1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 2.2 Multiplication and Division with Fractions and Mixed Numbers

2. 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.

3. 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 Find the product: Solution Change the mixed numbers to improper fractions, then multiply the fractions.

4. 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 Find Solution

5. 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 Multiply and reduce to lowest terms: Solution Using prime factors, we have

6. 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 Multiply and reduce to lowest terms: Solution Using prime factors, we have

7. 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 Multiply and reduce to lowest terms: Solution

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Multiplying and Reducing with Mixed Numbers Multiply and reduce to lowest terms: Solution Using prime factors, we have

9. 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 Reduce to lowest terms and multiply: Solution

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8: Multiplying and Reducing with Fractions Multiply and reduce to lowest terms: Solution

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Another approach is to use factors that are not all prime: Example 8: Multiplying and Reducing with Fractions (cont.)

12. 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 A study showed that of the members of a public 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?

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solution Example 9: Application of Multiplying and Reducing with Fractions (cont.) Thus there are 125 members in favor of the bylaw changes. We want to find of 200, so we multiply:

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Reciprocals The reciprocal of The product of a nonzero number and its reciprocal is always 1. Note: is undefined. That is, the number 0 has no reciprocal. Reciprocals

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10: Reciprocals The reciprocal of

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 11: Reciprocals The reciprocal of 10 is

17. 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, For example,

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 12: Dividing with Fractions Divide: Solution

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. First, change the mixed number to the improper fraction The reciprocal of is so multiply by and reduce by factoring as follows: Example 13: Dividing and Reducing with Fractions and Mixed Numbers Divide and reduce to lowest terms: Solution

20. 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 Divide and reduce to lowest terms: Solution

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. If the product of and another number is what is the other number? Example 15: Multiplying and Dividing with Fractions and Mixed Numbers Solution Divide the product by the given number to find the other number. is the other number.

22. 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:

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The maximum number of pieces of candy is more than 30. A box contains 30 pieces of candy. This is of the maximum amount of candy the box can hold. Example 16: Multiplying and Dividing with Fractions and Mixed Numbers a.Is the maximum amount of candy the box can hold more or less than 30 pieces? Solution

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The product would be less than 30. b.If you want to multiply times 30, would the product be more or less than 30? Example 16: Multiplying and Dividing with Fractions and Mixed Numbers (cont.) Solution

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. c.What is the maximum number of pieces of candy the box can hold? Solution To find the maximum number of pieces, divide: The maximum number of pieces the box will hold is 50. Example 16: Multiplying and Dividing with Fractions and Mixed Numbers (cont.)

26. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. 1.Perform the indicated operations and reduce to lowest terms. a.b.c. d.e.f. 2.If the product of two numbers is and one of the numbers is 3, what is the other number? Practice Problems

27. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1.a.b. 1c. d. e. undefinedf. 13 2.