miles. How much further do they have to go to complete the trail? 43. Jena walked 14 of a mile to a friend’s house, 2 34 miles to the store, and 15 of a mile back home. How far did Jena walk? 44. Cory uses 3 13 gallons of paint to mark one mile of this year’s spring road race. How many gallons will he use to mark the entire 5 12 mile course?
45. According to the ad below, how much could you save at Martin’s Department Store on an 2 item regularly priced at $108=00? 40. Elenita used 2 5 yards of material to recover one dining room chair. How much material would she need to recover all eight chairs?
Martin’s
41. The square tiles in Mr. Cooke’s math classroom measure 3 14 feet across. The students counted that the classroom was 6 38 tiles wide. How wide is Mr. Cooke’s classroom?
Department Store
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off all “Country Elegant” comforters and accessories
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