Three Types of Percent Problems

6.4 Three Types of Percent Problems 6.4 OBJECTIVES. 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 mount 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 Finding an Unknown mount NOTE Type : Finding an unknown amount. What is 8% of 300? We know the rate, 8%; and the base, 300; the amount is the unknown. Using the percent relationship, we can translate the problem to an equation. Rate Base Write 8% as the decimal 0.8 by the mount 0.8 300 rule of Section 6.. Then multiply to find the amount. 54 So 54 is 8% of 300. CHECK YOURSELF Find 65% of 200.. If the rate is less than %, the amount will be less than the base. 20 is 40% of 50 and 20 50 2. If the rate is greater than %, the amount will be greater than the base. 75 is 50% of 50 and 75 50 Let s consider a second type of percent problem involving an unknown rate. 495

496 CHPTER 6 PERCENTS Example 2 Finding an Unknown Percent NOTE Type 2: Finding an unknown percent. 30 is what percent of 50? We know the amount, 30, and the base, 50; the rate (what percent) is the unknown. gain using the percent relationship to translate to an equation, we have Base mount Rate 50 30 NOTE This will leave the rate alone on the left. We divide both sides by 50 to find the rate. Rate 30 50 5 0.20 20 20% 30 is 20% of 50. CHECK YOURSELF 2 75 is what percent of 300?. If the amount is less than the base, the rate will be less than %. 2. If the amount is greater than the base, the rate will be greater than %. 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 mount NOTE Notice that 40% is written as 0.40. 0.40 base 28 We divide both sides by 0.40 to find the base. Base 28 0.40 70 So 28 is 40% of 70. CHECK YOURSELF 3 70 is 35% of what number?

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. s 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 mount Base R R NOTE On the right, is the rate, and this proportion is equivalent to our earlier percent relationship. In symbols, B R Because in any percent problem we know two of the three quantities (, B,orR), 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 mount NOTE This is an unknownamount problem. is 30% of 50. R B Substitute the values into the percent proportion. R 50 30 The amount is the unknown term of the proportion. B We solve the proportion with the methods of Section 5.4. 50 30 4500 Divide by the coefficient,. 4500 45 The amount is 45. This means that 45 is 30% of 50. CHECK YOURSELF 4 Use the percent proportion to answer this question: What is 24% of 300?

498 CHPTER 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 problem. % of 400 is 72. R B Substitute the known values into the percent proportion. 72 400 R B Solving, we get 400R 7200 400R 400 7200 400 R, the rate, is the unknown term in this case. R 8 The rate is 8%. So 8% of 400 is 72. CHECK YOURSELF 5 Use the percent proportion to answer this question: What percent of 50 is 2.5? Finally, we use the same proportion to find an unknown base. NOTE This is an unknown-base problem. Example 6 Solving a Problem Involving an Unknown Base 40% of is 200. R B Substitute the known values into the percent proportion. R 200 B 40 Solving gives 40B 200 40B 40 20,000 40 In this case B, the base, is the unknown term of the proportion. B 500 The base is 500, and 40% of 500 is 200.

THREE TYPES OF PERCENT PROBLEMS SECTION 6.4 499 Remember that a percent (the rate) can be greater than. CHECK YOURSELF 6 288 is 60% of what number? Example 7 Solving a Percent Problem NOTE The rate is 25%. The base is 300. NOTE When the rate is greater than %, the amount will be greater than the base. What is 25% of 300? In the percent proportion, we have 300 25 So 300 25. Dividing by yields 37,500 375 So 375 is 25% of 300. CHECK YOURSELF 7 Find 50% of 500. We next look at two examples of solving percent problems involving fractions of a percent. Example 8 Solving a Percent Problem NOTE The amount is 34, the rate is 8.5%. We want to find the base. NOTE Divide by 8.5. 34 is 8.5% of what number? Using the percent proportion yields 34 B 8.5 Solving, we have 8.5B 34 or B 3400 8.5 400 So 34 is 8.5% of 400. CHECK YOURSELF 8 2.5% of what number is 75?

500 CHPTER 6 PERCENTS Example 9 Estimating Percentages Find 9.3% of 500. Round the rate to 20% 500 5 as a fraction, 5. n estimate of the amount is then Rounded rate Base Estimate of amount CHECK YOURSELF 9 Estimate the amount. 20.2% of 800 CHECK YOURSELF NSWERS. 30 2. 25% 3. 200 4. R 5. 300 24 B B 7200; 72 50R 250; R 25% 6. R 7. 750 8. 600 9. 60 288 B 60 60B 28,800; B 480 2.5 50 R

Name 6.4 Exercises Section Date Solve each of the following problems involving percent.. What is 35% of 600? 2. 20% of 400 is what number? 3. 45% of 200 is what number? 4. What is 40% of 200? 5. Find 40% of 2500. 6. What is 75% of 20? 7. What percent of 50 is 4? 8. 5 is what percent of 850? 9. What percent of 500 is 45? 0. 4 is what percent of 200?. What percent of 200 is 340? 2. 392 is what percent of 2800? 3. 46 is 8% of what number? 4. 7% of what number is 42? 5. Find the base if % of the base is 55. 6. 6% of what number is 92? 7. 58.5 is 3% of what number? 8. 2% of what number is 73.5? 9. Find 0% of 800. 20. What is 5% of 600? 2. What is 08% of 4000? 22. Find 60% of 2000. 23. 20 is what percent of 20? 24. What percent of 40 is 52? 25. 360 is what percent of 90? 26. What percent of 5,000 is 8,000? 27. 625 is 25% of what number? 28. 40% of what number is 350? 29. Find the base if 0% of the base is 935. 30. 30% of what number is 70? 3. Find 8.5% of 300. 32. 8 of 800 is what number? 4 % 33. Find 3 of 6000. 4 % 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 500? 39. What percent of 80 is 20? 40. What percent of 800 is 78? NSWERS. 2. 3. 4. 5. 6. 7. 8. 9. 0.. 2. 3. 4. 5. 6. 7. 8. 9. 20. 2. 22. 23. 24. 25. 26. 27. 28. 29. 30. 3. 32. 33. 34. 35. 36. 37. 38. 39. 40. 50

NSWERS 4. 42. 43. 44. 45. 46. 47. 48. 49. 50. 5. 52. 53. 4. What percent of 200 is 750? 42. 68 is what percent of 800? 43. 0.5% of what number is 420? 44. Find the base if of the base 2 % is 46. 45. 58.5 is 3% of what number? 46. 6.5% of what number is 325? 47. 95 is 7.5% of what number? 48. 2% of what number is 73.5? Estimate the amount in each of the following problems. 49. Find 25.8% of 4000. 50. What is 48.3% of 500? 5. 74.7% of 600 is what number? 52. 9.8% of 200 is what number? 53. Find 52% of 400. 54. What is 8% of 5000? 55. It is customary when eating in a restaurant to leave a 5% tip. (a) Outline a method to do a quick approximation for the amount of tip to leave. (b) Use this method to figure a 5% tip on a bill of $47.76. 54. 55. 56. 56. The dean of Enrollment Management at a college states, Last year was not a good year. 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. nswers. 20 3. 90 5. 0 7. 8% 9. 9%. 70% 3. 575 5. 500 7. 450 9. 880 2. 4320 23. 75% 25. 400% 27. 500 29. 850 3. 25.5 33. 705 35. 57.5 37. 7.5% 39. 66 2 4. 62.5% 43. 4000 45. 450 47. 2600 49. 0 3 % 5. 450 53. 600 55. 502