1. Chapter 6 Rates and Proportions © 2010 Pearson Education, Inc. All rights reserved.

2. 6.1 Ratios Objectives Slide 5.1- 2 1.Write ratios as fractions. 2. Solve ratio problems involving decimals or mixed numbers. 3. Solve ratio problems after converting units. Copyright © 2010 Pearson Education, Inc. All rights reserved.

3. Slide 5.1- 3 A ratio compares two quantities. Copyright © 2010 Pearson Education, Inc. All rights reserved.

4. a. Ratio of amount spent on salad to amount spent on bread. The ratio of salad to bread is b. Ratio of fish to bread. Parallel Example 1 Writing Ratios Slide 5.1- 4 Numerator (mentioned first) Denominator (mentioned second) Tamara spent $13 on fish, $8 on salad and $7 on bread. Write each ratio as a fraction. Copyright © 2010 Pearson Education, Inc. All rights reserved.

5. a. 80 days to 20 days. Divide the numerator and denominator by 20. b. 30 ounces to medicine to 140 ounces of medicine Parallel Example 2 Writing Ratios in Lowest Terms Slide 5.1- 5 Write each ratio in lowest terms. Copyright © 2010 Pearson Education, Inc. All rights reserved.

6. The price of a bag of dog food increased from $22.95 to $25.50. Find the ratio of the increase in price to the original price. Find the ratio of the increase in price to the original price. Now write the ratio as a ratio of whole numbers. Parallel Example 3 Using Decimal Numbers in a Ratio Slide 5.1- 6 new price – original price = increase $25.50  $22.95 = $2.55 Copyright © 2010 Pearson Education, Inc. All rights reserved.

7. Write each ratio as a comparison of whole numbers in lowest terms. a. 4 days to Write the ratio and divide out common units. Write as improper fractions. Parallel Example 4 Using Mixed Numbers in Ratios Slide 5.1- 7 Reciprocals Copyright © 2010 Pearson Education, Inc. All rights reserved.

8. b. Parallel Example 4 Using Mixed Numbers in Ratios Slide 5.1- 8 Copyright © 2010 Pearson Education, Inc. All rights reserved.

9. Write the ratio 34 days to 4 weeks. 4 weeks = 4  7 days = 28 days Parallel Example 5 Ratio Applications Using Measurement Slide 5.1- 9 Copyright © 2010 Pearson Education, Inc. All rights reserved.

10. Slide 5.1- 10 Copyright © 2010 Pearson Education, Inc. All rights reserved.

11. 6.2 Rates Objectives Slide 5.2- 11 1.Write rates as fractions. 2.Find unit rates. 3.Find the best buy based on cost per unit. Copyright © 2010 Pearson Education, Inc. All rights reserved.

12. Slide 5.2- 12 In a rate, you often find these words separating the quantities you are comparing: inforonperfrom 250 dollars for 15 hours 320 miles on 10 gallons A ratio compares two measurements with the same type of units, such as 8 yards to 20 yards (both length measurements). But many of the comparisons we make use measurements with different types of units, such as the following. Copyright © 2010 Pearson Education, Inc. All rights reserved.

13. Write each rate as a fraction in lowest terms. a. 8 gallons of antifreeze for $40. b. 192 calories in 6 ounces of yogurt Slide 5.2- 13 Parallel Example 1 Writing Rates in Lowest Terms Copyright © 2010 Pearson Education, Inc. All rights reserved.

14. Write each rate as a fraction in lowest terms. c. 84 hamburgers on 7 grills. Slide 5.2- 14 Parallel Example 1 continued Writing Rates in Lowest Terms Copyright © 2010 Pearson Education, Inc. All rights reserved.

15. Slide 5.2- 15 Use per or a slash mark (/) when writing unit rates. When the denominator of a rate is 1, it is called a unit rate. For example, you earn $16.25 for 1 hour of work. This unit rate is written: $16.25 per hour Copyright © 2010 Pearson Education, Inc. All rights reserved.

16. Find each unit rate. a. Slide 5.2- 16 Parallel Example 2 Finding Unit Rates 445.5 miles on 16.5 gallons of gas Divide to find the unit rate. The unit rate is 27 miles per gallon or 27 miles/gallon. Copyright © 2010 Pearson Education, Inc. All rights reserved.

17. Find each unit rate. b. Slide 5.2- 17 Parallel Example 2 continued Finding Unit Rates 413 feet in 14 seconds Divide to find the unit rate. The unit rate is 29.5 feet/second. Copyright © 2010 Pearson Education, Inc. All rights reserved.

18. A local store charges the following prices for jars of jelly. Slide 5.2- 18 Parallel Example 3 Determining the Best Buy The best buy is the container with the lowest cost per unit. All the jars are measured in ounces. Find the cost per ounce for each one by dividing the price of the jar by the number of ounces in it. Round to the nearest thousandth if necessary. 18 oz. 24 oz. 28 oz. $2.39 $3.09 $3.69 Copyright © 2010 Pearson Education, Inc. All rights reserved.

19. Parallel Example 3 continued Determining the Best Buy The lowest cost per ounce is $0.129, so the 24-ounce jar is the best buy. SizeCost per Unit (rounded) 18 ounces 24 ounces 28 ounces highest lowest Slide 5.2- 19 Copyright © 2010 Pearson Education, Inc. All rights reserved.

20. Juice is sold as a concentrated can as well as in a ready to serve carton. Which of the choices below is the best buy? Parallel Example 4 Solving Best Buy Applications 12 oz can makes 48 ounces of juice for $1.69 60 oz carton for $2.59 To determine the best buy, divide the cost by the number of ounces. Slide 5.2- 20 Copyright © 2010 Pearson Education, Inc. All rights reserved.

21. Parallel Example 4 continued Solving Best Buy Applications Concentrate Slide 5.2- 21 Carton Although, you must mix it yourself, the concentrated can of juice is the better buy. 12 oz can makes 48 ounces of juice for $1.69 60 oz carton for $2.59 Copyright © 2010 Pearson Education, Inc. All rights reserved.

22. 6.3 Proportions Objectives Slide 5.3- 22 1.Write proportions. 2. Determine whether proportions are true or false. 3. Find cross products. Copyright © 2010 Pearson Education, Inc. All rights reserved.

23. Slide 5.3- 23 A proportion states that two ratios (rates) are equivalent. The rate $20/4 hr is equivalent to $40/8 hr. As the amount of money doubles, the number of hours also doubles. Copyright © 2010 Pearson Education, Inc. All rights reserved.

24. Write each proportion. a.3 ft is to 9 ft as 12 ft is to 36 ft. b. $8 is to 3 cans as $32 is to 12 cans Parallel Example 1 Writing Proportions Slide 5.3- 24 Common units divide out and are not written. The units do not match so you must write them in the proportion. Copyright © 2010 Pearson Education, Inc. All rights reserved.

25. Determine whether each proportion is true or false by writing both ratios in lowest terms. a. b. Parallel Example 2 Writing Both Ratios in Lowest Terms Slide 5.3- 25 The proportion is false since the ratios are not equivalent. The proportion is true since the ratios are equivalent. Copyright © 2010 Pearson Education, Inc. All rights reserved.

26. Slide 5.3- 26 Copyright © 2010 Pearson Education, Inc. All rights reserved.

27. Use cross products to see whether each proportion is true or false. a. Parallel Example 3 Using Cross Products Slide 5.3- 27 Multiply along one diagonal, then multiply along the other diagonal. 9  12 = 108 4  27 = 108 Equal cross products; proportion is true. Copyright © 2010 Pearson Education, Inc. All rights reserved.

28. Use cross products to see whether each proportion is true or false. b. Parallel Example 3 Using Cross Products Slide 5.3- 28 Find the cross products. Unequal cross products; proportion is false. Copyright © 2010 Pearson Education, Inc. All rights reserved.

29. 6.4 Dividing Decimals Objectives Slide 5.4- 29 1.Find the unknown number in a proportion. 2.Find the unknown number in a proportion with mixed numbers or decimals. Copyright © 2010 Pearson Education, Inc. All rights reserved.

30. Slide 5.4- 30 Four numbers are used in a proportion. If any three of these numbers are known, the fourth can be found. Copyright © 2010 Pearson Education, Inc. All rights reserved.

31. Find the unknown number in each proportion. Round answers to the nearest hundredth when necessary. Parallel Example 1 Solving Proportions for Unknown Numbers Slide 5.4- 31 a. Ratios can be written in lowest terms. You can do that before finding the cross products. can be written in lowest terms as, which gives the proportion Copyright © 2010 Pearson Education, Inc. All rights reserved.

32. Parallel Example 1 continued Solving Proportions for Unknown Numbers Slide 5.4- 32 Show that the cross products are equivalent. Step 1 Find the cross products Step 2 Step 3 1 1 Copyright © 2010 Pearson Education, Inc. All rights reserved.

33. Parallel Example 1 continued Solving Proportions for Unknown Numbers Slide 5.4- 33 Step 4 Equal; proportion is true. Write the solution in the original proportion and check by finding cross products. The cross products are equal, so 25 is the correct solution. Copyright © 2010 Pearson Education, Inc. All rights reserved.

34. Parallel Example 1 continued Solving Proportions for Unknown Numbers Slide 5.4- 34 Show that the cross products are equivalent. Step 1 Find the cross products Step 2 Step 3 1 b. 1 Rounded to the nearest hundredth. Copyright © 2010 Pearson Education, Inc. All rights reserved.

35. Parallel Example 1 continued Solving Proportions for Unknown Numbers Slide 5.4- 35 Step 4 Very close, but not equal due to rounding. Write the solution in the original proportion and check by finding cross products. The cross products are very close, so 46.67 is the approximate solution. Copyright © 2010 Pearson Education, Inc. All rights reserved.

36. Find the unknown number in each proportion. Parallel Example 2 Solving Proportions with Mixed Numbers and Decimals Slide 5.4- 36 a. Find Find the cross products 1 12 Show the cross products are equivalent. Divide both sides by 8. Copyright © 2010 Pearson Education, Inc. All rights reserved.

37. Parallel Example 2 continued Solving Proportions with Mixed Numbers and Decimals Slide 5.4- 37 Equal The cross products are equal, so 30 is the correct solution. Copyright © 2010 Pearson Education, Inc. All rights reserved.

38. Find the unknown number in each proportion. Parallel Example 2 continued Solving Proportions with Mixed Numbers and Decimals Slide 5.4- 38 b. Show that cross products are equivalent. Divide both sides by 10.4. Copyright © 2010 Pearson Education, Inc. All rights reserved.

39. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Parallel Example 2 continued Solving Proportions with Mixed Numbers and Decimals Slide 5.4- 39 Equal 10.4 ∙ 8.06 = 83.824 12.4 ∙ 6.76 = 83.824 The cross products are equal, so 8.06 is the correct solution.

40. 6.5 Solving Application Problems with Proportions Objectives Slide 5.5- 40 1.Use proportions to solve application problems. Copyright © 2010 Pearson Education, Inc. All rights reserved.

41. Slide 5.5- 41 A proportion can be used to solve a wide variety of problems. Use the six problem-solving steps you learned in a previous section. Six Problem-Solving Steps Step 1 Read the problem. Step 2 Work out a plan. Step 3 Estimate a reasonable answer. Step 4 Solve the problem. Step 5 State the answer. Step 6 Check your work. Copyright © 2010 Pearson Education, Inc. All rights reserved.

42. Step 1 Read the problem. The problem asks for the number of circuit boards James can process in 252 minutes. Step 2 Work out a plan. Decide what is being compared. This example compares circuit boards to minutes. Write a proportion using the two rates. Be sure that both rates compare circuit boards to minutes in the same order. Parallel Example 1 Solving a Proportion Application Slide 5.5- 42 Copyright © 2010 Pearson Education, Inc. All rights reserved. On an assembly line, James can process 30 circuit boards in 108 minutes. At that rate, how many circuit boards can he process in 252 minutes?

43. On an assembly line, James can process 30 circuit boards in 108 minutes. At that rate, how many circuit boards can he process in 252 minutes? Step 3 Estimate a reasonable answer. Notice that 252 minutes is a little more than twice as much as 108 minutes, so James should process more than twice as many circuit boards. So use 2 30 = 60 circuit boards as our estimate. Parallel Example 1 Solving a Proportion Application Slide 5.5- 43 Matching units Copyright © 2010 Pearson Education, Inc. All rights reserved.

44. On an assembly line, James can process 30 circuit boards in 108 minutes. At that rate, how many circuit boards can he process in 252 minutes? Step 4 Solve the problem. Ignore the units while solving for x. Parallel Example 1 Solving a Proportion Application Slide 5.5- 44 Show that cross products are equivalent. Divide both sides by 108. Copyright © 2010 Pearson Education, Inc. All rights reserved.

45. On an assembly line, James can process 30 circuit boards in 108 minutes. At that rate, how many circuit boards can he process in 252 minutes? Step 5 State the answer. James can process 70 circuit boards in 252 minutes. Step 6 Check your work. The answer, 70 circuit boards is a little more than the estimate of 60 circuit boards, so it is reasonable. Parallel Example 1 Solving a Proportion Application Slide 5.5- 45 Copyright © 2010 Pearson Education, Inc. All rights reserved.

46. Step 1 Read the problem. The problem asks for the number of voters who will support the levy. Step 2 Work out a plan. You are comparing people who support the school levy to all the voters in town. Parallel Example 2 Solving a Proportion Application Slide 5.5- 46 Copyright © 2010 Pearson Education, Inc. All rights reserved. A local paper reported that 4 out of 5 voters surveyed stated that they supported the school levy. At that rate, how many of the 42,000 voters in town would you expect to support the school levy?

47. Step 3 Estimate a reasonable answer. To estimate the answer, notice that 4 out of 5 voters is more than half the total voters, but less than all the voters. Half of the 42,000 voters is 42,000 ÷ 2 = 21,000, so our estimate is between 21,000 and 42,000 voters. Parallel Example 1 Solving a Proportion Application Slide 5.5- 47 Copyright © 2010 Pearson Education, Inc. All rights reserved. A local paper reported that 4 out of 5 voters surveyed stated that they supported the school levy. At that rate, how many of the 42,000 voters in town would you expect to support the school levy?

48. Step 4 Solve the problem. Solve for the unknown number in the proportion. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Parallel Example 1 Solving a Proportion Application Slide 5.5- 48 Show that cross products are equivalent. Divide both sides by 5.

49. Step 5 State the answer. You would expect 33,600 voters to support the school levy. Step 6 Check your work. The answer, 33,600 voters is between 21,000 and 42,000, as called for in the estimate. Parallel Example 1 Solving a Proportion Application Slide 5.5- 49 Copyright © 2010 Pearson Education, Inc. All rights reserved. A local paper reported that 4 out of 5 voters surveyed stated that they supported the school levy. At that rate, how many of the 42,000 voters in town would you expect to support the school levy?