1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 7.4 Multiplication and Division with Real Numbers

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Multiply real numbers. o Divide real numbers. o Find the average (mean) of a set of real numbers.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Products of Positive and Negative Real Numbers

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Products of Negative Real Numbers

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Multiplication by 0 a. 6  0 = 0 b.  13  0 = 0

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Rules for Multiplication with Real Numbers If a and b are positive real numbers, then 1.The product of two positive numbers is positive: 2.The product of two negative numbers is positive: 3.The product of a positive number and a negative number is negative: Multiplication with Real Numbers

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Rules for Multiplication with Real Numbers (cont.) 4.The product of 0 and any number is 0: Multiplication with Real Numbers

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Division with Real Numbers For real numbers a, b, and x (where ), For real numbers a and b (where ), Division with Real Numbers Note:See Section 1.4 for a thorough discussion on division by 0.

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Division with Real Numbers

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Division with Real Numbers (cont.)

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Rules for Division with Real Numbers If a and b are positive real numbers (where ), 1.The quotient of two positive numbers is positive: 2.The quotient of two negative numbers is positive: Division with Real Numbers

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Division with Real Numbers Rules for Division with Real Numbers (cont.) 3.The quotient of a positive number and a negative number is negative:

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Division with Fractions and Decimals Perform each division. With fractions, reduce quotients to lowest terms. With decimals, round quotients to the nearest tenth.

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Division with Fractions and Decimals (cont.)

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Division with Real Numbers Notes The following common rules about multiplication and division with two nonzero real numbers are helpful in remembering the signs of answers. a.If the numbers have the same sign, both the product and quotient will be positive. b.If the numbers have different signs, both the product and quotient will be negative.

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Average (or Mean) Average The average (or mean) of a set of numbers is the value found by adding the numbers in the set and then dividing the sum by the number of numbers in the set.

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Average a.At noon on five consecutive days in Aspen, Colorado, the temperatures were  5 , 7 , 6 ,  7 , and 14  (in degrees Fahrenheit). (Negative numbers represent temperatures below zero.) Find the average of these noonday temperatures.

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solution First, add the five temperatures. Now divide the sum, 15, by the number of temperatures, 5. The average noon temperature was 3  F. Example 6: Average (cont.)

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Average (cont.) b.In a placement exam for mathematics, a group of ten students had the following scores: 3 students scored 75, 2 students scored 80, 1 student scored 82, 3 students scored 85, and 1 student scored 88. What was the mean score for this group of students? Solution To find the total of all the scores, we multiply and then add. This is more efficient than adding all ten scores.

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Average (cont.) The mean score on the placement test for this group of students was 81.

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Average (cont.) c.The following speeds (in miles per hour) of fifteen cars were recorded at a certain point on a freeway. 70756560616468 72596882767068 50 Find the average speed of these cars. (One car received a speeding ticket, while another had a broken muffler.)

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Solution Using a calculator, the sum of the speeds is 1008 mph. Dividing by 15 gives the average speed. Example 6: Average (cont.)

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems Find the following products.

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems (cont.) Find the quotients. 19.Find the mean of the set of integers  16, 20, 32, and 92.

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems Answers 1.  152.243.  324.0 5.186.  3007.  18.58.18.45 9.10.11.  312.  4 13.214.undefined15.016.  2.5 17.1.0818.219.32