Three Types of Percent Problems

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1 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 rule of Section 6.. Then multiply to find the amount. 54 So 54 is 8% of 300. CHECK YOURSELF Find 65% of If the rate is less than %, the amount will be less than the base. 20 is 40% of 50 and If the rate is greater than %, the amount will be greater than the base. 75 is 50% of 50 and Let s consider a second type of percent problem involving an unknown rate. 495

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.

2 496 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 NOTE This will leave the rate alone on the left. We divide both sides by 50 to find the rate. Rate % 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 base 28 We divide both sides by 0.40 to find the base. Base So 28 is 40% of 70. CHECK YOURSELF 3 70 is 35% of what number?

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.

3 THREE TYPES OF PERCENT PROBLEMS SECTION 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 The amount is the unknown term of the proportion. B We solve the proportion with the methods of Section Divide by the coefficient, 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?

In fact, many students find percent problems easier to approach with the proportion method. First, we will write what is called the percent proportion.

4 498 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 R B Solving, we get 400R R 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 B 40 20, 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.

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.

5 THREE TYPES OF PERCENT PROBLEMS SECTION Remember that a percent (the rate) can be greater than. CHECK YOURSELF 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 So Dividing by yields 37, 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 is 8.5% of what number? Using the percent proportion yields 34 B 8.5 Solving, we have 8.5B 34 or B So 34 is 8.5% of 400. CHECK YOURSELF 8 2.5% of what number is 75?

$We next look at two examples of solving percent problems involving fractions of a percent. Example 8 Solving a Percent Problem NOTE The amount is 34, the rate is 8.5%. We want to find the base.$

6 500 CHPTER 6 PERCENTS Example 9 Estimating Percentages Find 9.3% of 500. Round the rate to 20% 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 % R B B 7200; 72 50R 250; R 25% 6. R B 60 60B 28,800; B R

n estimate of the amount is then Rounded rate Base Estimate of amount CHECK YOURSELF 9 Estimate

7 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 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? is what percent of 2800? is 8% of what number? 4. 7% of what number is 42? 5. Find the base if % of the base is % of what number is 92? is 3% of what number? 8. 2% of what number is 73.5? 9. Find 0% of What is 5% of 600? 2. What is 08% of 4000? 22. Find 60% of is what percent of 20? 24. What percent of 40 is 52? is what percent of 90? 26. What percent of 5,000 is 8,000? is 25% of what number? % of what number is 350? 29. Find the base if 0% of the base is % of what number is 70? 3. Find 8.5% of of 800 is what number? 4 % 33. Find 3 of % 34. What is 3.5% of 500? 35. What is 5.25% of 3000? 36. What is 7.25% of 7600? is what percent of 800? is what percent of 500? 39. What percent of 80 is 20? 40. What percent of 800 is 78? NSWERS

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.

8 NSWERS What percent of 200 is 750? is what percent of 800? % of what number is 420? 44. Find the base if of the base 2 % is is 3% of what number? % of what number is 325? 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 What is 48.3% of 500? % of 600 is what number? % of 200 is what number? 53. Find 52% of 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 $ 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 % 9. 9%. 70% % % % % %

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.

5.4 Solving Percent Problems Using the Percent Equation

5.4 Solving Percent Problems Using the Percent Equation 5. Solving Percent Problems Using the Percent Equation In this section we will develop and use a more algebraic equation approach to solving percent equations. Recall the percent proportion from the last