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Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 1 Chapter 1 The Real Number System Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 2 1.6 Multiplying and Dividing Real Numbers Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 3 1.6 Multiplying and Dividing Real Numbers Objectives 1. 2. 3. 4. 5. 6. 7. Find the product of a positive number and a negative number. Find the product of two negative numbers. Use the reciprocal of a number to apply the definition of division. Use the rules for order of operations when multiplying and dividing signed numbers. Evaluate expressions involving variables. Translate words and phrases involving multiplication and division. Translate simple sentences into equations. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 4 1.6 Multiplying and Dividing Real Numbers Finding the Product of a Positive and Negative Number Multiplication Property of 0 For any real number a, a 0 0 a 0. 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. 6 3 18 and 6 3 18 Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 5 1.6 Multiplying and Dividing Real Numbers Finding the Product of a Positive and Negative Number Example 1 Find each product using the multiplication rule. (a) 9(–3) = –(9 · 3) = –27 (b) –6(8) = –(6 · 8) = –48 (c) –16(⅜) = –6 (d) 2.9(–3.2) = –9.28 Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 6 1.6 Multiplying and Dividing Real Numbers Finding the Product of Two Negative Numbers The product of two negative numbers is positive. Example 2 Find each product using the multiplication rule. (a) –5(–7) = 35 (b) –6(–12) = 72 (c) –2(3)(–1) = –6(–1) = 6 (d) 3(–5)(–2) = –15(–2) = 30 Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 7 1.6 Multiplying and Dividing Real Numbers Using a Reciprocal to Apply the Definition of Division Reciprocals Pairs of numbers whose product is 1 are called reciprocals of each other. 3 5 3 5 i.e. and are reciprocals because 1. 5 3 5 3 0 has no reciprocal. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 8 1.6 Multiplying and Dividing Real Numbers Using a Reciprocal to Apply the Definition of Division Division a of real numbers a and b, with b ≠ 0, is The quotient b a 1 a . b b Example : Note 8 1 8 2 4 4 0 If b 0, then 0. b Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 9 1.6 Multiplying and Dividing Real Numbers Using a Reciprocal to Apply the Definition of Division Example 3 Find each quotient. 15 1 (a) 15 5 3 3 8 (d) Undefined 0 3 2 3 3 9 (b) 4 3 4 2 8 0 (e) 0 5 1.8 1 1.8 (c) 6 0.3 0.3 Copyright © 2010 Pearson Education, Inc. All rights reserved. (f ) 4 3 3 4 1.6 – Slide 10 1.6 Multiplying and Dividing Real Numbers Using a Reciprocal to Apply the Definition of Division Example 4 Find each quotient. (a) 10 5 2 12 (b) 4 3 Copyright © 2010 Pearson Education, Inc. All rights reserved. 1 3 2 (c) 5 10 3 6.3 (d) 90 0.07 1.6 – Slide 11 1.6 Multiplying and Dividing Real Numbers Using a Reciprocal to Apply the Definition of Division Dividing Signed Numbers The 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 Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 12 1.6 Multiplying and Dividing Real Numbers Using the Order of Operations with Signed Numbers Example 5 Simplify. (a) –9(2) – (–3)(2) = Find all products, working from left to right. –9(2) – (–3)(2) = –18 – (–6) = –18 + 6 = –12 (b) –6(–2) –3(–4) = 12 – (–12) = 12 + 12 = 24 Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 13 1.6 Multiplying and Dividing Real Numbers Using the Order of Operations with Signed Numbers Example 5 (concluded) Simplify. 5(2) 3(4) (c) 2(1 6) Simplify the numerator and denominator separately. 5(2) 3(4) 10 12 2(1 6) 2(5) 22 10 Copyright © 2010 Pearson Education, Inc. All rights reserved. 11 5 1.6 – Slide 14 1.6 Multiplying and Dividing Real Numbers Evaluating Expressions Involving Variables Example 6 Evaluate each expression, given that x = –1, y = –2, and m = –3. (a) (3x + 4y)(–2m) Substitute the given values for the variables. Then use the order of operations to find the value of the expression. (3x + 4y)(–2m) = [3(–1) + 4(–2)][–2(–3)] = [–3 + (–8)][6] = (–11)(6) = –66 Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 15 1.6 Multiplying and Dividing Real Numbers Evaluating Expressions Involving Variables Example 6 (continued) Evaluate each expression, given that x = –1, y = –2, and m = –3. (b) 2x2 – 3y2 = 2(–1)2 – 3(–2)2 = 2(1) – 3(4) = 2 – 12 = –10 Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 16 1.6 Multiplying and Dividing Real Numbers Evaluating Expressions Involving Variables Example 6 (concluded) Evaluate each expression, given that x = –1, y = –2, and m = –3. 4 y 2 x 4(2) 2 (1) (c) m (3) 4(4) (1) 3 16 (1) 3 15 3 5 Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 17 1.6 Multiplying and Dividing Real Numbers Translating Words and Phrases Example Numerical Expression and Simplification Product of The product of –5 and –2 –5(–2) =10 Times 13 times –4 Twice (meaning “2 times”) Twice 6 2(6) =12 Of (used with ½ of 10 fractions) ½(10) =5 Word or Phrase Percent of 12 % of –16 Copyright © 2010 Pearson Education, Inc. All rights reserved. 13(–4) = –52 0.12(–16) =–1.92 1.6 – Slide 18 1.6 Multiplying and Dividing Real Numbers Translating Words and Phrases Word or Phrase Example Quotient of The quotient of –24 and 3 Divided by –16 divided by –4 Ratio of The ratio of 2 to 3 Copyright © 2010 Pearson Education, Inc. All rights reserved. Numerical Expression and Simplification 24 8 3 16 4 4 2 3 1.6 – Slide 19 1.6 Multiplying and Dividing Real Numbers Translating Words and Phrases Example 7 Write a numerical expression for each phrase, and simplify the expression. (a) Three fourths of the difference between 8 and –2 3 3 [8 (2)] (8 2) 4 4 3 (10) 4 15 7.5 2 Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 20 1.6 Multiplying and Dividing Real Numbers Translating Words and Phrases Example 7 (concluded) Write a numerical expression for each phrase, and simplify the expression. (b) 20% of the sum of 1200 and 400 0.20(1200 + 400) = 0.20(1600) = 320 Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 21 1.6 Multiplying and Dividing Real Numbers Translating Words and Phrases Example 8 Write a numerical expression for each phrase, and simplify the expression. (c) The quotient of 20 and the difference between –11 and –7 20 20 20 5 [11 (7)] [11 7] 4 Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 22 1.6 Multiplying and Dividing Real Numbers 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. x 6 8 Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 23 1.6 Multiplying and Dividing Real Numbers Translating Simple Sentences into Equations 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 translate as expressions, while in Example 9, the sentences translate as equations. Remember that an expression is a phrase, while an equation is a sentence. Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.6 – Slide 24