Multiplying and Dividing Fractions Chapter 2 Multiplying and Dividing Fractions © 2010 Pearson Education, Inc. All rights reserved.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.1 Basics of Fractions Objectives 1. Use a fraction to show which part of a whole is shaded. 2. Identify the numerator and denominator. 3. Identify proper and improper fractions. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Copyright © 2010 Pearson Education, Inc. All rights reserved. One way to write parts of a whole is with fractions. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 Identifying Fractions Write fractions for the shaded and unshaded portions of each figure. a. b. The figure has 8 equal parts. There are 5 shaded parts. shaded portion unshaded portion The figure has 12 equal parts. There are 6 shaded parts. shaded portion unshaded portion Copyright © 2010 Pearson Education, Inc. All rights reserved.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 Representing Fractions Greater Than 1 Use a fraction to represent the shaded part of each figure. a. b. An area equal to 7 of the ¼ parts is shaded. Write this as An area equal to 8 of the 1/6 parts is shaded. Write this as Copyright © 2010 Pearson Education, Inc. All rights reserved.

Copyright © 2010 Pearson Education, Inc. All rights reserved. In the fraction ¾, the number 3 is the numerator and the 4 is the denominator. The bar between the numerator and the denominator is the fraction bar. Numerator Fraction bar Denominator Copyright © 2010 Pearson Education, Inc. All rights reserved.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Identifying Numerators and Denominators Identify the numerator and denominator in each fraction. a. b. Numerator Denominator Numerator Denominator Copyright © 2010 Pearson Education, Inc. All rights reserved.

Copyright © 2010 Pearson Education, Inc. All rights reserved. Proper Fractions Improper Fractions Copyright © 2010 Pearson Education, Inc. All rights reserved.

Classifying Types of Fractions Parallel Example 4 Classifying Types of Fractions a. Identify all proper fractions in this list. Proper fractions have a numerator that is smaller than the denominator. The proper fractions are shown below. b. Identify all the improper fractions in the list above. A proper fraction is less than 1. An improper fraction is equal to or greater than 1. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.2 Mixed Numbers Objectives 1. Identify mixed numbers. 2. Write mixed numbers as improper fractions. 3. Write improper fractions as mixed numbers. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.2- 10

Copyright © 2010 Pearson Education, Inc. All rights reserved. Writing a Mixed Number as an Improper Fraction Change 3 ½ to an improper fraction. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.2- 11

Copyright © 2010 Pearson Education, Inc. All rights reserved. Use the following steps to write a mixed number as an improper fraction. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.2- 12

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 Writing a Mixed Number as an Improper Fraction Write as an improper fraction (numerator greater than denominator). Step 1 Multiply 5 and 9. Step 2 Add 8. The numerator is 53. 45 + 8 = 53 Step 3 Use the same denominator. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.2- 13

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.2- 14

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 Writing Improper Fractions as Mixed Number Write each improper fraction as a mixed number. a. Whole number part Divide 14 by 3. 12 2 Remainder The quotient 4 is the whole number part of the mixed number. The remainder 2 is the numerator of the fraction, and the denominator stays as 3. Remainder Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.2- 15

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 continued Writing Improper Fractions as Mixed Number Write each improper fraction as a mixed number. b. Whole number part Divide 48 by 6. 48 Remainder Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.2- 16

Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.3 Factors Objectives 1. Find factors of a number. 2. Identify prime numbers. 3. Find prime factorizations. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.3- 17

Copyright © 2010 Pearson Education, Inc. All rights reserved. Numbers that are multiplied to give a product are called factors. Because 2  8 = 16, both 2 and 8 are factors of 16. The numbers 1 and 16 are also factors of 16, because 1  16 = 16. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.3- 18

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 Using Factors Find all possible two-number factorizations of each number. a. 14 b. 80 1  14 = 14 2  7 = 14 The factors of 14 are 1, 2, 7, and 14. 1  80 = 80 2  40 = 80 4  20 = 80 5  16 = 80 8  10 = 80 The factors of 80 are 1, 2, 4, 5, 8, 10, 16, 20, 40 and 80. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.3- 19

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.3- 20

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 Identifying Composite Numbers Which of the following numbers are composite? a. 8 b. 13 c. 36 8 has factors of 2 and 4, numbers other than 1 or 8. The number is composite. 13 has only two factors, 13 and 1. It is not composite. A factor of 36 is 6, so 36 is composite. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.3- 21

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.3- 22

Which of the following numbers are prime? 3 7 9 13 29 33 Parallel Example 3 Finding Prime Numbers Which of the following numbers are prime? 3 7 9 13 29 33 The number 9 can be divided by 3, so it is not prime. Also, 33 can be divided by 3 so it is not prime. The other numbers can only be divided by themselves and 1, so they are prime. 3, 7, 13 and 29 are prime. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.3- 23

Copyright © 2010 Pearson Education, Inc. All rights reserved. The prime factorization of a number can be especially useful when we are adding or subtracting fractions and need to find a common denominator or write a fraction in lowest terms. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.3- 24

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 Determining the Prime Factorization Find the prime factorization of 24. Try to divide 24 by the first prime, 2. 24 ÷ 2 = 12 Try to divide 12 by the prime, 2. 12 ÷ 2 = 6 Try to divide 6 by the prime, 2. 6 ÷ 2 = 3 The prime factorization of 24 is 2  2  2  3. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.3- 25

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 5 Factoring by Using the Division Method Find the prime factorization of 56. Divide 56 by 2 (first prime) Divide 28 by 2. Divide 14 by 2. Divide 7 by 7. Continue to divide until the quotient is 1. The prime factorization of 56 is Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.3- 26

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 5 Using Exponents with Prime Factorization Find the prime factorization of 175. Divide 175 by 5. Divide 35 by 5. Divide 7 by 7. Continue to divide until the quotient is 1. The prime factorization of 175 is Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.3- 27

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 6 Factoring by Using a Factor Tree Find the prime factorization of 40. 40 = Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.3- 28

2.4 Writing a Fraction in Lowest Terms Objectives 1. Tell whether a fraction is written in lowest terms. 2. Write a fraction in lowest terms using common factors. 3. Write a fraction in lowest terms using prime 4. Determine whether two fractions are equivalent. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.4- 29

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.4- 30

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 Understanding Lowest Terms Are the following fractions in lowest terms? a. b. The numerator and denominator have no common factor other than 1, so the fraction is in lowest terms. The numerator and denominator have a common factor of 7, so the fraction is not in lowest terms. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.4- 31

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 Writing Fractions in Lowest Terms Write each fraction in lowest terms. a. b. The greatest common factor of 27 and 42 is 3. Divide both numerator and denominator by 3. Divide both numerator and denominator by 10. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.4- 32

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 continued Writing Fractions in Lowest Terms Write each fraction in lowest terms. c. Suppose we thought that 4 was the greatest common factor of 12 and 96. But is not in lowest terms, because 3 and 24 have a common factor of 3. The fraction could have been written in lowest terms in one step by dividing by 12. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.4- 33

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.4- 34

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Using Prime Factors Write each fraction in lowest terms. a. b. c. 1 Write the prime factorization of both numerator and denominator. Divide both numerators and denominators by any common factors. Write a 1 by each factor that has been divided. 1 Multiply the remaining factors in both numerator and denominator. 1 1 1 1 1 1 1 1 1 1 1 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.4- 35 1 1 1

Copyright © 2010 Pearson Education, Inc. All rights reserved. Writing fractions in lowest terms using prime factors. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.4- 36

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 Determining Whether Two Fractions Are Equivalent Determine whether each pair of fractions is equivalent. In other words, do both fractions represent the same part of a whole? a. 1 Equivalent 1 1 1 1 1 1 1 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.4- 37

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 continued Determining Whether Two Fractions Are Equivalent Determine whether each pair of fractions is equivalent. In other words, do both fractions represent the same part of a whole? b. 1 1 1 1 1 1 1 1 Not Equivalent 1 1 1 1 1 1 1 1 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.4- 38

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 continued Determining Whether Two Fractions Are Equivalent Determine whether each pair of fractions is equivalent. In other words, do both fractions represent the same part of a whole? c. 1 1 Equivalent 1 1 1 1 1 1 1 1 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.4- 39

2.5 Multiplying Fractions Objectives 1. Multiply fractions. 2. Use a multiplication shortcut. 3. Multiply a fraction and a whole number. 4. Find the area of a rectangle. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.5- 40

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.5- 41

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 Multiplying Fractions Multiply. Write answers in lowest terms. a. b. Multiply the numerators and multiply the denominators. Lowest terms Lowest terms Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.5- 42

Copyright © 2010 Pearson Education, Inc. All rights reserved. A multiplication shortcut that can be used with fractions is shown in the next example. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.5- 43

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 Using the Multiplication Shortcut Multiply Write answers in lowest terms. Not in lowest terms The numerator and denominator have a common factor other than 1, so write the prime factorization of each number. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.5- 44

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 Using the Multiplication Shortcut Multiply Write answers in lowest terms. Divide by the common factors 2 and 7. Or divide out common factors. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.5- 45

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Using the Multiplication Shortcut Use the multiplication shortcut to find each product. Write the answers in lowest terms and as mixed numbers where possible. a. Divide 8 and 6 by their common factor 2. Notice that 5 and 13 have no common factor. Then, multiply. 4 Lowest terms 3 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.5- 46

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Using the Multiplication Shortcut Use the multiplication shortcut to find each product. Write the answers in lowest terms and as mixed numbers where possible. b. c. Divide 9 and 18 by 9, and divide 10 and 16 by 2. 1 8 Lowest terms 5 2 6 7 2 3 5 1 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.5- 47

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.5- 48

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 Multiplying by Whole Numbers Multiply. Write answers in lowest terms and as whole numbers where possible. a. b. Write 9 as 9/1 and multiply. 3 1 5 2 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.5- 49

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.5- 50

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 5 Applying Fraction Skills a. Find the area of each patio block. Area = length  width Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.5- 51

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 5 continued Applying Fraction Skills b. Find the area of the rectangle. Area = length  width 1 1 4 1 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.5- 52

2.6 Applications of Multiplication Objectives 1. Solve fraction application problems using multiplication. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.6- 53

Copyright © 2010 Pearson Education, Inc. All rights reserved. Many application problems are solved by multiplying fractions. Use the following indicator words for multiplication. product double triple times of (when “of” follows a fraction) twice twice as much Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.6- 54

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 Applying Indicator Words Megan Paten has of her income deposited into a vacation fund. In one pay period she earns $966. How much money does she deposit in her vacation fund per pay period? Step 1 Read the problem. The problem asks us to find the amount of money she deposits into her vacation fund. Step 2 Work out a plan. The indicator word is of; Paten deposits of her income. When it follows a fraction, the word of indicates multiplication, so find the amount deposited by multiplying and $966. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.6- 55

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 continued Applying Indicator Words Megan Paten has of her income deposited into a vacation fund. In one pay period she earns $966. How much money does she deposit in her vacation fund per pay period? Step 3 Estimate a reasonable answer. Round the income of $966 to $1000. Then divide 1000 by ten to find of the income. Our estimate is $1000 ÷ 10 = $100. Step 4 Solve the problem. 138 1 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.6- 56

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 continued Applying Indicator Words Megan Paten has of her income deposited into a vacation fund. In one pay period she earns $966. How much money does she deposit in her vacation fund per pay period? Step 5 State the answer. Paten deposits $138 in her vacation fund per pay period. Step 6 Check. The exact answer $138, is close to the estimate of $100. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.6- 57

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 Solving a Fraction Application Problem Of the 48 kittens in the shelter, are males. How many male kittens are there? Step 1 Read the problem. The problem asks to find the number of male kittens in the shelter. Step 2 Work out a plan. Reword the problem to read Indicator word for multiplication when it follows a fraction. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.6- 58

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 continued Solving a Fraction Application Problem Of the 48 kittens in the shelter, are males. How many male kittens are there? Step 3 Estimate a reasonable answer. Round the number of kittens from 48 to 50. Then of 50 is 25. Since is more than , our estimate is that more than 25 kittens are males. Step 4 Solve the problem. 16 1 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.6- 59

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 continued Solving a Fraction Application Problem Of the 48 kittens in the shelter, are males. How many male kittens are there? Step 5 State the answer. There are 32 male kittens in the shelter. Step 6 Check. The exact answer 32, fits our estimate of more than 25. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.6- 60

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Finding a Fraction of a Fraction In Anne’s sport card collection, of all of her cards are baseball cards. Of those baseball cards, of them are of the New York Yankees. What fraction of the total cards are of the New York Yankees? Step 1 Read the problem. The problem asks for the fraction of the baseball cards that are the New York Yankees. Step 2 Work out a plan. Reword the problem to read Indicator word for multiplication when it follows a fraction Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.6- 61

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 continued Finding a Fraction of a Fraction In Anne’s sport card collection, 1/2 of all of her cards are baseball cards. Of those baseball cards, 1/3 of them are of the New York Yankees. What fraction of the total cards are of the New York Yankees? Step 3 Estimate a reasonable answer. If the cards are divided into 2 equal piles and each of these pile was divided into 3 equal parts, we would have 2 ∙ 3 = 6 equal parts. Our estimate is . Step 4 Solve the problem. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.6- 62

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 continued Finding a Fraction of a Fraction In Anne’s sport card collection, of all of her cards are baseball cards. Of those baseball cards, of them are of the New York Yankees. What fraction of the total cards are of the New York Yankees? Step 5 State the answer. The New York Yankees cards make up of Anne’s collection. Step 6 Check. The exact answer , matches our estimate. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.6- 63

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 Using Fractions with a Circle Graph The circle graph, or pie chart, shows where children 8 to 17 years of age make food purchases when away from home. If 2500 children were in the survey, find the number of children who buy food from the vending machine. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.6- 64

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 continued Using Fractions with a Circle Graph Step 1 Read the problem. The problem asks for the number of children who buy food from the vending machine. Step 2 Work out a plan. Reword the problem to read Indicator word for multiplication when it follows a fraction Step 3 Estimate a reasonable answer. ¼ of 2500 people is 625 people. 1/5 is less than ¼, so our estimate is less than 625 people. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.6- 65

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 continued Using Fractions with a Circle Graph 500 Step 4 Solve the problem. 1 Step 5 State the answer. 500 children buy food from the vending machine. Step 6 Check. The exact answer, 500 children, fits our estimate of “less than 625 children.” Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.6- 66

Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.7 Dividing Fractions Objectives 1. Find the reciprocal of a fraction. 2. Divide fractions. 3. Solve application problems in which fractions are divided. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.7- 67

Copyright © 2010 Pearson Education, Inc. All rights reserved. Reciprocal Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.7- 68

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 Finding Reciprocals Find the reciprocal of each fraction. a. b. c. d. 2 The reciprocal is The reciprocal is The reciprocal is The reciprocal is Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.7- 69

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.7- 70

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.7- 71

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 Dividing One Fraction by Another Divide. Write answers in lowest terms and as mixed numbers where possible. The reciprocal of 2 Reciprocals 1 Change division to multiplication Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.7- 72

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 Dividing One Fraction by Another Divide 1 4 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.7- 73

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Dividing with a Whole Number Divide. Write all answers in lowest terms and as whole or mixed numbers where possible. a. Write 9 as 9/1. Use the reciprocal of ¼ which is 4/1. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.7- 74

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Dividing with a Whole Number Divide. Write all answers in lowest terms and as whole or mixed numbers where possible. b. Write 4 as 4/1. The reciprocal of 4/1 is ¼. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.7- 75

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 Applying Fraction Skills The manager of a local feed mill must fill a 20-gallon container with feed. He only has a 5/8 gallon container to use. How many times must he fill the 5/8 gallon container and empty it into the 20-gallon container? Step 1 Read the problem. We need the number of times the manager needs to use a 5/8 gallon container in order to fill a 20-gallon container. Step 2 Work out a plan. We can solve the problem by finding the number of times 20 can be divided by 5/8. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.7- 76

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 Applying Fraction Skills The manager of a local feed mill must fill a 20-gallon container with feed. He only has a 5/8 gallon container to use. How many times must he fill the 5/8 gallon container and empty it into the 20-gallon container? Step 3 Estimate a reasonable answer. Round 5/8 to 1 gallon. He would need to use the container about 20 times. Step 4 Solve the problem. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.7- 77

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 Applying Fraction Skills The manager of a local feed mill must fill a 20-gallon container with feed. He only has a 5/8 gallon container to use. How many times must he fill the 5/8 gallon container and empty it into the 20-gallon container? Step 5 State the answer. The manager must fill the container 32 times. Step 6 Check. The exact answer is close to our estimate. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.7- 78

2.8 Multiplying and Dividing Mixed Numbers Objectives 1. Estimate the answer and multiply mixed numbers. 2. Estimate the answer and divide mixed numbers. 3. Solve application problems with mixed numbers. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.8- 79

Copyright © 2010 Pearson Education, Inc. All rights reserved. When multiplying mixed numbers, it is a good idea to estimate the answer first. Then multiply the mixed numbers by using the following steps. To estimate the answer, round each mixed number to the nearest whole number. If the numerator is half of the denominator or more, round up the whole number part. If the numerator is less than half the denominator, leave the whole number as it is. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.8- 80

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 Multiplying Mixed Numbers First estimate the answer. Then multiply to get an exact answer. Simplify your answers. a. Estimate the answer by rounding the mixed numbers. rounds to 4 and rounds to 3. 4 ∙ 3 = 12 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.8- 81

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 continued Multiplying Mixed Numbers a. To find the exact answer, change each mixed number to an improper fraction. Step 1 and Next multiply. Step 1 Step 2 Step 3 4 1 The estimated answer is 12 and the exact answer is . The exact answer is reasonable. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.8- 82

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 continued Multiplying Mixed Numbers First estimate the answer. Then multiply to get an exact answer. Simplify your answer. b. Estimate the answer by rounding the mixed numbers. rounds to 2 and rounds to 2. 2 ∙ 2 = 4 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.8- 83

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 continued Multiplying Mixed Numbers b. The exact answer is shown below. 5 1 The estimated answer is 4 and the exact answer is . The exact answer is reasonable. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.8- 84

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.8- 85

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 Dividing Mixed Numbers First estimate the answer. Then divide to get an exact answer. Simplify your answers. a. Estimate the answer by rounding the mixed numbers. rounds to 2 and rounds to 2. 2 ÷ 2 = 1 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.8- 86

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 continued Dividing Mixed Numbers a. To find the exact answer, change each mixed number to an improper fraction. Step 1 and Next, use the reciprocal of the second fraction and multiply. Step 2 Step 3 Step 4 1 1 The estimated answer was1, so the exact answer of is reasonable. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.8- 87

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 continued Dividing Mixed Numbers b. Estimate Now find the exact answer. Reciprocal Write 26 as . Our estimate was , so the exact answer of is reasonable. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.8- 88

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Applying Multiplication Skills A recipe for a batch of cookies call for cups of brown sugar. Madeline needs to make 6 times the original recipe for enough cookies for a luncheon. How many cups of brown sugar will she need? Step 1 Read the problem. The problem asks for the total cups of brown sugar needed for the recipe. Step 2 Work out a plan. Multiply the number of batches (6) and the amount of cups each batch needs ( cups). Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.8- 89

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 continued Applying Multiplication Skills A recipe for a batch of cookies call for cups of brown sugar. Madeline needs to make 6 times the original recipe for enough cookies for a luncheon. How many cups of brown sugar will she need? Step 3 Estimate a reasonable answer. Round cups to 1 cup. Multiply 1 cup by 6 batches to get an estimate of 6 cups. 2 Step 4 Solve the problem. 1 Step 5 State the answer. Madeline will need 8 cups of brown sugar. Step 6 Check. The exact answer 8 cups, is more than the estimate of 6 cups as is more than 1 cup. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 2.8- 90