6.4 Three Types of Percent Problems - McGraw Hill …

1 Three Types of Percent Problems 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 , 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.

2 Example 1. Finding an Unknown Amount What is 18% of 300? NOTE Type 1: Finding an We know the rate, 18%; and the base, 300; the amount is the unknown. Using the unknown amount. Percent relationship, we can translate the problem to an equation. Rate Base Write 18% as the decimal by the Amount 300 rule of Section Then multiply to find the amount. 54. So 54 is 18% of 300. CHECK YOURSELF 1. Find 65% of 200. 2001 McGraw -Hill Companies 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.

3 Let's consider a second type of Percent problem involving an unknown rate. 495. 496 CHAPTER 6 PERCENTS. Example 2. Finding an Unknown Percent 30 is what Percent of 150? NOTE Type 2: Finding an We know the amount, 30, and the base, 150; the rate (what Percent ) is the unknown. unknown Percent . Again using the Percent relationship to translate to an equation, we have Base Amount Rate 150 30. NOTE This will leave the rate We divide both sides by 150 to find the rate. alone on the left.. 30 1 1. Rate 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%.

4 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 28 is 40% of what number? NOTE Type 3: Finding an We know the amount, 28, and the rate, 40%. The base (what number) is the unknown. unknown base. From the Percent relationship we have Rate Amount NOTE Notice that 40% is base 28. written as We divide both sides by to find the base. 28. Base 70. 2001 McGraw -Hill Companies So 28 is 40% of 70. CHECK YOURSELF 3. 70 is 35% of what number? Three Types OF Percent Problems SECTION 497. We have now seen solution methods for the Three basic Types of Percent Problems : find- ing the amount, the rate, and the base.

5 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 iden- tical 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 A R. 100 . rate, and this proportion is B 100. equivalent to our earlier Percent relationship. Because in any Percent problem we know two of the Three quantities (A, B, or R), we can always solve for the unknown term.

6 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. The amount A is the unknown A 30. term of the proportion. 150 100. B. We solve the proportion with the methods of Section 100A 150 30. 100A 4500. Divide by the coefficient, 100. 1. 100 A 4500.. 100 100. 2001 McGraw -Hill Companies 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.

7 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 R, the rate, is the unknown term in this case. 400 100. 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 Finally, we use the same proportion to find an unknown base. Example 6. Solving a problem Involving an Unknown Base 40% of _____ is 200.

8 NOTE This is an unknown-base problem . R B A. Substitute the known values into the Percent proportion. A R. 200 40 In this case B, the base, is the unknown term of the proportion. B 100. 2001 McGraw -Hill Companies Solving gives 40B 200 100. 1. 40B 20,000.. 40 40. 1. B 500. The base is 500, and 40% of 500 is 200. Three Types OF Percent Problems SECTION 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 What is 125% of 300? NOTE The rate is 125%. The base is 300. In the Percent proportion, we have A 125. NOTE When the rate is greater.

9 Than 100%, the amount will be 300 100. greater than the base. So 100A 300 125. Dividing by 100 yields 37,500. A 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 of what number? Using the Percent proportion yields NOTE The amount is 34, the 34 rate is We want to find . B 100. the base. Solving, we have 34 100. or 2001 McGraw -Hill Companies 3400. NOTE Divide by B 400. So 34 is of 400. CHECK YOURSELF 8. of what number is 75? 500 CHAPTER 6 PERCENTS. Example 9. Estimating Percentages Find of 500.

10 1. Round the rate to 20% as a fraction, . An estimate of the amount is then 5. 1. 500 100. 5. Rounded rate Base Estimate of amount CHECK YOURSELF 9. Estimate the amount. of 800. CHECK YOURSELF ANSWERS. 1. 130 2. 25% 3. 200. 4. R 5. A. A 24 R.. 300 100 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 Name Exercises Section Date Solve each of the following Problems involving Percent . 1. What is 35% of 600? 2. 20% of 400 is what number? ANSWERS. 1. 2. 3. 45% of 200 is what number? 4. What is 40% of 1200? 3. 4. 5. Find 40% of 2500.