MATHEMATICS 5 by YOUG JI INTERNATIONAL SCHOOL

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Chapter 1 Understanding Whole Number Lesson 1:

Place Value

The place of the digit in a Hindu-Arabic numeral tells its value. This is because the Hindu-Arabic numeral system is a place value system.

Ones

Tens

Ones Period

Hundred Thousand s Ten Thousand s Thousand s Hundreds

Thousands Period Millions

Hundred Millions

Ten Millions

Millions Period

Ten Billions Billions

Billions Period Hundred Billions

The place value chart shows the value of each digit in a number. The place value chart below gives the values of 1 according to its position in a number. The value of the digit in a number depends on its position. Starting with the tens place, each place has a value of 10 times the place value to its right.

Ones 1 Tens 1 x 10 = 10 Hundreds 1 x 10 x 10 = 100 Thousands 1 x 10 x 10 x 10 = 1 000 Ten thousands 1 x 10 x 10 x 10 x 10 = 10 000 Hundred thousands 1 x 10 x 10 x 10 x 10 x 10 = 100 000 Millions 1 x 10 x 10 x 10 x 10 x 10 x 10 = 1 000 000 Ten millions 1 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10 000 000 Hundred millions 1 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 100 000 000 Billions 1 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 1 000 000 000 Ten billions 1 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10 000 000 000 Hundred billions 1 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 100 000 000 000 From the right, the digits are grouped by threes. Each group is called a period and has a special name â&#x20AC;&#x201C; ones period, thousands period, millions period, and billions period. The next three periods are the trillions, quadrillions, and quintillions. In each period are the ones, tens, and hundreds places.

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Ten Thousands

Thousands

Hundreds

Tens

Ones

Ones Period

Hundred Thousands

Thousands Period

Millions

Ten Millions

Hundred Millions

Millions Period

Billions

Ten Billions

Hundred Billions

Billions Period

The form showing the digits of a number, such as 5 036 004 708, is called standard form of a number. A number may also be written in other forms. The expanded form gives the sum of the values of each digit in the number. The expanded form of 5 036 004 708 is 5 000 000 000 + 30 000 000 + 6 000 000 + 4 000 + 700 + 8 In the expanded form, the place with a value of zero may be omitted. The number 5 036 004 708 may also be written in words: Five billion, thirty-six million, four thousand seven hundred eight Example: In the number 50 403 087 The digit 5 is in the ten millions place; The digit 4 is in the hundred thousands place; The digit 3 is in the thousands place; The digit 8 is in the tens place; and The digit 7 is in the ones place. The digit 5 has the value 50 000 000 The digit 4 has the value 400 000 The digit 3 has the value 3 000 The digit 8 has the value 80 The digit 7 has the value 7 The expanded form is: In words:

50 000 000 + 400 000 + 3 000 + 80 + 7

Fifty million, four hundred three thousand, eighty-seven

Practice Exercise 1.1 A. Write each number in words. 1. 10 010 010 100 2. 400 040 400 004 3. 3 521 643 812 4. 900 000 800 000 5. 20 012 548 075 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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B. Tell the place value of digit 5 in a given number. 1. 645 649 273 2. 56 238 473 483 3. 103 543 771 623 4. 582 272 794 286 5. 382 654 320 287 482 C. Write the digit in the given place value 197 345 634 582 1. Ten millions place 2. Hundred thousands place 3. Millions place 4. Thousands place 5. Hundred billions place

Lesson 2:

Rounding off Numbers

To round off a number is to get an approximate number that is close to the original number. We round off numbers so that they are easier to use and easier to remember. How do we round off numbers? To round off a number to a given place, we follow these rules: ď&#x201A;ˇ First, we locate the place to be rounded off and look at the digit to its right. If the digit to the right is 4, 3, 2, 1, or 0 we round down. This means we retain the number to be rounded off. If the number to the right is 5 or greater, we round up. This means we add 1 to the number to be rounded off. ď&#x201A;ˇ Then we replace with zeros the digit to the right of the digit rounded off. Example: Round off 8 475 to the nearest ten The tens digit is 7 The digit to the right is 5 We round up and replace the digit the right with zero.

8 475 8 480

Round off 8 475 to the nearest thousand The thousand digit is 8 8 475 The digit to the right is 4 and 4 < 5. We round down and replace the digits to the right with zeros 8 000 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Practice Exercise 1.2 1. Round off to the nearest hundred: 863 2. Round off to the nearest ten: 737 3. Round off to the nearest thousand: 131 840 4. Round off to the nearest million: 254 190 900 5. Round off to the nearest hundred: 1 854 6. Round off to the nearest hundred thousand: 518 437 7. Round off to the nearest hundred million: 1 948 294 763 8. Round off to the nearest thousand: 2 736 752 9. Round off to the nearest hundred: 497 583 674 10. Round off to the nearest ten thousand: 485 344 538

Lesson 3:

Adding Whole numbers

On the first day showing of a school play, 3,215 tickets were sold. On the second day, 1 403 tickets were sold and on the third day, 2 150 tickets. How many tickets were sold in all? To find the answer, we put the three numbers together. We add the numbers. 3 215 1 403 + 2 150 6 768

addends sum

In addition number sentence, the numbers to be added are called the addends. The result of addition is called the sum. The symbol for addition is the plus sign, +. The following properties of addition will help us find the sum easily and quickly.  Changing the order of the addends does not change the sum. This is called the Commutative Property of Addition. 8 + 16 = 24 16 + 8 = 24 This property tells us that the addends may be added in any order. The sum will always be the same.  When one of the addends is zero, the other addend is the sum. This called Identity property of addition. 0 + 52 = 52 This property tells us that a number added to zero is the same number.  When there are three addends or more, we may group them two at a time. Changing the grouping of the addends does not change the sum. This is called the associative Property of Addition. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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12 14 + 16

26 + 16 42

12 14 + 16

12 +30 42

Grouping together numbers that add up to 10 or its multiples ( 10, 20, 30, 1nd so on) makes addition easier. Pairs of numbers that add up to 10 are called compatible numbers of 10. Recognizing the compatible numbers make adding numbers easier. Example: 9 + 16 + 21 + 14 = ( 9 + 16 ) + ( 21 + 4 ) = 25 + 35 = 60

instead

9 + 16 + 21 + 14 = ( 9 + 21) + ( 16 + 14 ) = 30 + 30 = 60

Change the order first before grouping

Test Apply the property of addition to find the sum. Group the compatible numbers together to make the addition easier. 1. 412 + 223 + 28 = __________ 2. 237 + 315 + 23 = __________ 3. 57 + 44 + 23 + 36 = __________ 4. 78 + 11 + 39 + 22 = __________ 5. 118 + 414 + 212 = __________ 6. 813 + 104 + 216 + 97 = __________ 7. 678 + 49 + 412 + 11 = __________ 8. 311 + 198 + 549 + 212 = __________ 9. 216 + 107 + 814 + 713 = __________ 10. 919 + 265 + 846 + 115 = __________

What am I? I am a five-digit number. I am the result of adding 4 567, 5 432 and 1. What number am I? ________

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Adding large numbers In finding the sum of larger numbers, we align the digits by place values. Add the digits by column. Start adding the ones, then the tens, and so on. Regroup when the sum of the digits in a place value is more than 9. Example: Find the sum 1 11 2 345 1 462 22 346 + 34 26 187 Test Find the sum. 1. 2. 3. 4. 5.

1. 153 358 + 546 621 + 1 581 = __________ 579 730 + 225 682 + 211 278 = __________ 312 638 + 527 946 + 132 541 = __________ 936 453 + 175 462 + 69 706 + 43 532 = _____ 3 111 015 + 2 586 700 + 361 551 + 201 306 = __________

Estimating the sums When an exact answer is not necessary, we can estimate the sum. To estimate the sum is to find a number close to about the exact sum. One way of estimating the sum is to round off the addends to the greater place of the least number. 52 74 46 + 27

round off

About

50 70 50 + 30 200

If we round off to the nearest hundredths, we obtain a value much closer the exact sum. 1 872 + 2 635

1 900 +2 600 4 500

Exact sum 1 872 + 2 635 4 507

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Test: Estimate the sum of each set of numbers by rounding off the addends to the nearest thousand and to the nearest hundred. To the nearest Thousand __________ __________ + __________

to the nearest hundred __________ __________ + __________

1 675 3 365 + 2 135

5 175 + 2 735

___________ + ___________

__________ + __________

6 835 7 360 + 1 462

__________ __________ + __________

Lesson 4:

Subtracting Whole Numbers

Jerry, a professional player, has made 1 215 free throws in his career. Another basketball player; Jun, has made a total of 842 free throws. How many free throws more has Jerry made than Jun? To find the answer, we find the difference between the two numbers of free throws. We align the numbers by place values. Then start subtracting the ones, then the tens, and so on. 5 683 2 142 3 541

minuend subtrahend difference

In subtraction number sentence, the number we subtract from is the minuend, and the number we subtract with is the subtrahend. The result of subtraction is the difference. The symbol for subtraction is the minus sign, . When the minuend and subtrahend is the same, the difference is zero. 146 â&#x20AC;&#x201C; 146 = 0 What happens when the subtrahend is zero? 257 â&#x20AC;&#x201C; 0 = 257 When the subtrahend is zero, the difference is the same as the minuend. In a place value, when the digit of the minuend is less than that of the subtrahend, we regroup the next higher digit of the minuend so that there is enough to subtract from. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Example: Subtract

4 13 6 12 53 728 - 28 356 25 372

Test Find the difference 1. 2. 3. 4. 5.

974 – 321 = __________ 6 5 73 – 3 497 = ___________ 356 182 – 148 235 = __________ 7 777 – 888 = __________ 4 635 178 – 3 652 279 = __________

Subtracting with zeros in the minuend When there are zeros in the minuend, there would not be enough minuend to subtract from. We regroup as many times as needed before starting to subtract. Example: Subtract 125 from 500 More tens and ones are need. Regroup 5 hundreds to 4 hundreds, 10 tens. Regroup 10 tens to 9 tens and 10 ones. Now, start subtracting from ones. 4 10 10 500 - 125 375 What am I? I am a five-digit number. I am 6 hundred 6 less than fifty thousand, four hundred twenty-five. What number am I?

Test Find the difference 1. 2. 3. 4. 5.

60 – 19 = __________ 120 – 75 = __________ 402 – 125 = __________ 3 000 – 1 297 = __________ 10 000 – 6 437 = __________

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Estimating the differences To estimate the difference, round off both the minuend and the subtrahend. Then subtract using the rounded off numbers. Example: Estimate the difference: 3 875 – 2 162 Let us estimate to the nearest thousand 3 875 – 2 162

round off about

4 000 - 2 000 2 000

Remember this When estimating by rounding off to the highest place value and the result is zero, round off the numbers to the next place value. Test: Estimate the difference 1. 2. 3. 4. 5.

5 172 – 3 735 = _____ 4 386 – 1 875 = _____ 8 538 – 8 395 = _____ 41 950 – 39 502 = _____ 19 063 – 5 875 = _____

Lesson 5:

Multiplying Whole Numbers

Multiplication is a repeated addition. There are special names given to the parts of a multiplication sentence. 8 x 5 = 40. 8 and 5 are called factors and 40 is a called product. The numbers that we multiply are called factors while the result of multiplication is called a product. The symbol for multiplication is the times sign, X. The following properties of multiplication will help us in finding the product easily.     

Changing the order of the factors does not change the product. This property is called the communicative property of multiplication. 5 X 8 = 40 (8)(5)=40 When 1 is a factor, the product is equal to the other factor. This property is called identity Property of Multiplication. 75 x 1 = 75 When zero is a factor, the product is equal to zero. This property is called zero property of Multiplication. 48 x 0 = 0 Changing the grouping of the factors does not change the product. This property is called the associative property of multiplication. When the same factor is distributed across two or more addends, the product does not change. This property is called Distributive Property of Multiplication over addition.

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Practice exercise: 1. 2. 3. 4. 5.

48(0) = _____ 7 x 3 = ___ x 7 6 x (4 x 5)= (___ x 4) = _______ 9 x 1 =1 x _____ = _____ 6 x ( 4 + 5) = (_____ x 4) + ( _____ x 5) = _____ + _____ = _____

Multiplying large numbers The factors in the multiplication sentence have other special names. The number to be multiplied by another number is called the multiplicand; the number we multiply it with is the multiplier. 32 x3 96

multiplicand multiplier product

246 x 3 738

When the multiplier is a one â&#x20AC;&#x201C;digit factor, multiply it by each digit of the multiplicand, starting with the ones digit. Regroup when necessary. When the multiplier has more than one-digit multiply the multiplicand by each digit of the multiplier starting with ones digit. Be sure to align the place values of the partial products. Then add the partial products. 123 x 24 492 246 2952

Think: 24 = 20+4

4 x 123 20 x 123 Think: 2 168 = 2 000 + 100 + 60 + 8

15243 x 2168 121944 91458 15243 30486 33046824

8 x 15234 60 x 15234 100 x 15234 2 000 x 15234

Take not hat the number of digits in the multiplier is equal to the number of partial products. Test Find the product 1. 2. 3. 4. 5.

3 512 x 75 = _____ 64 528 x 193 = _____ 174 693 x 2 156 = _____ 2 513 428 x 4 162 = _____ 62 134 572 x 8435 = _____

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Zeroes in the multiplier When there are zeroes in the multiplier, the partial products will be reduced by the number of zeros in the multiplier. Be sure to align the place values of the partial products with those of the multiplier. Example: 1234 x 302 2468 37020 372668

2 x 1234 300 x 1234

4 236 x 700 2 965 200

51328 3004 205312 15398400 154189312 x

Test: Align the factors then multiply. 1. 2. 3. 4. 5.

675 x 203 = 72541 x 3000 = 213 x 102 = 67105 x 5040 = 23163 x 4000 =

Estimate the products To estimate the product, round off the factors to the highest place values. Then get the product of the rounded-off factors. 1. 468 x 7

round off about

2. 3561 round off x 63 about

500 x 7 3500

4000 x 60 24000

Test YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Estimate the product. 1. 345 x 9 = 2. 463 x 48 = 3. 419 x 82 = 4. 7312 x 251 = 5. 4518 x 135 =

Lesson 6 Factors and Multiples Factors of a numbers The factors of a given whole number are whole numbers that are multiplied to obtain the given number. 2=2x1 3 = 3x 1 6=1x6 6=2x3 12 = 1 x 12 12 = 2 x 6 12 = 3 x 4

The factors of 12 are 1,2,3,4,6,and 12

Whole numbers may be classified according to how many factors they have. A prime number is a whole number greater than 1 with only two distinct factors â&#x20AC;&#x201C; 1 ands itself. 2 and 3 are prime numbers. Can you name other prime numbers? Numbers that are not prime numbers are called composite numbers. Composite numbers are numbers that have more than two distinct factors. 12 is a composite number The number 12, 24, 16, 38, and 40 have something in common. One of their factor is 2. They are called even numbers. In general, an even number has 2,4,6,8,or 0 as its ones digit. Are all even numbers composite numbers? Why? A number that is not even is called odd number. In general an odd number has 1,3,5,7,9 as its ones digit. The numbers 11, 23, 45, 67 and 79 are odd numbers. Are all odd numbers composite numbers? Give an odd number that is prime number. Example:

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When asked to identify all the factors of a given number, you can write down its factor pairs to help you answer. Example: Find the factors of 36

= 1,2,3,4,6,9,12,18,36

The number 1 is neither a prime nor a composite number. When the factor of a composite number is a prime number, the factor is called prime factor. We can convert all the factors of a composite number into prime factors. The process is called prime factorization. We use a factor tree to find the prime factors of a number. These are factor trees for the number 2. Notice that we arrive at the same sewt of prime factors. 12

6 x

not prime 3 = 22 x 3

The prime factors of 12 are 3 and 22 2 x 2 may be written as 22. It is read as â&#x20AC;&#x153; 2 raised to the power of 2 or â&#x20AC;&#x153; 2 to the second power. 22

exponent base

The exponent tells us how many times the base is multiplied by itself. Let us list the factor of 9 and 12 Factors of 9: 1,3, and 9 Factors of 12: 1,2,3,4,6, and 12 Both 9 and 12 have 1 and 3 as factors. We call 1 and 3 the common factors of 9 and 12. A factor that is the same for two or more numbers is called a common factor of the numbers. What is the common factor of all prime numbers? Remember this The greatest common factor (GCF) of two or more numbers is the highest factor is common to the numbers. What is the greatest common factor of 36 and 42? Factors of 36: 1,2,3,4,6,9,12,18, and 36 Factors of 42: 1,2,3,6,7,14,21 and 42 Common factor: 1,2,3, and 6 The GCF of 36 and 42 is 6.

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Test A. Write / in the blank if the number is prime and x if it is composite. _____ 1. 7 _____ 2. 15 _____ 3. 22 _____ 4. 39 _____ 5. 45 _____ 6. 23 _____ 7. 75 _____ 8. 17 _____ 9. 31 _____ 10. 82 B. Identify the factors of the following pairs of numbers. Then, write the greatest common factor (GCF). factors GCF 11. 9 27 12. 8 32 13. 16 48 14. 11 22 15. 30 80

Multiples of a number

The multiples of a given number are numbers that have the given number as a factor. Is 30 a multiple of 3? Yes, since 30 = 3 x 10 and 3 is a factor of 30. List down the first ten multiples of 3 and 4 To list down the multiples of a number, we may skip count by that number from 0. Multiples of 3 = 0, 3,6,9,12,15,18,21, 24, 27 Multiple of 4: 0,4,8,12,16,20,24,28,32,36 Common multiples are nonzero multiples that are the same for two or more numbers. Do 3 and 4 have common multiples? Common multiples of 3 and 4 are 12 and 24â&#x20AC;Ś The least common multiple (LCM) of 3 and 4 is 12. Remember this The least common multiple or LCM of two or more numbers is the least nonzero multiple common to the numbers. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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To find the least common multiple of a set of numbers, we list the multiples of the largest number and check which one is the smallest multiple that has the other numbers as factors. Example: Find the least common multiple of 3,2, and 8? Multiples of 8: 8,16,24,32â&#x20AC;Ś 3 is not a factor of 8 and 16. 3,2,and 8 are factors of 24. So the LCM of 3,2 and 8 is 24.

Practice: A. Give the first nonzero multiples of each number a) 6 b) 11 c) 15 d) 17 e) 21 B. Give the first five multiples of the largest number. Tell whether the other numbers are factors of each multiple. Then state the LCM. 1. 6:_____, _____, _____ , _____, _____ 5: 3: LCM: _____ 2. 9: _____, _____, _____, _____, _____ 4: 2: LCM: _____ 3. 15: _____, _____, _____, _____, _____ 10: 6: LCM: _____ Multiples of 10, 100, 1000 The multiples of 10 are 10, 20, 30, 40, 40, 50, 60, and so on. We simply add one zero to the counting numbers to get the multiples of 10. What are the multiples of 100? The multiples of 100, are 100, 200, 300, 400 and so on.. we add or affix two zeros to the counting numbers to get the multiples of 100. What are the multiples of 1000? YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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The multiples of 100 are 1000, 2000, 3000, 4000 and so on. We add or affix three zeros to the counting numbers to get multiples of 1000. Test A. Fill in the blank to obtain multiples of 10. 1. 350, 360, _____, _____, 390, _____, _____, 420 2. 120, 130, _____, _____, 150, _____, _____, 180 3. 1450, 1460, _____, _____, _____, _____, 1510 B. Fill in the blanks to obtain the multiples of 100. 4. 1300, 1400, _____, _____, _____, _____, 1900 5. 2400, _____, _____, 2700, _____, _____, _____ C. Fill in the blanks to obtain multiples of 1000. 6. 11000, _____, _____, 14000, _____, _____ 7. 45000, _____, _____, 48000, _____, _____ D. Identify the factors of the following pairs of numbers. Then, write the greatest common factor (GCF). Factors GCF 8. 25 __________ _____ 35 __________ _____ 9. 72 __________ _____ 45 __________ _____ E. Write the first five multiples of the largest number. Check if the other numbers are factors of each multiple. Identify the least common multiple. 10. 5 8 10 _____, _____, _____, _____, _____ LCM: _____

Lesson 7 Dividing Whole Numbers

Divisibility Rules The divisibility rules can help you determine whether a number is divisible by another number without finding the quotient. Divisibility Rule Divisibility Rule for 2 Divisibility Rule for 3 Divisibility Rule for 4 Divisibility Rule for 5

A whole number is divisible by 2 if its ones digit is 0,2,4,6 or 8. A whole number is divisible by 3 if the sum of its digits is a multiple of 3 A whole number is divisible by 4 if the number formed by its last two digits is divisible by 4 A whole number is divisible by 5 if its ones digit is either 0 or 5

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Divisibility for 6 Divisibility for 8 Divisibility for 9 Divisibility for 10

Rule A whole number is divisible by 6 if it is divisible by both 2 and 3 Rule A whole number is divisible by 8 if the number formed by its last three digits is divisible by 8 Rule A whole number is divisible by 9 if the sum of its digit is divisible by 9 Rule A whole number is divisible by 10 if its ones digit is 0.

Example: Test the number for divisibility by 2, 3,4,5,6,8,9,10 a. 3,044 2 – 3,044 is divisible by 2 since it is an even number 3 – the sum of the digit is 11 ( 3+ 4+ 4 = 11) 11 is not a multiple of 3. Therefore it is not divisible by 3. 4 – the last 2 digit is 44 which is divisible by 4 5 – the number 3,044 does not end in 5 or 0 therefore it is not divisible by 5 6 – the number is divisible by 2 but not divisible by 3. 8 – the last 3 digit form the number 044 or 44 which is not divisible by 8 ( 44 ÷ 8 = 5 R 4) therefore, 3,044 is not divisible by 8. 9 – the sum of the digits is 11 and 11 is not divisible by 9 therefore 3,044 not divisible by 9. Practice: Write Y in the space if the given number is divisible by the number at the top of the column. Write N if it is not. Number

Divisibility by 2 3 4 5 6 8 9 10

390 2,128 1,172 3,580 28,260

Remember this:  When a number is divisible by 10, it is divisible by both 2 and 5  If a number is divisible by 9, it is also divisible by 3. If a number is divisible by 6, it is also divisible by 3 and 2.  When a number is divisible by 8, it is also divisible by 4 and 2.

Dividing with a one-digit divisor To divide a whole number by another whole number, follow these steps: 1. Estimate the quotient and divide 2. Multiply by the divisor 3. Subtract from the dividend YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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4. Bring down the next digit. We repeat these steps starting with the first digit from the left of the dividend. Example: Divide 456 by3 We may first test for divisibility by 3 to check if there is a remainder. 4 + 5 + 6 = 15, a multiple of 3 so there is no remainder in the division. Step 1. Estimate and divide. How many 3s in 4?

3 Step 2.

1 456

Multiply 1 x 3 = 3 1 456 3 Subtract 4 â&#x20AC;&#x201C; 3 = 1 3

Step 3.

Step 4.

1 456 3 1

Bring down the next digit

1 456 3_ 15

Divide 15 by 3 Step 1. How many 3s in 15?

Step 2.

5 x 3 = 15

Step 3.

15 456 3_ 15

15 456 3_ 15 15

15 â&#x20AC;&#x201C; 15 = 0

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15 456 3_ 15 15 0

Step 4. Bring down 6 15 3 456 3_ 15 15 06 Divide 6 by 3 Step 1. How many 3s in6

152 456 3_ 15 15 6

Step 2. 2 x 3 = 6

152 456 3_ 15 15 6 6

Step 3. 6 â&#x20AC;&#x201C; 6= 0

152 456 3_ 15 15 6 6 0 152 456

Let us some more examples. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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1304 R 3 6523 5 15 15 2 0 23 20 3

test for divisibility: 6523 does not end in 5 or 0. so it is not divisible by 5. This means there is a remainder.

Test 1. 2. 3. 4. 5.

9376 ÷ 6 896 ÷ 8 5263 ÷ 5 4685220 ÷ 8 812788 ÷ 7

Dividing by two-digit divisor When the divisor has two digits, start dividing with the first two digits of the dividend. Divide 9708 by 12 Divide 97 by 12 Step 1. Estimate and divide How many 12s in 97? 8 12 9708 Step 2.

Multiply 8 x 12 = 96 8 12 9708 96

Step 3.

Subtract. 97 – 96 = 1 8 12 9708 96 1

Step 4.

Bring down the next digit. 8

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12 9708 96 10 Divide 10 by 12 Step 1. How many 12s in 10? 80 12 9708 96 10 Step 2.

0 x 12 = 0 80 12 9708 96 10 0

Step 3.

10 â&#x20AC;&#x201C; 0 = 10 80 12 9708 96 10 0 10

Step 4. Bring down 8 80 12 9708 96 10 0 108 Divide 108 by 12 Step 1. How many 12s in 108? 809 12 9708 96 10 0 108 108 Step 2. 9 x 12 = 108 809 12 9708 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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96 10 0 108 108 Step 3. 108 – 108 = 0 809 12 9708 96 10 0 108 108 0

Test 1. 2. 3. 4. 5.

6 446 ÷ 11 18 753 ÷ 32 328 600 ÷ 40 410 890 ÷ 75 2 603 826 ÷ 11

Dividing with three-digit divisor When the divisor has three digits, start dividing with the first three digits of the dividend. Divide 703 945 by 153 Divide 703 by 153 Step 1. Estimate and divide How many 153s in 703? 4 153 703945

Step 2.

Multiply. 4 x 153 = 612 4 153 703945 612

Step 3.

Subtract 703 – 612 = 91

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4 153 703945 612 91 Step 4.

Bring down the next digit.

4 153 703945 612 919 Divide 919 by 153 Step 1.

How many 153s in 919? 46 153 703945 612 919

Step 2.

6 x 153 = 918 46 153 703945 612 919 918

Step 3.

919-918 = 1 46 153 703945 612 919 918 1

Step 4.

Bring down 4 46 153 703945 612 919 918 14 Divide 14 by 153 Step 1 how many 153s in 14? 4 60 153 703 945 612 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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91 9 91 8 14 Step 2.

0 x 153 = 0 4 60 153 703 945 612 91 9 91 8 14 0 145 0

Step 3.

Test 1. 2. 3. 4. 5.

145 – 0 = 145 4 600 153 703 945 612 91 9 91 8 14 0 145 0 145 4 600 R 145 153 703 945

27 446 ÷ 365 37 980 036 ÷ 120 883 240 767 ÷ 703 457 156 ÷ 812 17 926 128 ÷ 396

Lesson 8 Solving word problems involving whole numbers

In solving a word problem you may have to do addition, subtraction, multiplication, division or a combination of these operations. Example: A cellular phone company offers 900 minutes of free talk time to new subscribers. How many hours of free time is this? YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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What is asked? The number of hours of free time offered What is given? 900 minutes of free talk time Operations to be used: Division Solution: 900 minutes รท 60 minutes/hour = hours = 15 hours Complete answer: the cellular phone company offers 15 hours of free talk. Practice: 1. A basketball team scored 89 points in a game. The three best players in the team scored 24, 18, and 15 points. How many points did the rest of the team score? What is asked? ___________________________________ What are given? __________________________________ Operations to be used: ____________________________ Solution:

Complete answer:

2. Five buses were hired to carry students on an educational tour. A total of 235 students joined the tour. Each of the four buses was filled with 48 passengers. How many were on the fifth bus? What is asked? ___________________________________ What are given? __________________________________ Operations to be used: ____________________________ Solution:

Complete answer:

3. Find the total population of the following countries rounded off to the nearest million. Country Population France Thailand Poland

60 656 178 65 444 371 38 635 144

What is asked? ___________________________________ What are given? __________________________________ Operations to be used: ____________________________ Solution: YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Complete answer: 4. A family went shopping for clothes. The father bought two shirts at P 595.00 each. The mother bought a blouse at P 450.00 and a skirt at P 375.00. the son bought a pair of shoes at P 2,499.00 and a pair of socks at P 150.00. the daughter bought a t-Shirt at P 225.00 and a pair of pants at P 495.00 a. What was the total amount spent by the family? What is asked? ___________________________________ What are given? __________________________________ Operations to be used: ____________________________ Solution:

Complete answer: b. If the father bought only P 1,190.00 out of his P 6,000.00 budget, how much is left for snacks? What is asked? ___________________________________ What are given? __________________________________ Operations to be used: ____________________________ Solution:

Complete answer: _________________________________

Chapter 2 Understanding fractions Lesson 1 The meaning of fractions A fraction represents a part of a whole or a set. It may also indicate division. It has two parts: the numerator and the denominator.

The numerator tells the part of the whole or the set being considered. The denominator tells into how many parts the whole is divided or how many elements the set has. The following examples show fractions that represents parts of a whole. a. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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b. About

of the human body is water.

The following examples show fractions that represents parts of a set.

of the candles is black

of Mr. Medinaâ&#x20AC;&#x2122;s class are members of Math Club.

A fraction could also indicate division.

means 1 Ăˇ 4.

means 21 divided by 7. A fraction could also be used to compare two quantities. For example, there are two boys for every three girls in a class. The number of boys compared to the number of girls can be written as . The number of teacher compared to the number of pupils in an elementary school is represented by . A fraction may mean a part of a whole or a set. It may indicate division or a comparison of two numbers. Practice: A. Write in the blank the fraction indicated by the shaded portion of each figure. 1.

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B. Shade the region indicated by the given fraction. 1. 2.

C. Encircle the part of the set indicated by the given fraction. 4.

D. Compare the following using fractions: YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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6. The number of feet of a duck and of a dog 7. The number of legs of a grasshopper and a spider.

Lesson 2 Kinds of fractions A proper fraction is a fraction whose numerator is less than the denominator. The fractions are examples of proper fractions . A proper fraction may be a unit fraction or a common fraction. A unit fraction is a proper fraction whose numerator is 1. are unit fractions. A common fraction is a proper fraction whose numerator is greater than 1. are common fractions. An improper fraction is a fraction whose numerator is equal to or greater than the denominator. are improper fractions. A mixed number is a whole number plus a fraction.

are

mixed numbers. An improper fraction may be expressed as mixed number. We do this by dividing the numerator by the denominator. The quotient is the whole number part, the remainder becomes the denominator remains the same. Example Express

Hence

as a mixed numbers 2 4 11 8 3 =2

To express mixed number as an improper fraction, multiply first the denominator by the whole number. Add. The product to the numerator, the sum becomes the new numerator. The denominator stays the same. Example 3 to improper fraction. =

(

)

Similar fractions are fractions whose denominators are the same. are similar fractions. When the denominators of a set of fraction are not the same, the fractions are called dissimilar fractions. are dissimilar fractions. Test A. Write the correct answer in the blank. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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1. A unit fraction is also __________ fraction 2. Common fractions are also _____ fractions. 3. , are _____ fractions. 4.

are _________ fractions.

are __________ fractions.

B. Change the improper fraction to a mixed number. 1. _____ 2.

_____

C. Change each mixed number to improper fraction. 1. _____ 2. 5 _____ 3.

_____

_____ Lesson 3 Comparing and ordering fractions

Comparing fractions Two fractions are equal or equivalent if they represent parts that are equal in size. Apart from comparing the size of the fractional parts, there is another way of telling if two fractions are equal. Two fractions are equivalent if the product of the numerator of one fraction and the denominator of the other is equal to the product of the other pair of the numerator and denominator. The operation involved is called cross-multiplication and the pair of resulting products are called cross-product. Are the fractions and equal?

3 x 24 = 72 and 4 x 18 = 72 3 x 24 = 4 x 18 and

are equivalent fractions or

Two fractions are equivalent if their cross-products are equal. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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A fraction has many equivalent fractions. The following fractions are equivalent.

Any given fraction may be changed into an equivalent fraction with higher terms. To do this, multiply both the numerator and denominator if the given fraction by the same number.

are equivalent fractions Fractions that are not in lowest term s may be changed to equivalent fractions with lower terms. A fraction is not in its lowest terms if its numerator and denominator have a common factor other than 1. By dividing both its numerator and denominator by the common factor, an equivalent fraction is obtained. Is in its lowest terms? No, because 12 and 18 have common factors: 2, 3, and 6. Now let us reduce to lower terms.

Thus, these are equivalent fractions: in lowest terms.

. The fraction is the fraction

To compare them, we change them into similar fractions. In comparing similar fractions, we just look at the numerators. The one with the greater numerator is the greater fraction. Take note that

are dissimilar fractions.

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In changing two dissimilar fractions into similar fractions, we first assign a common denominator. For , we use the product of the denominators, 3 x 5 or 15 as the common denominator. Next, for each fraction, we multiply its numerator with the denominator of the other fraction. The product is the numerator. = = Clearly So Ordering fractions Now let us order the following fractions from greatest to least. In cases where more than two dissimilar fractions are being compared, we use the least common denominator or LCD as the common denominator. The LCD is the smallest number that is divisible by all the denominators of the dissimilar fractions. The LCD is equal to the least common multiple or LCM of the denominators. So we first get the LCM of the denominators 6, 8, 12. Multiples of 12: 12,24,36â&#x20AC;Ś 24 is the smallest multiple of the largest denominator, 12, that is also a multiple of 6 and 8. We use 24 as the LCD. Next, we now change the fractions into their equivalent fractions whose common denominator is 24. How do we do this? For each fraction, we divide the LCD by the denominator of the fraction. Then multiply the quotient with the numerator. The product is now the numerator of the equivalent fraction. (

(

)

(

Since 20> 18 > 14, then

)

and

Thus the fraction arranged from greatest to least: Test A. Use the cross products to compare each pair of dissimilar fractions. Write = in the box if the fractions are equivalent and if they are not. 1. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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2. 3. 4. 5. B. Arrange each set of dissimilar fractions from least to greatest. Use the LCD as the common denominator in changing them to similar fractions. 1. _______________ 2.

_______________

LESSON 4 Estimating and rounding off fractions We can estimate a fraction to be closer to 0 , , 1 Rules on estimating the value of a fraction  When the numerator is much less than the denominator, the value of the fraction is closer to 0. Thus, are fractions closer to 0  When the denominator is about half of the denominator, the value of the fraction is closer to are all closer to .  When the numerator is about equal to the denominator, the fraction is closer to 1. Thus, are all closer to 1. 

We use the symbol ~ to mean closer to or approximately equal to, thus we write

Test A. Tell whether the given fraction is closer to 0, ,or 1 1. 2. 3. 4.

_____ _____ _____ _____

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_____

B. Estimate each fraction. Tell if it is about 0, , or 1 1.

~ _____

Rounding off fractions Sometimes we need to round off mixed numbers to the nearest whole number. To do this, we take a look at the numerator and the denominator and compare them. To round off fractions, we follow these rules: ď&#x201A;ˇ We round down when the numerator is less than half the denominator. This means that we do not add 1 to the whole number. Thus, 3 is round down to 3, and 7 ď&#x201A;ˇ

is rounded down to 7.

When the numerator is equal to or greater than half the denominator, we round up. This means that we add 1 to the whole number. So, 2 is rounded up to 3, and 17

is rounded up to 18.

Practice Match the mixed number and its rounded off number. Write the letter of correct answer in the blank. _____ 1. 10 a. 36 _____ 2. 4

b. 12

_____ 3. 36

c. 70

_____ 4. 10

d. 5

_____ 5. 4

e. 13

_____ 6. 69 _____ 7. 11 _____ 8. 12

f. 11 g. 4 h. 10

_____ 9. 35 ____ 10. 69

i. 35 j. 69

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Lesson 5 Adding fractions Adding similar fractions In adding similar fractions, the numerator of the sum is the sum of the numerator. The denominator of the sum is the common denominator. Example

Following the rule of adding similar fractions, we obtain the sum of

Now suppose we have the following:

When the sum is an improper fraction, change it to a mixed number with the fraction in lowest term. Practice Add mentally 1. 2. 3. 4. 5. Adding dissimilar fractions To add dissimilar fractions, we must first change the addends into similar fractions. Then follow the rule of adding similar fractions. Add Step 1. Find the least common denominator of the fractions. Solving for the LCD of : Multiples of 6: 6,12,18 12 is the smallest multiple of 6 that is also divisible by 4 and 3. Thus the LCM of 3,4,and 6 is 12. The LCD of the fractions Is then 12.

Step 2. Change the addends into similar fractions with the LCD as the denominator. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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(

)

(

)

Step 3. Add the similar fractions Hence,

Practice A. Find the least common denominator of each set of fractions. 1.

LCD: _____

LCD: ____

B. Find the sum. 1. 2. 3. 4. 5.

Adding mixed numbers To find the sum of mixed numbers, first find the sum of the whole numbers. Then find the sum of the fractional parts by applying the rules on adding fractions. Example a. 1 + 2 = ( 1 + 2) + (

)=3+ =3

b. 2 + 1 = ( 2 + 1 ) + ( With LCD = 12, 3+(

)

)=3+( =

(

and

) =

)

=3+ =3 Practice Find the sums of the following. Change all sums to lowest terms. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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1. 2. 3. 1 4. 5. 6. 7. 2 8. 9. 10. 4 + 2 Test Solve the following problems 1. Gerry watched TV for 2 hours on Monday,

hour on Tuesday, 1

hours on

Wednesday, 2 hours on Thursday, and 3 hours on Friday. About how many hours did Gerry watch TV for the five days? 2. Dianne walked for 1 km on Saturday and 2 km on Sunday. About how many kilometers did Dianne walk for the weekend?

Lesson 6 Subtracting fractions Subtracting similar fractions In subtracting similar fractions, the numerator of the difference is the difference of the numerator of fractions. The denominator is the common denominator of the fractions. Example

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Practice Find the difference. Express in lowest terms 1. 2. 3. 4. 5.

Subtracting dissimilar fractions In subtracting dissimilar fractions, change the fractions first to similar fractions with the LCD as the common denominator. Then apply the rule on subtracting similar fractions. Example 1 The LCM of 2 and 3 is 6. Using 6 as the LCD,

and

= Example 2 The LCM of 4 and 10 is 20. The LCD is thus 20, so = Practice Change to similar fractions and subtract. Write your answers in lowest term. 1. = 2. 3. 4. 5. 6. 7. 8. 9. 10.

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Subtracting from mixed numbers In subtracting a fraction from a mixed number, simply subtract the subtrahend from the fractional part of the minuend. If the denominators are different, change the fractions into similar fractions. Example 12 - = 12 +

= 12 + = 12 (

)

With LCD = 10 , (

)

(

)

Sometimes the fractional part of the minuend is less than that of the subtrahend. In this case, we regroup the minuend. Example Subtract (

) (

(

)

(

)

) (

)

In subtracting a mixed number from another mixed number, subtract the whole number first. Then following the rules on subtracting similar or dissimilar fractions, subtract the fractional parts. Example (

)

(

)

In subtracting a whole number from a mixed number, find the difference between the whole numbers. Then carry over the fractional part. Example (

)

Test Find the difference 1. 2. 3. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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4. 5. 6. 7. 8. 9. 10. Subtracting from whole numbers In subtracting a mixed number or a fraction from a whole number, we regroup the minuend so that it will have a fractional part that will be subtracted by the fractional part of the subtrahend. Example Find the difference Step 1. Regroup the minuend into a mixed number whose fractional part is a similar fraction of the subtrahend. Think 4 = 3 + 1 = 3 + = 3 Step 2. Subtract the whole numbers, then subtract the fractional parts. ( ) 4 â&#x20AC;&#x201C;1 =2+ =2 Another example 5â&#x20AC;&#x201C;2 =4 Test Find the difference 1. 4 = 2. 8

3. 6

4. 20

5. 37 6. 10

= =

7. 25 8. 13 9. 7 10. 12

= = = = Lesson 7 Estimating sums and differences of mixed numbers

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In our lesson on estimating proper fractions, we rounded them off to the nearest, 0, , or 1. When dealing with mixed number it may be more convenient to round off the fractional part into either 1 or 0 since there is a whole number component. Any fractional part equal to or greater than is rounded up to 1. Any fractional part less than is rounded down to 0. It is easy to estimate sums of differences involving . For example 2 + 1 is easily seen as 4. Making a good estimate of a sum or difference is a useful skill. A good estimate gives us an idea of a ballpark around which the exact value lies. It also allows us to do a lot of estimation. You might have estimated how much money you will need to buy materials for school project before going to a school supplies store. Your parents might have used estimation in finding out how much money they need to save month to buy as new appliance. So a systematic way of estimation is part of our lives. Practice Estimate the sum or difference by rounding off the mixed numbers. 1. 3 2. 3. 4. 5.

Lesson 8 Word problems: addition and subtraction of fractions In an elderly ladyâ&#x20AC;&#x2122;s will, one-third of her property will be left to her favorite niece, one-fourth to her favorite nephew, and the rest to her only sister. What part of the property will her sister get? What is asked? The part of the property that will go to the sister What are given? One-third part of property that will go to the niece One-fourth part will go to the nephew Operations to be used: 1st step: Addition â&#x20AC;&#x201C; Get the sum of the parts for the niece and the nephew. 2nd step: Subtraction-subtract the sum from the 1st step from the whole portion to get the part for the sister. Test Solve the following problems. If answer is a fraction, express it in lowest terms. 1. A cargo jeep is carrying fruits. Bananas make up half of the load; mangoes compose one-third of the weight; and the rest are papayas. What part is made up of papayas? YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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What is asked? _________________________ What are given? ________________________ Operations to be used: __________________ Solution:

Complete answer: _________________________ For items 2 to 6, refer to the following table. continent Fraction of world’s land area Africa Antarctica Asia Australia Europe North America South America

2. Arrange the continents from least to greatest. What is asked? _________________________ What are given? ________________________ Operations to be used: __________________ Solution:

Complete answer: _________________________ 3. How much larger is the biggest continent than the smallest continent? Express the difference as a fraction of the world’s land area. What is asked? _________________________ What are given? ________________________ Operations to be used: __________________ Solution:

Complete answer: _________________________ 4. What part of the world’s area is occupied by north and South America? YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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What is asked? _________________________ What are given? ________________________ Operations to be used: __________________ Solution:

Complete answer: _________________________ 5. What part of the world’s land area is occupied by Asia, Europe, and Africa? What is asked? _________________________ What are given? ________________________ Operations to be used: __________________ Solution:

Complete answer: _________________________ 6. Which is greater- the combined areas of Asia and Europe or the combined areas of North America and South America? What is asked? _________________________ What are given? ________________________ Operations to be used: __________________ Solution:

Complete answer: _________________________ Chapter 3 Understanding Multiplication and division of fractions

Lesson 1 Multiplying fractions

To multiply fraction by another fraction,  Multiply the numerators to get the numerator of the product.  Multiply the denominators to get the denominator of the product.  Write the product in the lowest terms. Example: x =

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We may cancel out GCF of numerators and denominators before getting the product of the fractions. Example:

GCF of 8 and 2 is 2. Practice: Find the product and reduce to lowest terms. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

To multiply fraction and a whole number   

First write the whole number as an improper fraction whose denominator is 1. Multiply the denominators to get the denominator of the product. Express the product in lowest terms by cancelling GCF of numerators and denominators  When the product is an improper fraction, write it as a whole number or a mixed number in lowest terms. Example: GCF of 18 and 16 is 2. Multiplying a fraction by a whole number is another way of finding the fractional part of a whole. Example: Father gave Alma ₱25.00 she spent of it to buy a small notebook. How much did the notebook cost? To find the cost of the notebook, we first find

of ₱25.00.

x 25 = The small notebook costs ₱15.00. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Practice: Find the fractional part. 1.

of ₱50.00 =

of ₱24.00 =

of ₱90.00 =

of 100 m =

of 1 dozen eggs are painted red =

Find each product 6. 9 x = 7.

x 12 =

8. 9. 10. Lesson 2 Multiplying mixed number

To multiply a fraction and a mixed number:  First, write the mixed number as an improper fraction.  Cancel out the GCF of numerators and denominators.  Multiply the numerators to obtain the numerator of the product.  Multiply the denominators to obtain the denominator of the product.  When the product is improper fraction, write it as a whole number or a mixed number in lowest terms.  You may also express the mixed number as the sum of the whole numbers and a fraction and apply the distributive property of multiplication. Example 1: ( ) ( ) ( )

Example 2:

Practice: 1. 2. 3. 4. 5. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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6. 7. 8. 9. 10. Multiplying mixed number by another mixed number To multiply a mixed number by another mixed number or a whole number:  First write the factors as improper fractions.  Cancel out the GCF of both numerators and denominators.  Multiply the numerators to get the numerator of the product.  Multiply the denominators to get the denominator of the product.  When the product is improper fraction, write it as a whole number or a mixed number in lowest terms. Example:

Practice: 1. 2. 3. 4. 5. 6. 5 7. 6 8. 9. 10. Lesson 3 Dividing fractions Dividing a whole number by a fraction means finding out how many of the fractions are in that number. Example: How many s are there in 4? By counting the number of s, we know that there are eight how many

in 4. Tasking

in 4 means dividing by

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That is, 4

Another example: How many three fourths are there in 3? That is, 3 ÷ = 4 Reciprocal of a number When a product of two numbers is 1, we say that the two numbers are reciprocals of a unit. 5 x =1

4 and are reciprocal

6 x =1

5 and are reciprocal

To find the reciprocal of a number,  First write the number as a fraction.  Invert the fraction by exchanging the position of the numerator and denominator.  Check to see whether the product is 1. Example: Find the reciprocal of 15. 15 =

to check

15 and

are reciprocals of each other.

Example 2. What is the reciprocal of each number? a.

and

=1 =1

Therefore: and

are reciprocals are reciprocals

Study the following example 4÷ = x = =8 3÷ =

= =4

To obtain the quotient, we change the operation from division to multiplication. We multiply the dividend by the reciprocal of the fraction. To divide a whole number by a fraction:  Write the whole number as a improper fraction with denominator 1.  Multiply the dividend by the reciprocal of the divisor.  Cancel GCF whenever possible.  Multiply the numerators and the denominators.  When the product is an improper fraction, express it as a whole number or a mixed number in lowest term. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Test. I.

Answer each question

1. How many s are there in 4? 2. How many s are there in 3? 3. How many s are there in 5? 4. How many s are there in 6? 5. How many s are there in 6? II. 6. 6

Divide =

7. 24 8. 18

= =

9. 9

10. 4

Dividing fraction by a whole number To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number. Example

4 Practice Divide 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

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Dividing a fraction by another fraction Division of a fraction by another fraction means finding out how many of the second fraction are in the first fraction. To divide a fraction by another fraction:  Multiply the dividend by the reciprocal of the divisor.  Multiply the numerators and denominators  When the result is an improper fraction, change to mixed numbers. Example How many one-ninths are there in ? To obtain the answer, we find the quotient: x This means that there are six s in . Practice 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Dividing a mixed number by a fraction To divide a mixed number by a fraction, change the mixed number into improper fraction first. Then follow the rules on dividing by fractions. Example Practice Find the quotient 1. 2. 3. 4.

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5. 6. 7. 8. 9. 10. Dividing a mixed number by another mixed number We change the dividend and divisor into improper fractions and then apply the rule on dividing a fraction by another. Practice Divide 1. 2. 3. 4. 5. 6. 7. 8. 5 9. 10. Lesson 4 Estimating products and quotients of mixed numbers In estimating the products or quotients of mixed number we will also follow the technique used in estimating sums and differences of mixed numbers. We round off the fractional part to either 1 or 0. Example Estimate the product of

times 3 .

Since is nearer to 1, then 4 is rounded off to 5. And since is nearer to 0, then 3 is rounded off to 3. So the estimated product is x 3 ~ 5 x 3 = 15 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Example 2

Practice Estimate the product or quotient 1. 2. 3. 4. 5. Lesson 6 Word problems: Multiplication and Division of fractions A school band needs new uniforms for its 24 members. A pair of pants uses up about 1

meters (m) of cloth. How many meters of cloth is needed for sewing pants

for all band members? What is asked? The total length of cloth needed to sew 24 pants What given? 1 m â&#x20AC;&#x201C;length of cloth need per pair of pants 24 â&#x20AC;&#x201C; number of pants to be sewn Operation to be used: multiplication. Solution: ?

Complete answer: to sew pants for all band members, 42 m of cloth is needed. Practice Solve the following problems 1. On the average, a pupil studies his lesson for 1 hour and 45 minutes in a day. How many hours does he spend on studying in 1 week of 5 class days? What is asked? __________________________ What are given? _________________________ Operations to be used: ____________________ Solution:

Complete answer: ________________________ 2. Data shows that, on the average,

of tornados in the United States occur in

the months of April, May, June, and July. If around 80 tornados occur during YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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the year, how many tornados do you expect to occur during the rest of the year? What is asked? __________________________ What are given? _________________________ Operations to be used: ____________________ Solution:

Complete answer: ________________________ 3. Three fourths of the stocks of T-shirts was sold by the department store during the Christmas season. Half of those sold are large in size. If 200 Tshirts were sold, how many large-sized T-shirts were sold? What is asked? __________________________ What are given? _________________________ Operations to be used: ____________________ Solution:

Complete answer: ________________________ Chapter 4 Understanding decimals Lesson 1 Writing and Reading Decimals The number 0.1 and 0.25 are called decimals. Decimals are numbers based on the number 10. A fraction or a mixed number in which the denominator is a power of 10, for example 10, 100, 1000 is called a decimal fraction. Examples: a) b) c) d)

= 0.5 read as “five tenths” = 0.04 read as “ four hundredths” = 0.003 read as “ three thousandths” = 6.125 read as “ six and one hundred twenty-five thousandths”

A fraction in which the denominator is not a power of 10 can be written as a decimal by first making it a decimal fraction. Find the number to which you can multiply the denominator such that the product is a power of 10. Multiply also the numerator by this number so that you have a decimal fraction equivalent to the original fraction. Example: a)

= 0.2

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x =

= 0.68

As in whole numbers, we can show decimals in a place value chart. The decimal places are to the right of the decimal point. The value of each digit in a decimal depends on its place in the decimal. Note that each place is ten times greater than the place to its right. The following are examples a)

= 0.5 read as “ five tenths”

ones point tenths 0 . 5

= 0.04 read as “ four

hundredths” ones point tenths hundredths 0 . 0 4 Let us consider the mixed numbers. ones tenths hundreds thousandths 3 . 0 2 5 3 is 3 ones 0 is 0 tenths 2 is 2 hundredths 5 is 5 thousandths In standard form, we write this number as = 3.025 read as “ three and twenty-five thousandths”

Practice: A. Write the decimals in words a) 0.2 b) 0.34 c) 1.12 d) 2.43 e) 10.275 B. Express the following fraction in decimal a) b) c) 8 d) e) 15 Lesson 2 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Comparing and Ordering decimals In comparing decimals, check first if the whole number part of the numbers you are comparing is equal. If so, compare the decimal part digit by starting the digits in the tenths place. Example: 0.725 > 0.275 since 7 > 2 in the tenths place 2.834 < 2.836 since 4 < 6 in the thousandths place 3.06 = 3.060 Practice: Compare each pair of decimals a) 0.50 0.05 b) 0.75 0.57 c) 2.5 2.25 d) 3.12 3.012 e) 5.078 5.708 Lesson 3 Adding Decimals The following are the steps in adding and subtracting decimals; Step 1 Write the decimals in a column. Make sure digits having the same place value are aligned. Step 2 Add and subtract decimals as you would with whole numbers. Annex zeros to the decimals when necessary. Remember to align the decimal point of the answer with the decimals being added or subtracted. Example: Find the sum of 0.456 and 0.748 Step 1 Align the decimal points of the addends 0.456 + 0.748 Step 2 Add the digits in the thousands place. Regroup in the hundredths place. Add the digits in the hundredths place. Regroup in the tenths place. Add the digits in the tenths place. Regroup in the ones place. 1 11 0.456 + 0.748 1.204 Practice: Find the sum YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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a) b) c) d) e)

0.457 + 0.738 = 0.357 + 0.624 + 0.372 = 0.135 + 0.241 + 0.330 + 0.211 = 15.243 + 12.542 + 4.253 = 73.435 + 63.583 + 90.345 = Lesson 4 Subtracting Decimals

In subtracting decimals, align the decimal points of the minuend and the subtrahend. In doing this, you put in the same column the tenths, the hundredths, the thousandths and so on. Then subtract by columns starting with the rightmost digit. Subtract the digits in the thousandths, the hundredths, tenths place. Put the decimal point directly under the decimal points of the minuend and the subtrahend To check the answer, add the difference and the subtrahend. You should get the minuend. Example: Find the difference: 0.634 – 0.376 0.634 - 0.376 0.258 Practice: Find the difference: a) 0.684 – 0.352 = b) 0.756 – 0.543 = c) 0.534 – 0.326 = d) 0.628 – 0.455 = e) 0.967 – 0.754 = Subtracting decimals from whole numbers Example find the difference of 7 – 3.5 Rewrite 7 as 7.0 (based on the decimal place of the subtrahend). Align the decimal points. 7.0 6 + 1.0 3.5 3 + 0.5 3 + 0.5 = 3.5 To check: 3.5 + 3.5 7.0 Practice: Subtract. a) 4 – 2.6 = b) 5 – 3.7 = YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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c) 3 – 0.18= d) 7 – 1.75 = e) 8 – 2.43 = Lesson 5 Word Problems: Addition and Subtraction of Decimals There are many practical problems involving decimals because many decimal numbers are the result of measurement. For example, we can speak of 2.5 meters (m) as the length of the dining table, 1.5 kilogram (kg) as the weight of a dressed chicken, and 1.65 m as the height of a person. Even money is expressed as a decimal for amounts less than one peso. The cost of a pencil may be ₱9.75; the cost of a floor tile may be ₱19.50; the cost of pad paper may be ₱24.25. Steps in problem solving: 1. Understand what is asked. 2. Determine what are the given information and conditions. 3. What operation(s) is/are needed to arrive at the answers. 4. Check the correctness of the solution and give the answer. Let us begin by solving the following sample problem. From a market vendor, Aling Saria bought a kilogram of whole dressed chicken and a kilogram of pork. If chicken costs ₱125.75 per kg and pork costs ₱156.50 per kg, how much did Aling Saria pay for what she bought? What is asked? The total cost of meat What is given? Weight of chicken: 1 kg; weight of pork: 1 kg cost of chicken ₱ 125.75 per kg and pork costs ₱156.50 per kg. Operation(s) to be used: Addition Solution: ₱125.75 ₱156.50 ₱282.25 Complete Answer: The total cost of meat is ₱282.25. Suppose in the problem, Aling Saria gave a ₱500-bill to the vendor, how much change will she get? Solution: ₱500 - ₱282.25 ₱217.75 Since addition is the inverse operation of subtraction then we can check the answer by adding the difference to the subtrahend to obtain the minuend ₱217.75 - ₱282.25 ₱500 Remember that in adding or subtracting decimals, the decimal points must be aligned. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Answer this: Answer this: Give the complete solution to each of the following problems. a) From the bookstore, Chito bought a pencil at ₱11.75 and an intermediate pad paper at ₱24.25. Find the total cost of his purchase. What is asked: _____________________________________________________ What are given: ____________________________________________________ Operations to be used: _______________________________________________ Number Sentence: __________________________________________________ Solution:

Complete Answer: ______________________________________ b) Dona Ana became a widow. She gave part of a large residential lot to her two children, Ching and Paul. The rectangular lot measures 30.5 m by 24.5 m. of the lot was given to Ching and to Paul. Find the dimensions – length and width, and perimeter of each lot. What is asked: _____________________________________________________ What are given: ____________________________________________________ Operations to be used: _______________________________________________ Number Sentence: __________________________________________________ Solution:

Complete Answer: _________________________________________________ c) The school is 7.4 kilometers (km) from where Lita and Bobby live. To exercise, Bobby walks for 3.5 km while Lita rides a bike to cover the same distance. Then Bobby takes a bike and Lita walks all the way to school. What is the total distance travelled by Bobby and Lita using the bike? What is asked: _____________________________________________________ What are given: ____________________________________________________ Operations to be used: _______________________________________________ Number Sentence: __________________________________________________ Solution:

Complete Answer: _________________________________________________

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d) The family of Mr. Cruz ate in a restaurant and ordered the following the following food: a plate of rice at ₱38.50, a bowl of nilaga at ₱74.50, an order of adobo at ₱65.75, an order of pancit at ₱55.50 and an order of fried fish at ₱86.25. He added four soft drinks which cost ₱47.00. if Mr. Cruz gave a ₱500-bill, how much is the change? What is asked: _____________________________________________________ What are given: ____________________________________________________ Operations to be used: _______________________________________________ Number Sentence: __________________________________________________ Solution:

Complete Answer: _________________________________________________ e) Benny bought a pair of maong pants at ₱975.75 and a pair of rubber shoes that costs ₱1,569.50. he gave three 1 000-pesos bills to the cashier. How much change did he receive? What is asked: _____________________________________________________ What are given: ____________________________________________________ Operations to be used: _______________________________________________ Number Sentence: __________________________________________________ Solution:

Complete Answer: _________________________________________________ Test yourself: Choose the correct answer. 1. When converted into decimal, equals _____ a. 0.5

b. 0.2

c. 0.05

d. 0.02

2. The decimal number 0.25 is equal to a.

3. Expressed as a decimal, 1 a. 1.40

b. 1.45

equals c. 1.20

d. 1.35

4. The decimal number 5.33 is equivalent to a. 5

b. 5

c. 5

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5. In a class of 50, 20 are girls. The ratio of girls to boys is a.

Chapter 5 Understanding multiplication and division of decimals Lesson 1 Multiplying Decimals We learned that

is written as 0.4 in decimal form and

as 0.6. What is the

product of 0.4 x 0.6? Before answering the question, let us recall how we multiply fractions. = Since

(

)

(

)

, then 0.4 x 0.6 = 0.24 (24 hundredths)

0.4 1 decimal place X 0.6 1 decimal place 0.24 2 decimal places To multiply two decimals;  Multiply the factors as whole numbers.  Count the total number of decimal places of the factors.  Move the decimal point to the left of the product equal to the number of places in step 2 Other example: Decimal places of the product 0.12 x 0.2 = 0.024 2+1=3 0.12 x 0.02 = 0.0024 2+2=4 0.12 x 0.002 = 0.00024 2+3=5 Remember this: In multiplication involving decimals multiply the factors as whole numbers, The decimal point of the product is the total number of the decimal places of the factors. Practice: Find the product of each pair of decimals 1) 0.6 x 0.7 = _____ 2) 0.2 x 0.08 = _____ 3) 0.45 x 0.9 = _____ 4) 0.9 x 0.32 = _____ 5) 0.05 x 0.7 = _____ Multiplying decimals by whole numbers The following gives the equivalent of $1 in different countries. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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$1 = â&#x201A;ą48.75 = 32.90 Thai baht = 3.75 Saudi Arabian Riyal = 0.751 Euro = 0.515 UK pounds How much is $5 in pesos? In baht? In riyal? In euro? In pounds? To answer the questions, we have to multiply a whole number (5) by decimals. To find the value of $5 I pesos, 5 x 48.75 =? We can write the expression in vertical form then multiply as if multiplying whole numbers. 48.75 X 5 243.75 To find the decimal places of the product, count the number of decimal places in the decimal factor. Thus, $5 = â&#x201A;ą243.75 In multiplying whole number by a decimal, the number of decimal places of the product is equal to that of the decimal factor. Practice: Find the product 1) 12 x 0.6 = 2) 16 x 0.8 = 3) 8 x 0.07 = 4) 7 x 0.3 = 5) 6 x 0.004 = Multiplying mixed numbers Find the product: 4.2 x 3.5 4.2 (1 decimal place) X 3.5 (1 decimal place) 210 126 1 4 .7 0 ( 2 decimal places) To multiply mixed decimals, multiply as whole numbers and count the number of decimal places of the factors. Practice: 1) 0.4 x 12 = 2) 15 x 0.2 = 3) 7 x 2.5 = 4) 9 x 1.25 = 5) 2.44 x 3.56 = 6) 6 .08 x 4.5 = 7) 3.16 x 4.12 = 8) 5.55 x 4.44 = YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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9) 15 x 2.65 = 10)15.05 x 10.05 = Lesson 2 Mental Multiplication Involving Decimals Multiplying Decimals by 10, 100, and 1000 In multiplying a number by 10, we move the decimal point one place to the right In multiplying a number by 100, we move the decimal point two places to the right. To multiply a number by 1000 means to move the decimal point three places to the right. If necessary, place zeros between the decimal point and the last digit of the given number. The same rule in multiplying whole numbers by 10, 100, or 1 000 applies even when the other factor is a decimal. 1.4 x 10 = 14 1.4 x 100 = 140 1.4 x 1 000 = 1 400 Likewise, 0.07 x 10 = 0.7 0.07 x 100 = 7 0.07 x 1 000 = 70 Practice: Find the product 1) 0.6 x 10 = 2) 0.05 x 100 = 3) 0.75 x 10 = 4) 1.2 x 10 = 5) 4.1 x 100 = Multiplying decimals by 0.1, 0.01, and 0.001 In multiplying a number by 0.1, 0.01, or 0.001, we move the decimal point to the number 0ne, two, three decimal places to the left respectively. If necessary, place zeros between the decimal point and the first digit of the given number. Example: 4.2 x 0.1 = 0.42 4.2 x 0.01 = 0.042 4.2 x 0.001 = 0.0042 Practice: Multiply. 1) 0.2 x 0.1 = 2) 0.25 x 0.01 = 3) 5.3 x 0.1 = YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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4) 7.35 x 0.01 = 5) 8.24 x 0.01 = Lesson 3 Dividing Decimals To divide a decimal by a whole number, we follow these steps:  We write the decimal point above the decimal point of the dividend.  Then divide the same way as in whole numbers. Regroup when needed.  To check, multiply the quotient by the divisor. The product should be the dividend. Example: 0.64 ÷ 4 = 0.16 0.16 4 0.64 0 6 4 24 24 0 Practice: Find the quotient 1) 4 8.4 2) 3

15.9

3) 5

35.5

4) 3

4.5

5) 2

27.6

Dividing by Tenths To divide a decimal by another decimal, we change the divisor into a whole number. To change the divisor to a whole number, we multiply both numerator and denominator by a power of 10. Then divide as with whole numbers. Example: 2.5 17.5 7 2 5 17 5 17 5

move the decimal point one place to the right.

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0 Practice: 1) 1.5 3.75 2) 1.2 5.52 3) 0.2 0.72 4) 3.1 17.98 5) 3.5 21.7 Mental Division by 0.1, 0.01, and 0.001 Divide 0.8 ÷ 0.1 Since 0.1 =

, then

0.8 ÷ 0.1 = 0.8 ÷

= 0.8 x (10) = 8

And since 0.01 =

and 0.001 =

0.8 ÷ 0.01 = 0.8 ÷ 0.8 ÷ 0.001 = 0.8 ÷

then it follows that

= 0.8 x (100) = 80 = 0.8 x (1000) = 800

Remember this To divide a number by 0.1 means to multiply it by 10. To divide a number by 0.01 means to multiply it by 100. To divide a number by 0.001 means to multiply it by 1000. Practice: Find the quotient mentally 1) 5 ÷ 0.1 = 2) 0.5 ÷ 0.1 = 3) 60 ÷ 0.1 = 4) 0.063 ÷ 0.1 5) 47 ÷ 0.01 = 6) 5.2 ÷ 0.01 = 7) 0.98 ÷ 0.001 = 8) 26 ÷ 0.001 = 9) 0.09 ÷ 0.001= 10) 0.12 ÷ 0.001 = Lesson 4 Word Problems: Multiplication and Division of Decimals Problem 1 A basketball team with 12 members needs a pair of shorts for the uniform of each member. To make a pair of shorts, 0.75 m of cloth is needed. How many meters of cloth in all are needed? YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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What is asked? The total of length of cloth needed What are given? 0.75 – length of cloth needed to sew a pair of shorts 12 – number of shorts needed Operation to be used: Multiplication Solution: 0.75 X 12 150 75 9.00 Complete answer: a total of 9 m of cloth is needed Problem 2 The car of Mr. Tan can cover a distance of 9.5 kilometers (km) for every liter (L) of gasoline consumed. If Mr. Tan drove from Manila to Quezon Province covering a distance of 125.5 km, how many liters of gasoline used up by his car? What is asked? The number of liters of gasoline used up by Mr.Tan’s car What are given? 9.5 km – distance traveled by car for every liter of gasoline consumed 12.5 km – distance traveled by the car Operation to be used: Division Solution: 125.5 ÷ 9.5 = 13.21 Complete answer: the car used up 13.2 L of gasoline. Practice: Answer the problem: 1) The circumference of the Earth is 40074 km. What is the radius of the Earth if the radius is equal to the circumference divided by the product 2 x 3.14 What is asked: _____________________________________________________ What are given: ____________________________________________________ Operations to be used: _______________________________________________ Number Sentence: __________________________________________________ Solution:

Complete Answer: _________________________________________________ 2) There are 39.4 inches in 1 m. How many meters are there in 196.8 in? What is asked: _____________________________________________________ What are given: ____________________________________________________ Operations to be used: _______________________________________________ YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Number Sentence: __________________________________________________ Solution:

Complete Answer: _________________________________________________ 3) There are 3.28 feet in 1 meter. How many feet are there in 12 m? What is asked: _____________________________________________________ What are given: ____________________________________________________ Operations to be used: _______________________________________________ Number Sentence: __________________________________________________ Solution:

Complete Answer: _________________________________________________ 4) There are 12 inches in 1 ft. and there are 3 ft. in 1 yard (yd). How many yards are there in 2 250 in? What is asked: _____________________________________________________ What are given: ____________________________________________________ Operations to be used: _______________________________________________ Number Sentence: __________________________________________________ Solution:

Complete Answer: _________________________________________________ 5) If the peso (₱) – US dollar ($) exchange rate on a certain day is ₱48.95 to 1$, how many US dollars is ₱18,000.00 equivalent to? What is asked: _____________________________________________________ What are given: ____________________________________________________ Operations to be used: _______________________________________________ Number Sentence: __________________________________________________ Solution:

Complete Answer: _________________________________________________ 6) An ordinary worker is paid ₱37.50 per hour and works 8 hours per day. How much salary does he receive in a 5-day work week? YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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What is asked: _____________________________________________________ What are given: ____________________________________________________ Operations to be used: _______________________________________________ Number Sentence: __________________________________________________ Solution:

Complete Answer: _________________________________________________ 7) Mr. and Mrs. Duque paid US$9,000 for fare and hotel bills in a vacation to Europe. If total expenses amounted to ₱530,000.00 and the average exchange rate was ₱48.82 per US dollar, how much money in dollars did they have left for other expenses? What is asked: _____________________________________________________ What are given: ____________________________________________________ Operations to be used: _______________________________________________ Number Sentence: __________________________________________________ Solution:

Complete Answer: _________________________________________________ 8) The radius of Jupiter is 139 822 km. what is its circumference? What is asked: _____________________________________________________ What are given: ____________________________________________________ Operations to be used: _______________________________________________ Number Sentence: __________________________________________________ Solution:

Complete Answer: _________________________________________________ 9) A slice of cake costs ₱12.50. How many slices of cake can mother buy with ₱100? What is asked: _____________________________________________________ What are given: ____________________________________________________ Operations to be used: _______________________________________________ Number Sentence: __________________________________________________ Solution:

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Complete Answer: _________________________________________________ 10) A class of 42 pupils went on an outing to a beach. Each pupil’s initial contribution was ₱50.50. The total expenses, however, amounted to ₱2,562.00. How much should each pupil pay? What is asked: _____________________________________________________ What are given: ____________________________________________________ Operations to be used: _______________________________________________ Number Sentence: __________________________________________________ Solution:

Complete Answer: _________________________________________________ Chapter 6 Understanding Ratio, Proportion and Percent Lesson 1 The meaning of Ratio In an animal farm, there are 5 horses and 20 cows. We can compare the number of horses to the number of cows and form a ratio. A ratio is a way of comparing two numbers or quantities. The ratio of the number of horses to the number of cows is 5 to 20. There are there ways of writing a ratio: 5 to 20 or 5:20 or We read each ratio as “ five to twenty.” The numbers in a ratio are the terms of the ratio. From the given number of animals, we observe the following. The number of horses: 5(a part) The number of cows: 20 (a part) The total number of animals: 25 ( the whole) Other comparisons may also be formed from these numbers of animals. Horses : all animals 5 to 25 or 5:25 or Cows:horses

20 to 5 or 20: 5 or

( part: whole)

(part:part)

All animals: cows 25 to 20 or 25:20 or

(whole:part)

When written as a fraction, the ratio may be expressed in simplest form. This is the same as writing a fraction in the lowest terms. To express a fraction in lowest terms, we divide each number by their greatest common factor(GCF). Thus the ratio

This ratio may be interpreted as – for every one horse there are four cows. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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To express a ratio in simplest terms, write it as a fraction and divide the numbers by their GCF. Write these ratios in simplest terms: 12:30

= or 2:5 or 2 to 5

Practice Reduce each ratio in simplest form 1. 4: 10 = _____ 2.

= _____

5. EQUAL RATIOS When ratios have the same value, they are called equal ratios. Equal ratios can be written as equivalent fractions. Recall that we form equivalent fractions by multiplying or dividing both numerator and denominator by the same number. Consider the following example: =

Thus, = That is, 6:9, 2:3 and 12:18 are equal ratios. Practice Write the ratios that are equivalent to the given ratio 1. 2 : 5 = _____ 2. 4 : 3 = ______ 3. 1 : 7 = _____ 4. 18 : 12 = _____ 5. 12 : 36 = _____ Lesson 2 The meaning of proportion A proportion is a number sentence showing that two ratios are equal. The following are examples of proportions: 3 is to 5 as 9 is to 15.

3:5 = 9 : 15

How we can tell if two ratios are proportion? One method is to express the two ratios in their simplest form. The ratios form a proportion when they can be simplified into the same fraction. Example: Is

a proportion? =

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Since both fractions are reduce to lowest terms, therefore

is a proportion.

This proportion can also be written as 18: 24 = 9 : 12 The numbers 18, 24, 9, and 12 are the terms of proportion. We call the first and fourth terms, 18 and 12, the extremes. We call the second and third terms, 24 and 9, the means. Another way of testing if a number sentence is a proportion is by using the cross product rule. Two ratios form a proportion if their cross products are equal. The cross product rule states that the product of the means equals the product of the extremes. Tell whether 3:5 = 9:15 is a proportion by the cross product rule. The product of the means 5(9) = 45 The product of the extremes: 3 (15) = 45 Since the two products are equal, then 3:5 = 9: 15 Practice Tell whether these ratios are proportions by expressing them in their simplest forms. Write Yes or No. 1. 5 : 25 and 4 : 20 2. 6 : 12 and 4 : 8 3. 8 : 12 and 9 : 15 4. 20 : 25 and 36 : 45 5. 12 : 24 and 21 : 18 Missing numbers in proportion When one term in a proportion is missing, we may form equal ratios to solve for it. Example

n = 30

n=8 n=1

Practice Find the missing number in each proportion by forming equal ratios. 1. 2. 3. 4. 5.

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Lesson 3 Ratios, fractions, and percent The meaning of percent Percent is a ratio of some number to 100. The word percent means â&#x20AC;&#x153;per one hundredâ&#x20AC;?. The symbol for percent is %. A fraction whose denominator is 100 may be expressed in percent. Example = 23 % = 65 % When a fraction does not have 100 as denominator, we can still change it to percent? The answer is yes. But first we have to rename so that the denominator is 100. Example Remember the compatible numbers of 100 are 2 x 50 4 x 25 5 x 20 10 x 10 Practice Rename each fraction so that the denominator is 100. 1. 2. 3. 4. 5. Changing percent to fraction To change percent to fraction 1. We first write it as a fraction whose denominator is 100. 2. Then reduce it to lowest term by dividing both numerator and denominator by their GCF Example 55% = 36% = 70% =

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Practice Change each percent to fraction in lowest terms 1. 90% 2. 5% 3. 50% 4. 75% 5. 15% Lesson 4 Percent and decimals To change percent to decimals 1. First change to a fraction whose denominator is 100 2. Then write the fraction as a decimals Example 62% = 9% =

= 0.62 = 0.09

There is shorter way to change percent to decimal. Move the decimal point to two places to the left. Then drop off the percent sign. If there is no ones digits, we write 0 in its places. Example 62% 62 0.62 9% 09 0.09 Practice 1. 12% 2. 7% 3. 86% 4. 63% 5. 1% From decimal to percent To change decimal to percent: 1. First change to fraction with denominator of percent 100. 2. Write the fraction as percent and affix the percent symbol (%) Decimal fraction percent 0.54

54%

0.03

There is a quicker way to change decimal to percent. Move the decimal point two places to the right. Then affix the percent symbol %. Decimal percent 0.54 0.54 54% 0.03 0.03 3% YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Practice A. Change each decimal to a fraction with denominator of 100, then to percent. 1. 0.57 2. 0.43 3. 0.06 4. 0.039 5. 0.08 B. Write as percent 1. 0.4 2. 0.99 3. 0.6 4. 0.02 5. 0.25 Lesson 5 Finding the percent The percent of a number In a class of 40 pupils, 45% are boys. How many boys are there? To solve the problem, we find the percent (P ) of a number: 45% of 40 = P 45% or

is the rate ÂŽ

40 is the base(B) There are two ways to find the percent of a number. 1. Express the percent as a fraction. Then multiply by the number. 45% of 40 =

x 40 = 18

2. Express the percent as a decimal. Then multiply by the number. 45% of 40 = 0.45 x 40 = 18 There are 18 boys in the class. Let us have another example. a. 50% of 70 =

x 70 =

b. 5% of 60 =

x 60 = 3

= 35

c. 40% of 15 = 0.40 x 15 = 6 d. 8% of 90 = 0.08 x 90 = 7.2 Practice A. Solve for the percent of the number by writing the percent as a fraction. 1. 5% of 200 = _____ 2. 32% of 150 = _____ 3. 10% of 70 = _____ YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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4. 25% of 80 = _____ 5. 3% of 90 = _____ B. Solve for the percent of a number by writing the percent as a decimal 1. 50% of 86 = _____ 2. 25% of 50 = _____ 3. 20% of 80 = _____ 4. 60% of 40 = _____ 5. 15% of 300 = _____

Chapter 7 Understanding concepts in Geometry Lesson 1 Lines and angles Lines A line is a set of points on a straight path that extends indefinitely in opposite directions, as indicated by arrow heads. A line segment is part of a line that has two end points indicating a definite length. A ray is also a part of a line that has an endpoint and an arrowhead at the other end. It extends indefinitely in one direction. Beam of light extends indefinitely from a source. It best represents a ray. When lines do not meet however you extend them, they are called parallel lines. The lines of a pad paper are parallel lines. The symbol for parallel is . When lines meet at a point, they are intersecting lines. When two lines, line segments, or rays intersect such that they form a square corner, they are called perpendicular lines. The symbol for perpendicular line is ⊥. Angles When two rays meet at a point, an angle is formed. The rays form the sides of the angle;’ the point where the two rays meet is the vertex of the angle. The symbol for angle is . An angle maybe represented by its vertex, or by the three points that form it. In writing the three letters, we write the vertex at the middle. A B C The sides are ⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗ the vertex is B. Angle may be represented as ABC, CBA, or B The unit of measure of an angle is the degree (0 ). We use a protractor to measure the number of degrees of an angle. Angles maybe classified according to their measures An acute angle is less than 900 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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A right angle measures 900. Note the small square indicates 900

An obtuse angle is more than 900 and less than 1800

A straight angle measures 1800.

A reflex angle is more than 1800

Angles that have the same measure are called congruent angles. All right angles are congruent. We use the symbol m P to indicate the measure of angle P. Lesson 2 Properties of polygons A plane figures is a figure that lies in a flat surface called the plane. A plane extends indefinitely in all directions. A polygon is a closed figure in a plane that is made up of line segments that intersect only at their endpoints. The line segments of a polygon are called the sides; their endpoints are called the vertices (singular:vertex) There are special names given to some polygons. A triangle has 3 sides and 3 vertices A quadrilateral has 4 sides and 4 vertices A pentagon has 5 sides and 5 vertices A hexagon has 6 sides and 6 vertices A heptagon has 7 sides and 7 vertices An octagon has 8 sides and 8 vertices A nonagon has 9 sides and 9 vertices YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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A decagon has 10 sides and 10 vertices Properties of a triangle A triangle is a polygon having 3 sides and 3 angles. Equilateral triangle – all sides equal Isosceles triangle – two sides equal Scalene – no sides equal Acute triangle – all angles acute Right triangle – one right angle Obtuse triangle – one obtuse angle In a triangle, angles opposite equal sides are congruent. Thus an equilateral triangle is also equiangular. The angles opposite the equal sides of an isosceles triangle are also equal. A triangle may belong to two or more types of angles depending on the measures of the sides and angles. For example, a right triangle may have two equal sides. This triangle is called an isosceles right triangle. The sum of the three angles in a triangle is 180. Properties of a quadrilateral A quadrilateral is a polygon that has four sides and four angles. The different types of quadrilaterals include the following: Quadrilateral – four sides Trapezoid – a pair of parallel sides Parallelogram – 2 pairs of parallel sides Rectangle – 2 pairs of parallel sides. All angles are congruent (900) Rhombus – 2 pairs of parallel sides. All sides are congruent. Square – 2 pairs of parallel sides. All sides are congruent. All angles are congruent. Properties of regular polygons When all the sides and all the angles of a polygon are congruent, it is called a regular polygon. A regular triangle has three equal sides and three congruent angles. The sum of three angles in a triangle is 1800. The measure of each angle in a regular triangle is then m

= 60

A square has four sides and four equal angles. Thus a square is a regular quadrilateral. The diagonal of a square divides it into two triangles. The sum of the angles in a square is then (1800) 2 = 3600 Each angle in a square has a measure of m

= 900

A regular pentagon has five equal sides and angles. If we draw diagonals from one vertex to the other vertices of the polygon, how many triangles are formed? Three triangles are formed. Thus the sum of the angles in a regular pentagon is 180 0 x 3 = 5400. Each angle in a regular pentagon has a measure of m YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

= 1080 Page 76

Can you see a pattern? Suppose a regular polygon has n sides. The sum of its angles is found by the formula: (1800) ( n - 2) Each angle of a regular polygon has a measure of

(

)

A regular polygon has 15 sides. What is the sum of the angles? Find the measure of each angle. Sum of the angles = 1800 ( 15 â&#x20AC;&#x201C; 2 ) = 1800 (13) = 23400 Each angle measures =

= 1560

Test Write T if the statement is true and F if the statement is false. 1. A quadrilateral has four vertices. _____ 2. A rhombus is a square. _____ 3. The sum of the angles in a pentagon is 5400._____ 4. A regular polygon has congruent sides and angles. _____ 5. A decagon has nine sides. _____ 6. A trapezoid has no parallel sides. _____ 7. An equilateral triangle has three equal angles. ______ 8. A circle is a polygon. _____ 9. Each angle in a regular heptagon measures 1200. _____ 10. A polygon has line segments as sides. _____ Lesson 3 Circle and its parts A circle is a closed plane figure formed by the points that are the same distance from a given point called the center. When points on the circle or in its center are connected, line segments are formed. These line segments are given special names. A line segment that connects the center with a point on a circle is called the radius. A line segment that connects two points on the circle is called the chord. A chord that passes through the center of the circle is called the diameter is twice the radius. The arc is a part of the circle. The angle formed where the vertex is at the center and the sides are radii of the circle is called a central angle. D radius A C angle diameter B chord YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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test A. Write sometimes, always, or never in the blanks. 1. 1. A diameter is __________ longer than the chord. 2. A diameter is __________ longer than the radius. 3. A diameter has __________ two radii. 4. A chord is __________ longer than the radius. B. Fill in the blanks. 5. The longest chord in a circle is the __________. 6. If the radius of the circle is 5 cm, the diameter is __________ cm. 7. If the diameter of the circle is 8 cm, the radius is __________ cm. C. Name the parts of a circle 8. __________ 9. __________ 10. __________ Lesson 4 Congruent polygons Figures that have exactly the same size and shape are congruent figures. In congruent polygon, all corresponding sides and angles are congruent. This means that the length of the corresponding sides is equal and the measures of the corresponding angles are equal. We use the symbol for congruent.

Congruent polygons are found in the design of homes and other buildings. Congruent squares and rectangles are found in bathrooms: in floors of homes, school, office buildings, churches, and other structures. Congruent triangles like bridges and windows of large buildings. Remember this: Geometric figures of the same size and shape are congruent. All corresponding angles and sides of congruent figures are congruent. Do this. Look for congruent polygons in pictures in magazines. Paste the pictures on an album. Bring your album to class and explain what congruent figures are found in the pictures. Lesson 5 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Similar Figures Figures that have the same shape are similar figures. The figures may or may not have the same size. These figures are similar figures.

These figures are not similar. They do not have the same shape.

Test Answer the following questions with yes or no. if your answer is no, give an example why it is no. 1. Are all circles congruent? __________ 2. Are all circles similar? __________ 3. Are all squares congruent? __________ 4. Are all equilateral triangles similar?__________ 5. Are all rectangles similar? __________

Lesson 5 Spatial figures A figure that encloses a part of a space is called spatial figure. It is also called a solid figure. In a spatial figure, the flat surface is called faces. The line segments YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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where the faces meet are called the edges. The point where three or more edges meet is called the vertex. Many familiar objects we see around us have the shape of a spatial figure. These objects are three-dimensional.

Cube

rectangular prism

Cylinder

cone

A solid figure whose sides are polygons is called a polyhedron. Two common polyhedrons are the prisms and the pyramids. Prisms A prism has the following characteristics:  The two congruent polygons that lie parallel to each other are called the bases. A prism is named by the shape of its bases.  Each face of the prism that is not a base is called a lateral face. Each lateral face of a prism is formed by a parallelogram.

Triangular prism

pentagonal prism

An oblique prism Pyramids A pyramid has the following characteristics:  A pyramid has only one base and it is always a polygon. The polygon at the base gives the name of the pyramid.  All the faces of the pyramid except for the base intersect at one point called the vertex. They are called lateral faces. Each lateral face of a pyramid is a triangle.  The lateral faces meet at a line segment called the lateral edge.  The altitude of a pyramid is a line segment from the vertex perpendicular to the base. A triangular pyramid has four faces, four vertices, and six edges. A square pyramid has five faces, five vertices, and eight edges.

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Triangular pyramid

square pyramid

Other spatial figures There are spatial figures that have curved surfaces. They do not have faced or edges A cone has one curved surface, one circular base, and one vertex. A cylinder has one curved surface, two circular bases, and no vertices.

A sphere has no vertices or edges. All of the points of a sphere have the same distance from a point called the center.

Test True or false write T if the statement is true about the spatial figure, F if it is not true. 1. The prism has two bases. 2. The prism has the same number of lateral faces as vertices. 3. The pyramid has only one vertex. 4. The lateral faces of a pyramid are rectangles. 5. The pyramid has always an even number of edges. 6. The lateral edge of a right prism is its altitude. 7. The cylinder has an edge. 8. The lateral faces of a right prism are rectangles. 9. A sphere has no vertices, no edges, and no faces. 10. A cone has more than one vertex. Test Choose the best answer. Circle the letter of your choice. 1. A triangle is equilateral if __________. a. Two sides are equal. b. One side is perpendicular to another. c. Three sides are equal. d. One angle is a right angle. 2. A triangle is isosceles if __________. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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a. Two sides are equal b. One side is perpendicular to another. c. Three sides are equal. d. One angle is a right angle 3. One angle of an equiangular triangle is __________. a. 450 b. 600 c. 750 d. 900 4. The measure of an angle in a regular decagon is __________. a. 1200 b. 1440 c. 1500 d. 1000 5. The sum of the angles in a regular heptagon is __________. a. 9000 b. 12600 c. 10800 d. 7600 6. A regular polygon has 20 sides. Each angle measures __________. a. 32400 b. 1500 c. 1620 d. 1800 7. Each angle in a regular octagon measures __________. a. 1350 b. 1400 c. 1200 d. 1500 8. An angle that measures 750 is __________. a. Acute b. Right c. Obtuse d. Reflex 9. Two points on a circle are joined by a line segment called __________. a. Diagonal b. Center c. Chord d. Arc 10. The distance from the center to any point on the circle is called the __________. a. Diameter b. Radius c. Chord YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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d. Arc 11. The lateral edges of a pyramid meet at the __________ a. Center b. Vertex c. Face d. Altitude 12. Which solid figure has no vertices but have two circular bases? a. Cone b. Cylinder c. Sphere d. Prism 13. The vertices of a rectangle measure __________ a. 600 b. 900 c. 1200 d. 1500 14. The sum of the angles of a quadrilateral is __________. a. 1800 b. 3600 c. 5400 d. 7300 15. Each angle of an equilateral triangle is equal to __________. a. 600 b. 900 c. 5400 d. 7200 Chapter 8 Understanding Measurement Lesson 1 The metric units of length, perimeter, area and volume The standard unit of length in the metric system is the meter(m). The width of a door is about 1 m. A smaller unit of length is the centimeter (cm). The width of your fingernail is about 1 cm. centimeter is used to measure objects smaller than a meter. 100cm = 1 m A much smaller unit of length is a millimeter (mm). a P5 coin is about 2mm thick. 10 mm = 1 cm. For long distances like the distance between two towns, we use the kilometer (km). When you walk for 15 minutes, you cover about 1 km. 1000 m = 1 km. The perimeter is the length around a geometric figure. The unit of measure for the perimeter is the unit of measure for length â&#x20AC;&#x201C; cm, m, or km. Measures of Area YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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The area is a measure of the space enclosed by plane geometric figures. The standard unit for measuring area is a square whose side is a unit length. If the length of one side of the square is 1 cm, then the unit area is: Area = 1 cm x 1 cm = 1 square centimeter 1cm = sq cm 1 cm 2 = 1 cm If the unit of length is a meter, then the unit area is 1 sq m, or 1 m 2. The sq m or m2 is used for measuring areas of figures whose length range up to several meters. We speak of the area of a classroom in square meter. The area of a residential lot is best expressed in sq m. If the unit of length is kilometer, then the unit area is 1 sq km or 1 km 2the sq km is used for larger areas. Examples include the area of a city, a province, a country, an ocean, or a continent. Measures of volume Volume is the measure of the space enclosed by a geometric solid or spatial figure. The unit for measuring volume is a cube of a unit length. The volume is in cubic units. The volume of a cube whose side is 1 cm is Volume = 1 cm x 1 cm x 1 cm = 1 cubic centimeter 1 cm = 1 cu cm 1 cm 3 = 1 cm The cubic centimeter (cm3) is used for measuring small geometric solids like a shoebox or a gift package. If the length of one side of the cube is 1 metre, the volume is in cubic metres, cu m or m3. The cubic metre is used for measuring relatively large volumes like the amount of water consumed by a family, a school, or a factory. If the length is in kilometers, then the volume is in kilometers, cu km or km 3. The cubic kilometers is used for measuring very large volumes like the volume of water in an ocean and the volume of air in the atmosphere. Test A. Write mm, cm, m or km to choose the unit of measure of length. 1. Thickness of a plate _____ 2. Length of a blackboard _____ 3. Width of a shoe box _____ 4. Distance to the next city _____ 5. Depth of a swimming pool _____ 6. Distance of the earth _____ 7. Diameter of a basketball _____ 8. Thickness of a coin _____ 9. Length of a pencil _____ 10. Width of a window _____ YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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B. Write cm2, m2, of km2 to choose the best unit of measure of area. 11. Area of vegetable garden _____ 12. Area of television screen _____ 13. Area of basketball court _____ 14. Area of the Pacific Ocean _____ 15. Area of a pad paper _____ C. Write cm3, m3, of km3 to choose the best unit of measure of volume. 16. Volume of medicine bottle. _____ 17. Volume of a room _____ 18. Volume of a swimming pool _____ 19. Volume of the earth _____ 20. Volume of powdered milk in a can _____

Lesson 2 Perimeters and areas of polygons Perimeter is the length around the figure. In a rectangle, two pairs of sides are parallel and congruent. Perimeter = length (l) + length (l) + width (w) + width (w) = 2 length + 2 width = 2L + 2w Formulas for perimeter of some polygons a. For a rectangle Perimeter = 2 length + 2 width or P = 2L + 2w b. A square is a polygon whose four sides are equal Perimeter = side + side + side + side or P = 4S A square is 3 cm on one side. What is the perimeter? P = 4S = 4 ( 3 cm ) = 12 cm c. A triangle has three sides. For triangle a

b Perimeter = side + side + side =a+b+c Find the perimeter of a triangle whose sides are 4 cm, 7 cm, and 5 cm P = 4 cm + 5 cm + 7 cm = 16 cm d. A trapezoid has one pair of parallel sides. a YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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b c

P=a+b+c+d Find the perimeter of a trapezoid with the given dimensions. 9 cm 5 cm

6 cm

7 cm P = 9 cm + 6 cm + 7 cm + 5 cm = 27 cm Solve the following problems 1. A rectangular residential lot is 15 m long and 12 m wide. Find its perimeter. 12 m 15 cm 2. A rectangular kitchen is made of tiles; 15 tiles in its length and 10 tiles in its width, making a total of 150 tiles. Each tile is a square of 30 cm in width. Find the perimeter of the kitchen in centimeters.

15 tiles 30 cm 30 cm 3. A square is 5 cm on a side. Five squares are arranged in the following manner. Find its perimeter. 5 cm 4. Six equilateral triangles are arranged like the one below. If one side of the triangle is 8 cm, what is the perimeter of the figure? 8 cm YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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5. A square and a right triangle are arranged to form a trapezoid. If ̅̅̅̅ = 6 cm, ̅̅̅̅ = 8, and ̅̅̅̅ = 10 cm; find the perimeter of the trapezoid. A

10 cm

6 cm

E 8 cm D C 6. Two congruent right triangles are arranged to form a rectangle. If the two sides are 3 cm and 4 cm respectively, find the perimeter of the rectangle. 3 cm 4 cm 7. Five congruent squares are arranged like the figure below. One side of the square is 7 cm. find the perimeter of the figure.

8. Five rectangles whose dimensions are shown here are placed together to form one big rectangles. Find the perimeter of the big rectangle. 8 cm

8 cm

8cm

5 cm 5 cm 9. A trapezoid is formed by a rectangle and two congruent right triangles with the dimensions as shown. What is the perimeter of the trapezoid? 10 cm 8 cm 5 cm 10. A rectangle and an isosceles triangle are arranged like that shown below. Find the perimeter of the figure. 10 cm 6 cm

6 cm

12 cm 6 cm Areas of Some Polygons To find the area of a figure, we find the number of square units inside the figure. In a metric system, the standard units we use to measure area are the square YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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millimeters (mm2), square centimeters ( cm2), square meters (m2), or square kilometers (km2) Formulas to find the areas of common figures a. The area of a rectangle is the product of the length (l) and the width (w).

Area = length x width The floor area of the classroom is 10 m long and 7 m wide. What is its area? Area of classroom = 10 m x 7 m = 70 m2 The area of a rectangle is 18 m2. If one side is 3 m, find the dimension of the other side. Given: area = 18 m2 Let the width = 3 m Area = l x w 18 = length x 3 m Length =

=6m

b. The area of a square is the square of one side (s).

Area = side x side A = s2 Find the area of a square handkerchief whose side has a length of 35 cm. Area = s2 = (35 cm)2 = 1225 cm2 c. The area of a triangle is one-half the product of the base (b) and the altitude (h). The altitude is the perpendicular line from a vertex to the line containing the base. The height is the length of the altitude. h b Area = ( base x height) = bh Find the area of a triangle whose base is 13cm and height is 7 cm. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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A = bh = (13 cm) (7 cm) = 45.5 cm2 d. The area of a parallelogram is the product of the base and the height or altitude h. as in a triangle, the altitude of a parallelogram is the line perpendicular from a vertex to the line containing the base. h b Find the area of a parallelogram whose base is 7 cm and height is 4 cm. A = bh = (7 cm)(4 cm) = 28 cm2 e. The area of a trapezoid is one-half the product of the height and the sum of the two bases. That is, (

Area =

)

Example In trapezoid MELT, b1 = 5 cm, b2 = 3 cm, and h = 2.5 cm. find the area. What is asked? Area of MELT What are given? b1 = 5 cm, b2 = 3 cm, and h = 2.5 cm Solution: (

Area = (

) )(

)

Answer: the area of the trapezoid is 10 cm2. Example 2 The area of a trapezoid is 30 cm2. The two bases are 6 cm and 4 cm. what is the height or altitude? What is asked? Altitude, h What are given? Area = 30 cm2, b1 = 6 cm, b2 = 4 cm Solution: We substitute the given measures in the formula for the area of a trapezoid. A=

(

30 =

(

) )

(

)

30 = 5h h=

= 6 cm

Answer: the height of the trapezoid is 6 cm. Test Answer the following 1. Find the area of a trapezoid whose bases are 3 cm and 4 cm, and altitude is 5 cm. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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What is asked? _____________________________________________ What are given? ____________________________________________ Operations to be used: _______________________________________ Solution:___________________________________________________

Complete answer: ____________________________________________ 2. The area of a trapezoid is 35 cm2. If the bases are 6 cm and 8 cm, find its altitude. What is asked? _____________________________________________ What are given? ____________________________________________ Operations to be used: _______________________________________ Solution:___________________________________________________

Complete answer: ____________________________________________ 3. The area of a trapezoid is 32 cm2. The altitude is 4 cm and one of the bases is 9 cm. find the length of the other base. What is asked? _____________________________________________ What are given? ____________________________________________ Operations to be used: _______________________________________ Solution:___________________________________________________

Complete answer: ____________________________________________ Lesson 3 Circumference and Area of a Circle The circumference of a circle The perimeter of a circle is called the circumference. The circumference is the distance around the circle. If you wrap around a can of juice and stretch the string to make a straight line, its length is the circumference. To actually find the circumference of a round object, we may use a tape measure and wrap it around the object. The circumference of a circle is related to its diameter. The longer the diameter, the greater is the circumference. The Greek mathematician, Archimedes (287-212 B.C.) discovered that the ratio of the circumference C to the diameter d is constant. This constant is denoted by the Greek letter ( pi, pronounced as â&#x20AC;&#x153;pieâ&#x20AC;?) and is approximately equal to 3.14. C= YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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The circumference of a circle is about 3.14 times as long as the diameter. The ratio of the circumference ( c ) to the diameter (d) of any circle is constant ( ).

Example 1 What is the circumference of a circle whose diameter is 5 m. What is asked? Circumference, C What are given? Diameter, d = 5 m Solution: C = 3.14 d = 3.14 ( 5 cm) = 15.7 Answer: the circumference of the circle is 15.7 m The diameter is twice the radius, or d = 2r. hence the circumference is C= =2( ) C=2 Where r is the radius Example 2 Find the circumference of a circle of radius 3 cm. What is asked? Circumference, C What is given? Radius, r = 3 cm Solution: C=2 = 2 (3.14)(3cm) = 18.84 Answer: the radius is 18.84 cm Example 3 If the circumference of a circle is 12 cm, find its diameter and radius. What are Asked? Diameter, d and radius, r What is given? Circumference, C = 12 cm Solution: C= = 3.82 cm

Answer: the diameter is 3.82 and the radius is 1.91 cm. Test Solve the following problems. 1. The circumference of a circle is 282.6 km. find its diameter and radius. What is asked? _____________________________________________ What are given? ____________________________________________ Operations to be used: _______________________________________ YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Solution:___________________________________________________

Complete answer: ____________________________________________ 2. A large can of juice has a diameter of 15 cm. what is the circumference of the can? What is asked? _____________________________________________ What are given? ____________________________________________ Operations to be used: _______________________________________ Solution:___________________________________________________ Complete answer: ____________________________________________ 3. The circumference of a small pizza pie is 62.8 cm. what is the diameter of a pizza pie? What is asked? _____________________________________________ What are given? ____________________________________________ Operations to be used: _______________________________________ Solution:___________________________________________________

Complete answer: ____________________________________________ 4. The diameter of a compact disc or CD is 12 cm. find the circumference? What is asked? _____________________________________________ What are given? ____________________________________________ Operations to be used: _______________________________________ Solution:___________________________________________________

Complete answer: ____________________________________________ 5. The radius of a coin is about 1.3 cm. find its diameter and its circumference. What is asked? _____________________________________________ What are given? ____________________________________________ Operations to be used: _______________________________________ Solution:___________________________________________________

Complete answer: ____________________________________________ 6. A bicycle wheel has a radius of 34 cm. what is the circumference of the wheel. What is asked? _____________________________________________ What are given? ____________________________________________ YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Operations to be used: _______________________________________ Solution:___________________________________________________

Complete answer: ____________________________________________ 7. The diameter of the moon is 3 475 km. what is the moonâ&#x20AC;&#x2122;s circumference? What is asked? _____________________________________________ What are given? ____________________________________________ Operations to be used: _______________________________________ Solution:___________________________________________________

Complete answer: ____________________________________________ 8. The circumference of the earth is around 40 000 km. What is asked? _____________________________________________ What are given? ____________________________________________ Operations to be used: _______________________________________ Solution:___________________________________________________

Complete answer: ____________________________________________

Area of a Circle

Half of the circumference YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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To find the area of a circle, we can cut a circle into equal sections as shown in the figure below. These congruent sections can then be rearranged to form the figure that is very much like a parallelogram. The length of the base of this â&#x20AC;&#x153;parallelogramâ&#x20AC;? is about one-half the circumference of the circle. The height of this parallelogram is equal to the radius of the circle. We can now derive the area of a circle from the area of a parallelogram. Area = base x height = Cxr =

(

)

= A= The area of a circle is equal to the square of the radius multiplied by the constant . If = 3.14, then, Area = (3.14)r2 Examples Problem 1 Find the area of a circle whose radius is 6 cm. What is asked? Area , A What is given? Radius, r = 6 cm Solution: A = = (3.14) (6)2 = 3.14 x 36 = 113.04 cm Answer: the area of the circle is 113.04 cm2 Problem 2 Diameter of a circle is 8 m. find the area. What is asked? Area, A What is given? Diameter, d = 8 Solution: since d = 2r, then r = r= =4m A= = (3.14) (4)2 = 3.14 x 16 = 50.24 m2 Answer: the area is 50.24 m2 Problem 3 The circumference of a circle is 20 cm. find the area of the circle. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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What is asked? Area of the circle What is given? Circumference, C = 20 cm Solution: We need to solve for the radius first since C = 2 ( ) Area = = 3.14 ( 3.18)2 = 31.75 cm2 Answer: the area of the circle is 31.75 cm2. Practice Answer the following problem. 1. A plate has a diameter of 26 cm. what is the area of the plate? What is asked? _________________________________ What is given? _________________________________ Solution:

Complete answer: _______________________________ 2. A round table top has a radius of 65 cm. what is the area of the table top? What is asked? _________________________________ What is given? _________________________________ Solution:

Complete answer: _______________________________ 3. A round mirror has a radius of 50 cm. find the area of the mirror. What is asked? _________________________________ What is given? _________________________________ Solution:

Complete answer: _______________________________ 4. The circumference of a circle is 10 . Find the area. What is asked? _________________________________ What is given? _________________________________ Solution:

Complete answer: _______________________________ YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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5. The lens of a microscope has a diameter of 6 mm. find the area of the lens of the microscope. What is asked? _________________________________ What is given? _________________________________ Solution:

Complete answer: _______________________________ 6. The radius of a circular swimming pool is 4 meters. Find the area of the pool. What is asked? _________________________________ What is given? _________________________________ Solution:

Complete answer: _______________________________ 7. The diameter of a basketball ring is 45.7 cm. what is the area of the basketball ring? What is asked? _________________________________ What is given? _________________________________ Solution:

Complete answer: _______________________________ 8. Find the circumference and area of the circle whose radiuys is 25 cm. What is asked? _________________________________ What is given? _________________________________ Solution:

Complete answer: _______________________________ 9. Find the circumfenece and area of the circle whose diameter is 14 m. What is asked? _________________________________ What is given? _________________________________ Solution:

Complete answer: _______________________________ Lesson 4 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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The surface Area of a Rectangular Prism The surface area of a polyhedron is the sum of the areas of its faces. The unit of measure is the square units. In a rectangular prism, each face is a rectangle. Since a rectangular prism has six faces â&#x20AC;&#x201C; lateral faces and two bases, then its surface area is the sum of the six rectangles that compose it. 3 cm 2 cm 4 cm The surface area of the rectangular prism is Top area = 4 x 2 = 8 Back area = 4 x 3 = 12 Bottom area = 4 x 2 = 8 Front Area = 4 x 3 = 12 Left side Area = 3 x 2 = 6 Right side area = 3 x 2 = 6 Area of rectangular prism = 52 A cube is a rectangular prism whose faces are squares. So we only need to know the lenth of one edge to find its surface area. Find the surface area of the cube at the right. One square has an area of 5 x 5 = 25 cm2. Since there are six faces of the cube, then the surface area of the cube is 7 x 25 = 150 cm2

Practice 1. 4m 4m 4m 2.

4m 1m 7m Lesson 5 The volume of a rectangular Prism

The volume is the amount of space that a spatial figure contains. This unit for measuring volume is a cube of unit length. If the unit length is centimeter (cm), then, YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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the unit of volume is cubic centimeter (cm3). The volume of a right prism is the area of the of the base multiplied by the height h. In a right rectangular prism, the base is a rectangle and the lateral edge is its height. h w l The area of a rectangle is the product of its sides l x w . the height of a rectangular prism is h. the volume of a right rectangular prism is Volume = V = l x w x h

15 cm 10 cm 25 cm The volume of the rectangular prism is 25 cm x 10 cm x 15 cm = 3750 cm3 The volume of a cube whose lateral edge is s cm is then Volume = V = s3 The volume of a cube whose lateral edge is 4 cm is V = 4 cm x 4 cm x 4 cm = (4 cm)3 = 64 cm3 Problem: A road is to be made of concrete. It is 10 m wide and 50 cm deep. How much concrete is needed for a portion of the road that is km long? First we need to rewrite each measure in the same units. We express our measures in cubic meters. Since 100 cm = 1 m, then 50 cm = m Since 1 km = 1 000 m, then km = 500 m Then we now have the following dimensions: Length = km = 500 m Width = 10 m Height = 50 cm = m The volume of the portion of the road: Volume = length x width x height = 500 m x 10 m x m = 2 500 m3 or 2 500 cubic meters Practice YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Find the volume of the rectangular prism 1. Length _____ Width _____ 4 cm Height _____ 3 cm Volume: _____ 5 cm 2. Length _____ idth _____ Height _____ Volume: _____

3 cm 4 cm 12 cm

3. Length _____ Width _____ Height _____ Volume: _____

10 cm 1cm 7 cm

4. Length : 6 m Width: 4 m

Height 3 cm Volume: __?___ 5. Length: 4 cm Width: 4 cm Height: 4 cm Volumne: __?___

Lesson 6 Temperature Do you sometimes listen to the weather reports over the radiuo or watch the weather forecast on television? Have you noticed that the reports usually include temperature readings in cities around the Philippines or around the world? Look at the sample weather forecast for selected cities in the Philippines. Cities

Metro Manila

24-Hr Weather, Winds & Sea Condition May 09 240C Light to modrate to East

YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

Extended Weather Outlook May May May May 10 11 12 13 0 0 0 25 C 25 C 24 C 240C to to to to Page 99

330C

Bagiuo City

Tagaytay City Puerto Princessa

160C to 210C 230C to 320C 260C to 320C

Manila Bay: Moderate to occ. Strong Moderate to occ. Strong EastSoutheast Light to modrate East-Moderate Light to modrate Northeast-Variable Coastal waters: slight to moderate

340C

350C

340C

170C to 240C 220C to 330C 250C to 320C

170C to 250C 210C to 320C 240C to 330C

170C to 250C 210C to 320C 240C to 320C

160C to 240C 220C to 310C 240C to 320C

Metro Cebu 250C to 330C

Light to modrate Northeast-Variable Coastal waters: slight to moderate

250C to 330C

240C to 320C

240C to 330C

Cagayan de Oro city

250C to 330C

Light to modrate Northeast-Variable Coastal waters: slight to moderate

240C to 320C

230C to 320C

220C to 320C

Metro Davao

240C to 330C

Light to modrate Northeast-Variable Coastal waters: slight to moderate

240C to 320C

230C to 330C

230C to 320C

These expected temperature readings tell us whether it will be cold or warm in particular places. Temperature is a measure of how cold orhot an object is. A glass of orange juice is cold; a cup of cholate drink is hot. If an object is cold, we say it has a loew temperature. If it is hot, we say it has a high temperature. Instrument for measuring temperature consist of column of liquid mercury enclosed in glass tube. Mercury expandfs or rises in the tube whenthe temperature is high and contracts or goes down when the temperature is low. The instrument is called thermometer. The emptytop portion is a vacuum. The mercury rises if dipped in a hot object and goes down if dipped in a cold object. The column of mercury serves as the reference in the scale to be used. The most commonly used scales for measuring temperature are the degrees Celcius (0C) and the degrees Fahrenheit (0F). in the metric system, the standard unit for measuring temperature is the degrees Celcius. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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The Celsius and Fahrenheit In the celsius scale, which is used in the Philippines and in many other countries, the freezing point of water is 00C while the boiling point is 1000C. The Fahrenheit scale is mused in the USA in the United Kingdom. In this scale, the freezing point of water is 320F while the boiling point of water is 2120F. Since the temperature in cities around the world is sometimes reported in the Celsius or in Faherenheit scale, it would be useful to know how to convert one to the other. The conversion formula is (

)

For example, change 860F to the Celsius scale. (

) (

)

= 300C Change 300C to the Fahrenheit scale.

= 9 (6) + 32 = 860F A speciall kind of thermometer is used to measure the temperature of the human body. It has a neck to constrict the flow of mercury. This helps in reading the thermometer after the temperature of the body has been taken because the level of mercury remains the same. In a body thermometer, the scale is from 35 0C to 420C only. The scale is narrow in range because the temperature of the human body cannot go lower than 350C nor heigher than 420C. the body lies at those temperatures. The normal body temperature is 370C, you can have slight fever, and when it reaches 390C, you have a fever. Test A. Tell whether each temperature readinbg is cold, warm, or hot. 1. 50C 2. 450C 3. 750C 4. 100C 5. 330C 6. 820C 7. 350C YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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8. 600C 9. 1200C 10. 1600C B. Change the temperature in the Fahrenheit scale to Celsius scale or in the Celsius scale to Fahrenheit scale 1. 750C 2. 200C 3. 1040C 4. 400C 5. 680C C. Solve the following problems 1. On a summer day, the temperature in New York is reported as 90 0F. what is it in degrees Celsius? 2. At 2 oâ&#x20AC;&#x2122;clock a.m. the temperature in Bagiuo city is reported to be 13 0C. what is it in degrees Fahrenheit?

Chapter 9 Reading and Interpreting Line Graphs

temperature (0C)

Line graphs show the amount of change over a certain period of time. It is used in making comparisons. A line graph has a scale on the horizontal axis and the vertical axis. The horizontal axis shows the period of time. The vertical axis displays the amount of change. To read a line graph, we do the following steps: 1. Find the given number or time on one axis. 2. Find the unknown number or time on the other axis. Answer each question using the line graph below. 36 35 34 33 32 31 30 29 28 27 Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sept

Oct

Nov

Dec

1. Which is the warmest month? May â&#x20AC;&#x201C; it has the highest point on the graph. 2. Which months are the coldest? January and December. They have the lowest points on the graph 3. Which months have about the same average temperature? August, September, October, and November. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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4. Between what months was the highest increase in temperature? February and March – the line between the dots is the longest and steepest. Practice Read the graph and answer the questions that follow: Mila’s savings from June to December 250 210

200 175 150

160

150

145

145 130

100

0 Jun

Jul

Aug

Sept

Oct

Nov

Dec

Months

1. 2. 3. 4. 5.

Which month did Mila save the least? How much did she save on this month? Which months did Mila save the same amount? How much did she save on these months? What is the highest amount that Mila save? Lesson 2 Constructing the line Graph

To construct a line graph, we follow these steps:  Use the data from the given table to determine the appropriate scale.  Draw and label the scale of the horizontal axis ( )  Draw and label the scale of the vertical axis ( ) at the near left. Start at 0.  Locate the dots on the graph.  Connect the points or data in order with line segment.  Write the title of the line graph. We usually set the time data on the horizontal axis and the data that change over time on the vertical axis. In marking the scale of the vertical axis, the scale should begin with zero. Choose an interval that will best represent the data and mark off equal spaces along the vertical line. Be sure to fit on the graph the lowest and the YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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highest numbers. To save space, we may draw zigzag line to represent a jump in the interval from 0 to the lowest data value. Let us now construct the line graph of the table below Air temperature in Baguio City Time Temperature ( 0C) 12 midnight 15 2 a.m. 13 4 a.m. 11 6 a.m. 14 8 a. m. 17 12 n.n. 23 The table shows the temperature varies with time. The line graph is the best presentation for this type of data. First we look at the data. Time ranges from 12 midnight to 12 noon. The temperature is from 11 to 23 degrees. C Draw the horizontal line and label it time. Under the line, we write the periods of time. Draw vertical line and set up the temperature scale starting at 0. 25

Temperature ( 0)

0 12:00 midnight

2:00

4:00

6:00

8:00

10:00

12:00 noon

Time

Let us answer some questions about the graph 1. What was the lowest temperature between 12 midnight and 12 noon? 11 0C 2. At what time did the lowest temperature occur? At 4 a.m. 3. Can you tell the temperature at 9 a.m.? To estimate the temperature at a given time, we draw a line vertical designating 9 a.m. then draw a horizontal line from the graph to the temperature scale and read the corresponding scale. At 9 a.m. the temperature was 18.5 0C. Practice Draw a line graph for each set of data. Then answer the questions that follow. YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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A. Average grade of grade 5 students in Math Grading period Average grade 1st 78 nd 2 82 3rd 80 4th 83 Use 1-grade interval starting from 77 on the vertical axis. 1. Which grading period has the lowest average grade? 2. What is the lowest average grade? 3. Which grading period has the highest average grade? 4. What is the highest average grade?

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