1. 6 Chapter Rational Numbers and Proportional Reasoning
Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

2. 6-3 Multiplication Division, and Estimations with Real Numbers
Students will be able to understand and explain
• Multiplication and division of rational numbers.
• Properties of multiplication and division of rational numbers.
• Estimation of multiplication and division with rational numbers.
• Extension of exponents to include negative integers.

3. Multiplication of Rational Numbers
Multiplication as repeated addition
Multiplication as part of an area

4. Multiplication of Rational Numbers
If are any rational numbers, then

5. Example 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.

6. Properties of Multiplication of Rational Numbers
Multiplicative Identity
The number 1 is the unique number such that
for every rational number

7. Properties of Multiplication of Rational Numbers
Multiplicative Inverse (Reciprocal)
For any nonzero rational number , is the
unique rational number such that

8. Properties of Multiplication of Rational Numbers
Distributive Property of Multiplication Over Addition
If are any rational numbers, then

9. Properties of Multiplication of Rational Numbers
Multiplication Property of Equality
If are any rational numbers such that
, and if is any rational number, then

10. Properties of Multiplication of Rational Numbers
Multiplication Property of Inequality

11. Properties of Multiplication of Rational Numbers
Multiplication Property of Zero
If is any rational number, then

12. Example
A bicycle is on sale at of its original price. If the sale price is $330, what was the original price?
Let x = the original price. Then the sale price.
The original price was $440.

13. Multiplication with Mixed Numbers
Using Improper Fractions
Using the Distributive Property

14. Division of Rational Numbers
How many are in 3 wholes?
How many are in ?

15. Division of Rational Numbers
How many are in
The bar of length is made up of 6 equal-size pieces of length .

16. 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 .

17. 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

18. 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.

19. Example
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.

20. Example (continued) b. How many -min. public service announcements
can be allowed in the 36 minutes?
announcements are allowed.

21. Example
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.

22. Example (continued) b. How much material will be left over?
Because the division was by the amount of material left over is

23. Example
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.

24. Example (continued) 30 books can be placed on the shelf.
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?
30 books can be placed on the shelf.

25. 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.

26. Example Estimate each of the following: a.
The product will be between 21 and 32. b.
The quotient will be between 5 and 6.

27. Extending the Notion of Exponents1. 2. 3.

28. Extending the Notion of Exponents4. 5. 6.

29. Extending the Notion of Exponents7. 8. 9.

30. Example
Use properties of exponents to justify the equality or inequality: a.

31. Example (continued) b. c.

32. Example (continued) d. e.

33. Example
Write each of the following in simplest form using positive exponents in the final answer: a. b.

34. Example (continued) c. d.