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Section 2.2 Multiplication and Division with Fractions and Mixed Numbers HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Multiply mixed numbers. o Multiply and reduce with fractions and mixed numbers. o Understand the term reciprocal. o Learn to divide with fractions and mixed numbers. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Multiplication with Mixed Numbers 1 1 Find the product: 1 2 2 5 Solution Change the mixed numbers to improper fractions, then multiply the fractions. 3 1 1 3 11 3 11 33 or 3 1 2 2 5 2 5 2 5 10 10 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Multiplication with Fractions and Mixed Numbers 3 3 Find of 5 . 5 4 Solution 9 3 3 3 23 69 or 3 5 5 4 5 4 20 20 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Multiplying and Reducing with Fractions 15 7 Multiply and reduce to lowest terms: 28 9 Solution Using prime factors, we have 15 7 15 7 3 5 7 5 5 . 28 9 28 9 2 2 7 3 3 2 2 3 12 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Multiplying and Reducing with Fractions 9 25 44 Multiply and reduce to lowest terms: 10 32 33 Solution Using prime factors, we have 9 25 44 9 25 44 3 3 5 5 2 2 11 10 32 33 10 32 33 2 5 2 2 2 2 2 3 11 3 5 15 . 2 2 2 2 16 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 5: Multiplying and Reducing with Fractions 55 8 91 Multiply and reduce to lowest terms: . 26 44 35 Solution 55 8 91 55 8 91 26 44 35 26 44 35 5 11 2 2 2 7 13 2 13 2 2 11 5 7 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 1 1 1 Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Multiplying and Reducing with Mixed Numbers 2 1 1 Multiply and reduce to lowest terms: 4 1 2 3 7 16 Solution Using prime factors, we have 2 1 1 14 8 33 4 1 2 3 7 16 3 7 16 2 7 2 2 2 3 11 11 11. 3 72222 1 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7: Multiplying and Reducing with Fractions 9 25 8 Reduce to lowest terms and multiply: 10 64 3 Solution 3 5 1 9 25 8 15 = 10 64 3 16 2 8 1 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 3 divides both 3 and 9. 5 divides both 25 and 10. 8 divides both 8 and 64. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8: Multiplying and Reducing with Fractions 36 14 15 Multiply and reduce to lowest terms: 49 75 18 Solution 2 2 1 36 14 15 4 = 49 75 18 35 7 5 1 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 18 divides both 36 and 18. 7 divides both 14 and 49. 15 divides both 15 and 75. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8: Multiplying and Reducing with Fractions (cont.) Another approach is to use factors that are not all prime: 36 14 15 4 9 2 7 15 4 49 75 18 7 7 5 15 2 9 35 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 9: Application of Multiplying and Reducing with Fractions 5 A study showed that of the members of a public 8 service organization were in favor of a new set of bylaws. If the organization had a membership of 200 people, how many were in favor of the changes in the bylaws? HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 9: Application of Multiplying and Reducing with Fractions (cont.) Solution 5 We want to find of 200, so we multiply: 8 5 5 200 5 2 2 2 5 5 5 5 5 125. 200 8 1 2 2 2 1 1 8 Thus there are 125 members in favor of the bylaw changes. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Reciprocals Reciprocals a b The reciprocal of is (a 0 and b 0). The product b a of a nonzero number and its reciprocal is always 1. a b 1 b a 0 1 Note: 0 , but is undefined. 1 0 That is, the number 0 has no reciprocal. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10: Reciprocals 2 3 The reciprocal of is . 3 2 2 3 23 1 3 2 32 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 11: Reciprocals 1 The reciprocal of 10 is . 10 1 10 1 10 1 1 10 10 1 10 1 10 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Division with Fractions and Mixed Numbers To Divide Fractions To divide by any nonzero number, multiply by its reciprocal. In general, a c a d where b, c , d 0. b d b c 1 4 1 3 3 For example, . 2 3 2 4 8 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12: Dividing with Fractions 3 2 Divide: 4 3 Solution 2 3 3 The reciprocal of is , so we multiply by . 3 2 2 3 2 3 3 9 4 3 4 2 8 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 13: Dividing and Reducing with Fractions and Mixed Numbers 16 8 Divide and reduce to lowest terms: 2 27 9 Solution 8 First, change the mixed number 2 to the improper 26 26 9 9 9 fraction . The reciprocal of is , so multiply by 26 9 9 26 and reduce by factoring as follows: 16 8 16 26 16 9 8 29 8 2 27 9 27 9 27 26 3 9 2 13 39 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 14: Dividing and Reducing with Mixed Numbers 1 1 Divide and reduce to lowest terms: 3 19 4 2 Solution 13 2 1 1 1 1 13 39 13 2 3 19 4 2 4 2 4 39 2 2 3 13 6 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 15: Multiplying and Dividing with Fractions and Mixed Numbers 1 5 If the product of 1 and another number is , what is 2 18 the other number? Solution Divide the product by the given number to find the other number. 5 1 5 3 5 2 52 5 1 18 2 18 2 18 3 2 9 3 27 5 is the other number. 27 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 15: Multiplying and Dividing with Fractions and Mixed Numbers (cont.) Check: 1 5 3 5 3 5 5 1 . 2 27 2 27 2 3 9 18 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 16: Multiplying and Dividing with Fractions and Mixed Numbers 3 A box contains 30 pieces of candy. This is of the 5 maximum amount of candy the box can hold. a. Is the maximum amount of candy the box can hold more or less than 30 pieces? Solution The maximum number of pieces of candy is more than 30. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 16: Multiplying and Dividing with Fractions and Mixed Numbers (cont.) 3 b. If you want to multiply times 30, would the 5 product be more or less than 30? Solution The product would be less than 30. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 16: Multiplying and Dividing with Fractions and Mixed Numbers (cont.) c. What is the maximum number of pieces of candy the box can hold? Solution To find the maximum number of pieces, divide: 3 30 5 2 3 5 5 50 50 30 1 3 1 5 1 3 The maximum number of pieces the box will hold is 50. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems 1. Perform the indicated operations and reduce to lowest terms. 2 15 17 27 7 5 5 a. b. c. 2 6 21 22 100 34 9 8 8 5 2 f. 3 7 7 9 2. If the product of two numbers is and one of the 10 numbers is 3, what is the other number? d. 8 2 25 15 e. 6 0 7 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 5 77 d. 12 or 2 2 5 5 3 2. 10 1. a. b. 1 c. 9 or 2 1 4 4 e. undefined f. 13 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.