1. Chapter Rational Numbers and Proportional Reasoning 6 6 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

2. 6-3Multiplication and Division of Rational Numbers  Multiplication of Rational Numbers  Properties of Multiplication of Rational Numbers  Multiplication with Mixed Numbers  Division of Rational Numbers  Estimation and Mental Math with Rational Numbers  Extending the Notion of Exponents Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

3. Multiplication of Rational Numbers Multiplication as repeated addition Multiplication as part of an area Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

4. If are any rational numbers, then Multiplication of Rational Numbers Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

5. Example 6-12 If of the population of a certain city are college graduates and of the city’s college graduates are female, what fraction of the population of that city is female college graduates? of the population is female college graduates. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

6. Multiplicative Identity The number 1 is the unique number such that for every rational number Properties of Multiplication of Rational Numbers Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

7. Multiplicative Inverse (Reciprocal) For any nonzero rational number, is the unique rational number such that Properties of Multiplication of Rational Numbers Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

8. Properties of Multiplication of Rational Numbers If are any rational numbers, then Distributive Property of Multiplication Over Addition Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

9. Multiplication Property of Equality If are any rational numbers such that, and if is any rational number, then Properties of Multiplication of Rational Numbers Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

10. Multiplication Property of Inequality Properties of Multiplication of Rational Numbers Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

11. Multiplication Property of Zero Properties of Multiplication of Rational Numbers If is any rational number, then Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

12. A bicycle is on sale at of its original price. If the sale price is $330, what was the original price? Example 6-14 Let x = the original price. Then the sale price. The original price was $440. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

13. Multiplication with Mixed Numbers Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

14. Multiplication with Mixed Numbers Using the Distributive Property Using Improper Fractions Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

15. Division of Rational Numbers How many are in 3 wholes? How many are in ? Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

16. Division of Rational Numbers How many are in The bar of length is made up of 6 equal-size pieces of length. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

17. Division of Rational Numbers There is at least one length of in. If we put another bar of length on the number line, we see there is 1 more of the 6 equal-length segments needed to make. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

18. Division of Rational Numbers If are any rational numbers and is not zero, then if and only if is the unique rational number such that Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

19. If are any rational numbers and is not zero, then Division of Rational Numbers Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

20. Division of Rational Numbers When two fractions with the same denominator are divided, the result can be obtained by dividing the numerator of the first fraction by the numerator of the second. To divide fractions with different denominators, we rename the fractions so that the denominators are equal. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

21. Example 6-16 A radio station provides 36 min for public service announcements for every 24 hr of broadcasting. a.What part of the broadcasting day is allotted to public service announcements? There are 60 · 24 = 1440 minutes in a day. So, of the day is allotted for announcements. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

22. Example 6-16 (continued) b.How many -min. public service announcements can be allowed in the 36 minutes? announcements are allowed. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

23. Example 6-17 We have yards of material available to make towels. Each towel requires yards of material. a.How many towels can be made? Find the integer part of the answer to 94 towels can be made. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

24. Example 6-17 (continued) b.How much material will be left over? Because the division was by the amount of material left over is Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

25. Example 6-18 A bookstore has a shelf that is 37 ½ in. long. Each book that is to be placed on the shelf is 1 ¼ in. thick. How many books can be placed on the shelf? We need to find how many 1 1/4s there are in 37 ½. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

26. Example 6-18 A bookstore has a shelf that is 37 ½ in. long. Each book that is to be placed on the shelf is 1 ¼ in. thick. How many books can be placed on the shelf? Copyright © 2013, 2010, and 2007, Pearson Education, Inc. 30 books can be placed on the shelf.

27. Estimation and Mental Math with Rational Numbers Estimation and mental math strategies that were developed with whole numbers can also be used with rational numbers. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

28. Example 6-20 Estimate each of the following: a. b. The product will be between 21 and 32. The quotient will be between 5 and 6. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

29. Extending the Notion of Exponents 1. 2. 3. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

30. Extending the Notion of Exponents 4. 5. 6. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

31. Extending the Notion of Exponents 7. 9. 8. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

32. Example 6-21 Use properties of exponents to justify the equality or inequality: a. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

33. Example 6-21 (continued) b. c. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

34. Example 6-21 (continued) d. e. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

35. Example 6-22 Write each of the following in simplest form using positive exponents in the final answer: a. b. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

36. Example 6-22 (continued) c. d. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.