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6.4 Three Types of Percent Problems 6.4 OBJECTIVES 1. Find the unknown amount in a percent problem 2. Find the unknown rate in a percent problem 3. Find the unknown base in a percent problem From your work in Section 6.3, you may have observed that there are three basic types of percent problems. These depend on which of the three parts—the amount, the rate, or the base—is missing in the problem statement. The solution for each type of problem depends on the percent relationship. Rules and Properties: Percent Relationship Amount rate base We will illustrate the solution of each type of problem in the following examples. Let’s start with a problem in which we want to find the amount. Example 1 Finding an Unknown Amount NOTE Type 1: Finding an unknown amount. What is 18% of 300? We know the rate, 18%; and the base, 300; the amount is the unknown. Using the percent relationship, we can translate the problem to an equation. Rate Base Amount 0.18 300 Write 18% as the decimal 0.18 by the rule of Section 6.1. Then multiply to find the amount. 54 So 54 is 18% of 300. CHECK YOURSELF 1 © 2001 McGraw-Hill Companies Find 65% of 200. 1. If the rate is less than 100%, the amount will be less than the base. 20 is 40% of 50 and 20 50 2. If the rate is greater than 100%, the amount will be greater than the base. 75 is 150% of 50 and 75 50 Let’s consider a second type of percent problem involving an unknown rate. 495 496 CHAPTER 6 PERCENTS Example 2 Finding an Unknown Percent NOTE Type 2: Finding an unknown percent. 30 is what percent of 150? We know the amount, 30, and the base, 150; the rate (what percent) is the unknown. Again using the percent relationship to translate to an equation, we have Base Amount Rate 150 30 NOTE This will leave the rate alone on the left. We divide both sides by 150 to find the rate. Rate 30 1 1 0.20 20 20% 150 5 100 30 is 20% of 150. CHECK YOURSELF 2 75 is what percent of 300? 1. If the amount is less than the base, the rate will be less than 100%. 2. If the amount is greater than the base, the rate will be greater than 100%. Let’s look at a percent problem involving an unknown base in Example 3. Example 3 Finding an Unknown Base NOTE Type 3: Finding an unknown base. 28 is 40% of what number? We know the amount, 28, and the rate, 40%. The base (what number) is the unknown. From the percent relationship we have Rate NOTE Notice that 40% is Amount 0.40 base 28 written as 0.40. Base 28 70 0.40 So 28 is 40% of 70. CHECK YOURSELF 3 70 is 35% of what number? © 2001 McGraw-Hill Companies We divide both sides by 0.40 to find the base. THREE TYPES OF PERCENT PROBLEMS SECTION 6.4 497 We have now seen solution methods for the three basic types of percent problems: finding the amount, the rate, and the base. As you will see in the remainder of this section, our work in Chapter 5 with proportions will allow us to solve each type of problem in an identical fashion. In fact, many students find percent problems easier to approach with the proportion method. First, we will write what is called the percent proportion. Rules and Properties: The Percent Proportion Amount R Base 100 In symbols, R NOTE On the right, is the 100 rate, and this proportion is equivalent to our earlier percent relationship. R A B 100 Because in any percent problem we know two of the three quantities (A, B, or R), we can always solve for the unknown term. Consider in Example 4 the use of the percent proportion. Example 4 Solving a Problem Involving an Unknown Amount NOTE This is an unknown- ________ is 30% of 150. amount problem. A R B Substitute the values into the percent proportion. R A 30 150 100 The amount A is the unknown term of the proportion. B We solve the proportion with the methods of Section 5.4. 100A 150 30 100A 4500 Divide by the coefficient, 100. © 2001 McGraw-Hill Companies 1 4500 100 A 100 100 1 A 45 The amount is 45. This means that 45 is 30% of 150. CHECK YOURSELF 4 Use the percent proportion to answer this question: What is 24% of 300? 498 CHAPTER 6 PERCENTS The same percent proportion will work if you want to find the rate. Example 5 Solving a Problem Involving an Unknown Rate NOTE This is an unknown-rate ______% of 400 is 72. problem. R B A Substitute the known values into the percent proportion. A 72 R 400 100 R, the rate, is the unknown term in this case. B Solving, we get 400R 7200 1 400R 7200 400 400 1 R 18 The rate is 18%. So 18% of 400 is 72. CHECK YOURSELF 5 Use the percent proportion to answer this question: What percent of 50 is 12.5? Finally, we use the same proportion to find an unknown base. Example 6 Solving a Problem Involving an Unknown Base NOTE This is an unknown-base 40% of ______ is 200. problem. R B A Substitute the known values into the percent proportion. R 200 40 B 100 In this case B, the base, is the unknown term of the proportion. Solving gives 40B 200 100 1 20,000 40B 40 40 1 B 500 The base is 500, and 40% of 500 is 200. © 2001 McGraw-Hill Companies A THREE TYPES OF PERCENT PROBLEMS SECTION 6.4 499 Remember that a percent (the rate) can be greater than 100. CHECK YOURSELF 6 288 is 60% of what number? Example 7 Solving a Percent Problem NOTE The rate is 125%. The base is 300. NOTE When the rate is greater than 100%, the amount will be greater than the base. What is 125% of 300? In the percent proportion, we have A 125 300 100 So 100A 300 125. Dividing by 100 yields A 37,500 375 100 So 375 is 125% of 300. CHECK YOURSELF 7 Find 150% of 500. We next look at two examples of solving percent problems involving fractions of a percent. Example 8 Solving a Percent Problem 34 is 8.5% of what number? Using the percent proportion yields NOTE The amount is 34, the rate is 8.5%. We want to find the base. 34 8.5 B 100 Solving, we have © 2001 McGraw-Hill Companies 8.5B 34 100 or NOTE Divide by 8.5. B 3400 400 8.5 So 34 is 8.5% of 400. CHECK YOURSELF 8 12.5% of what number is 75? CHAPTER 6 PERCENTS Example 9 Estimating Percentages Find 19.3% of 500. Round the rate to 20% as a fraction, 1 . An estimate of the amount is then 5 1 500 100 5 Rounded rate Base Estimate of amount CHECK YOURSELF 9 Estimate the amount. 20.2% of 800 CHECK YOURSELF ANSWERS 1. 130 4. 2. 25% R 3. 200 5. A A 24 300 100 12.5 R 50 100 B B 100A 7200; A 72 50R 1250; R 25% 6. A R 7. 750 8. 600 9. 160 288 60 B 100 60B 28,800; B 480 © 2001 McGraw-Hill Companies 500 Name 6.4 Exercises Section Date Solve each of the following problems involving percent. 1. What is 35% of 600? 2. 20% of 400 is what number? 3. 45% of 200 is what number? 4. What is 40% of 1200? 5. Find 40% of 2500. 7. What percent of 50 is 4? 9. What percent of 500 is 45? 11. What percent of 200 is 340? 1. 2. 3. 4. 5. 6. 8. 51 is what percent of 850? 7. 8. 10. 14 is what percent of 200? 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 6. What is 75% of 120? 12. 392 is what percent of 2800? 13. 46 is 8% of what number? 14. 7% of what number is 42? 15. Find the base if 11% of the base is 55. 16. 16% of what number is 192? 17. 58.5 is 13% of what number? 18. 21% of what number is 73.5? 19. Find 110% of 800. 20. What is 115% of 600? 21. What is 108% of 4000? 22. Find 160% of 2000. 23. 210 is what percent of 120? 24. What percent of 40 is 52? 25. 360 is what percent of 90? 26. What percent of 15,000 is 18,000? 27. 625 is 125% of what number? ANSWERS 28. 140% of what number is 350? 33. 29. Find the base if 110% of the base is 935. 30. 130% of what number is 1170? 31. Find 8.5% of 300. 32. 8 % of 800 is what number? 1 4 34. 35. © 2001 McGraw-Hill Companies 36. 3 4 33. Find 11 % of 6000. 34. What is 3.5% of 500? 35. What is 5.25% of 3000? 36. What is 7.25% of 7600? 37. 60 is what percent of 800? 38. 500 is what percent of 1500? 37. 38. 39. 40. 39. What percent of 180 is 120? 40. What percent of 800 is 78? 501 ANSWERS 41. 42. 41. What percent of 1200 is 750? 42. 68 is what percent of 800? 43. 10.5% of what number is 420? 44. Find the base if 11 % of the base 1 2 is 46. 43. 45. 58.5 is 13% of what number? 46. 6.5% of what number is 325? 45. 47. 195 is 7.5% of what number? 48. 21% of what number is 73.5? 46. Estimate the amount in each of the following problems. 47. 49. Find 25.8% of 4000. 50. What is 48.3% of 1500? 51. 74.7% of 600 is what number? 52. 9.8% of 1200 is what number? 53. Find 152% of 400. 54. What is 118% of 5000? 44. 48. 49. 50. 55. It is customary when eating in a restaurant to leave a 15% tip. 51. (a) Outline a method to do a quick approximation for the amount of tip to leave. (b) Use this method to figure a 15% tip on a bill of $47.76. 52. 53. 54. 55. 56. 56. The dean of Enrollment Management at a college states, “Last year was not a good year. Answers 1. 210 15. 500 27. 500 2 3 51. 450 39. 66 % 502 3. 90 5. 1000 7. 8% 9. 9% 11. 170% 13. 575 17. 450 19. 880 21. 4320 23. 175% 25. 400% 29. 850 31. 25.5 33. 705 35. 157.5 37. 7.5% 41. 62.5% 53. 600 43. 4000 55. 45. 450 47. 2600 49. 1000 © 2001 McGraw-Hill Companies Our enrollments were down 25%. But this year we increased our enrollment by 30% over last year. I think we have turned the corner.” Evaluate the dean’s analysis.