copyright © 2010 pearson education, inc. all rights reserved. 1.6 – slide 1


The Real Number System

Chapter 1


1.6

Multiplying and Dividing

Real Numbers


Objectives

1. Find the product of a positive number and a negative number.

2. Find the product of two negative numbers.3. Use the reciprocal of a number to apply the

definition of division.4. Use the rules for order of operations when

multiplying and dividing signed numbers.5. Evaluate expressions involving variables.6. Translate words and phrases involving

multiplication and division.7. Translate simple sentences into equations.

1.6 Multiplying and Dividing Real Numbers



Multiplication Property of 0For any real number a,

Finding the Product of a Positive and Negative Number

Since multiplication can also be considered repeated addition, the product 3(–1) represents the sum –1 + (–1) + (–1) = –3.

Add –1 three times.

The product of a positive number and a negative number is negative.

0 0 0.a a

6 3 18 and 6 3 18


Example 1

Find each product using the multiplication rule.

(a) 9(–3) =

1.6 Multiplying and Dividing Real NumbersFinding the Product of a Positive and Negative Number

–(9 · 3) =

(b) –6(8) = –(6 · 8) =

(c) –16(⅜) = –6

(d) 2.9(–3.2) = –9.28

–27

–48


Example 2

Find each product using the multiplication rule.

(a) –5(–7) =

1.6 Multiplying and Dividing Real NumbersFinding the Product of Two Negative Numbers

35

(b) –6(–12) = 72

(c) –2(3)(–1) = –6(–1) =

(d) 3(–5)(–2) = –15(–2) =

The product of two negative numbers is positive.

6

30



ReciprocalsPairs of numbers whose product is 1 are called reciprocals of each other.

Using a Reciprocal to Apply the Definition of Division

3 5 3 5i.e. and are reciprocals because 1.

5 3 5 3

0 has no reciprocal.



DivisionThe quotient


Note

a

bof real numbers a and b, with b ≠ 0, is

.1a

ab b

8 1: 8 2

4 4Example

0If 0, then 0.b

b


Example 3

Find each quotient.

1.6 Multiplying and Dividing Real NumbersUsing a Reciprocal to Apply the Definition of Division

15(a)

3

115

3

3 2(b)

4 3 3 3

4 2

1.8(c)

0.3

1

1.80.3

8(d)

0

Undefined

0(e)

5

0

5

9

8

6 4 3(f )

3

4


Example 4

Find each quotient.

1.6 Multiplying and Dividing Real NumbersUsing a Reciprocal to Apply the Definition of Division

10(a)

2

1 3

(c)5 10

6.3(d)

0.07

90

5

12(b)

3

4

2

3



Dividing Signed NumbersThe quotient of two numbers having the same sign is positive. The quotient of two numbers have different signs is negative.

For any positive real numbers a and b,

.a a a

b b b

For any positive real numbers a and b,

.a a

b b



(b) –6(–2) –3(–4) =

Example 5

Simplify.

1.6 Multiplying and Dividing Real NumbersUsing the Order of Operations with Signed Numbers

Find all products, working from left to right.

(a) –9(2) – (–3)(2) =

–9(2) – (–3)(2) = –18 – (–6)

= –18 + 6

= –12

12 – (–12)

= 12 + 12

= 24


Example 5 (concluded)

Simplify.


5( 2) 3(4)(c)

2(1 6)

Using the Order of Operations with Signed Numbers

Simplify the numerator and denominator separately.5( 2) 3(4) 10 12

2(1 6) 2( 5)

22

10

11

5


Example 6

Evaluate each expression, given that x = –1, y = –2, and m = –3.

1.6 Multiplying and Dividing Real NumbersEvaluating Expressions Involving Variables

Substitute the given values for the variables. Then use the order of operations to find the value of the expression.

(a) (3x + 4y)(–2m)

(3x + 4y)(–2m) = [3(–1) + 4(–2)][–2(–3)]

= [–3 + (–8)][6]

= (–11)(6)

= –66


Example 6 (continued)



(b) 2x2 – 3y2 = 2(–1)2 – 3(–2)2

= 2(1) – 3(4)

= –10

= 2 – 12





24(c)

y x

m

24( ) ( )2

( )

1

3

4(4) ( 1)

3

16 ( 1)

3

15

3

5


Translating Words and Phrases

Word or Phrase

ExampleNumerical

Expression and Simplification

Product of The product of –5 and –2 –5(–2) =10

Times 13 times –4 13(–4) = –52

Twice (meaning “2 times”)

Twice 6 2(6) =12

Of (used with fractions)

½ of 10 ½(10) =5

Percent of 12 % of –16 0.12(–16) =–1.92




Word or Phrase

ExampleNumerical

Expression and Simplification

Quotient of The quotient of –24 and 3

Divided by –16 divided by –4

Ratio of The ratio of 2 to 3


248

3

164

4

2

3




Example 7

Write a numerical expression for each phrase, and simplify the expression.

(a) Three fourths of the difference between 8 and –23

[8 ( 2)]4

3

(8 2)4

3(10)

4

157.5

2






(b) 20% of the sum of 1200 and 400

0.20(1200 + 400) =

= 320

0.20(1600)




Example 8


(c) The quotient of 20 and the difference between –11 and –7

20

[ 11 ( 7)]

20

[ 11 7] 20

4

5


Translating Simple Sentences into Equations


Example 9

Write each sentence in symbols, using x to represent the number.

(a) Five times a number is 40.

5x = 40

(b) The quotient of a number and –8 is 6.

68

x


CAUTION

It is important to recognize the distinction between the

types of problems found in Examples 7 and 8 and those

in Example 9. In Example 7 and 8, the phrases trans-

late as expressions, while in Example 9, the sentences

translate as equations. Remember that an expression

is a phrase, while an equation is a sentence.

Translating Simple Sentences into Equations


copyright © 2010 pearson education, inc. all rights reserved. 1.6 – slide 1

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