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

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.

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

Multiplication of Rational Numbers If are any rational numbers, then

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.

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

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

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

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

Properties of Multiplication of Rational Numbers Multiplication Property of Inequality

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

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.

Multiplication with Mixed Numbers Using Improper Fractions Using the Distributive Property

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

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

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 .

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

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.

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.

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

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.

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

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.

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.

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.

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

Extending the Notion of Exponents 1. 2. 3.

Extending the Notion of Exponents 4. 5. 6.

Extending the Notion of Exponents 7. 8. 9.

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

Example (continued) b. c.

Example (continued) d. e.

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

Example (continued) c. d.