5.4 Solving Percent Problems Using the Percent Equation

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1 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 section: percent 100 = percentage base Considering that percent 100 of it in the equation: is merely a percent divided by 100, we can consider the decimal form percent = percentage base base percent = base percentage base This last statement is called the percent equation. Remember that the term percent refers to the decimal (or fraction) form, which has already been divided by 100. This percent equation involves the familiar quantities from the past section: percent, base, and percentage. Once identified, the method of solving the equation utilizes our steps for solving equations from Chapter. Suppose we are asked to find 18% of 50. The percent is 18% = 0.18 (converted to a decimal), and the base is 50. x = x = 5 Thus 18% of 50 is 5. Let s use this approach to re-solve Example 1 from the last section. Example 1 Find the following percentages. a. 8% of 10 b. 5% of 86 c. 7.5% of 1500 d % of 00 e. 150% of

2 Solution a. The percent is 8% = 0.8 (converted to decimal) and the base is 10. x = x = 57.6 Thus 8% of 10 is b. The percent is 5% = 0.5 (converted to decimal) and the base is 86. x = x = 18.7 Thus 5% of 86 is c. The percent is 7.5% = (converted to decimal) and the base is x = x = Thus 7.5% of 1500 is d. The percent is 15 1 % = 15.5% = (converted to decimal) and the base is 00. x = x = 37 Thus 15 1 % of 00 is 37. e. The percent is 150% = 1.50 (converted to decimal) and the base is 60. x = x = 390 Thus 150% of 60 is

3 The second type of percent problem involved one in which the percentage is given and the base is unknown. Suppose we know that 68% of a number is 0. The percent is 68% = 0.68, converted to a decimal. This time the percentage is 0, and the base is unknown. Using the percent equation: 0.68x = 0 x = = 300 Thus 68% of 300 is 0. Note that the percent equation provides an easy check of our answer, since = 0, verifying our answer. As more practice, let s re-solve Example from the last section. Example Solve the following percent problems. a. % of what number is equal to 10? b. 35% of what number is equal to 51.8? c. 7.5% of what number is equal to 70? d. 6 1 %of what number is equal to 75? e. 15% of what number is equal to 800? Solution a. The percent is % = 0. (converted to decimal) and the percentage is x = 10 x = = 500 Thus % of 500 is 10. Checking the value: = 10. b. The percent is 35% = 0.35 (converted to decimal) and the percentage is x = 51.8 x = = 18 Thus 35% of 18 is Checking the value: =

4 c. The percent is 7.5% = (converted to decimal) and the percentage is x = 70 x = = 3600 Thus 7.5% of 3600 is 70. Checking the value: = 70. d. The percent is 6 1 % = 6.5% = (converted to decimal) and the percentage is x = 75 x = = 00 Thus 6 1 % of 00 is 75. Checking the value: = 75. e. The percent is 15% = 1.5 (converted to decimal) and the percentage is x = 800 x = = 60 Thus 15% of 60 is 800. Checking the value: = 800. The third type of percent problem is where both the percentage and base are given, but the percent is unknown. Suppose we want to know what percent 08 represents out of 30. The percentage is 08, the base is 30, and the percent is unknown. p 30 = 08 p = p = 0.65 = 65% 398

5 Thus 08 represents 65% of 30. Again note that we can check the equation: = 08. Note the use of the variable p in this equation. It was used as a reminder that p represents a percent, and thus must be converted back to a percent for the final step. This is a good habit to get into in solving percent equations. Now let s re-solve Example 3 from the last section. Example 3 Find the following percents. a. 36 is what percent of 80? b. 13 is what percent of 165? c. 10 is what percent of 180? d. 00 is what percent of 160? Solution a. The percentage is 36, the base is 80, and the percent is unknown. p 80 = 36 p = p = 0.5 = 5% Thus 36 is 5% of 80. Checking the value: = 36. b. The percentage is 13, the base is 165, and the percent is unknown. p 165 = 13 p = p = 0.80 = 80% Thus 13 is 80% of 165. Checking the value: = 13. c. The percentage is 10, the base is 180, and the percent is unknown. p 180 = 10 p = p = 3 = 66 3 % 399

6 Thus 10 is 66 % of 180. Note in this problem we did not convert to a 3 decimal, since a repeating decimal would have resulted. Checking the value: 180 = d. The percentage is 00, the base is 160, and the percent is unknown. p 160 = 00 p = p = 1.5 = 15% Thus 00 is 15% of 160. Checking the value: = 00. Often students find the percent equation easier to use than the proportion method. Keep in mind that both methods require you to recognize the percent, the base, and the percentage. We will re-solve Example from the previous section, however this time we use the percent equation approach, rather than percent proportions. Example Solve the following percent problems. a. 1 %of 60 is what number? b. 35% of what number is 30.1? c. 11 is what percent of 10? d. 97 represents 5% of what number? e. 19.8% of 100 is what number? f. 0 represents what percent of 180? 00

7 Solution a. The percent, base, and percentage are: percent = 1 % =.5% = 0.05 base = 60 percentage = x (unknown) x = x = 7.9 Thus 1 % of 60 is 7.9. b. The percent, base, and percentage are: percent = 35% = 0.35 base = x (unknown) percentage = x = 30.1 x = = 86 Thus 35% of 86 is Checking the value: = c. The percent, base, and percentage are: percent = p (unknown) base = 10 percentage = 11 p 10 = 11 p = p = 0.80 = 80% Thus 11 is 80% of 10. Checking the value: =

8 d. The percent, base, and percentage are: percent = 5% = 0.5 base = x (unknown) percentage = x = 97 x = = 660 Thus 97 represents 5% of 660. Checking the value: = 97. e. The percent, base, and percentage are: percent = 19.8% = base = 100 percentage = x (unknown) x = x = 37.6 Thus 19.8% of 100 is f. The percent, base, and percentage are: percent = p (unknown) base = 180 percentage = 0 p 180 = 0 p = p = 3 = % Thus 0 represents % of 180. Checking the value: 180 = 0 3 Note again we used fractions here to avoid using repeating decimals. 0

9 Terminology percent equation Exercise Set 5. Solve the following percent problems. Remember that you will need to recognize the percent, the base, and the percentage in order to set up the percent equation % of 00 is what number?. 65% of 18 is what number? 3. 35% of what number is 80?. 55% of what number is 75? is what percent of 160? is what percent of 00? 7. 37% of 150 is what number? 8. 56% of 10 is what number? 9. 5 is what percent of 60? is what percent of 50? % of what number is 109.6? 1. 57% of what number is 86.5? % of,000 is what number? % of 1,300 is what number? % of what number is 37.5? % of what number is 316.8? is what percent of 7? is what percent of 900? %of 900 is what number? %of 800 is what number? 1. 10% of what number is 91?. 160% of what number is 83.? is what percent of 150?. 500 is what percent of 360? % of 100 is what number? %of 80 is what number? % of what number is 8.075? %of what number is 90.3? is what percent of 0? is what percent of 50? % of 30 is what number? 3. 8% of 50 is what number? %of what number is 100? 3. 9 % of what number is 17.9? is what percent of 86? is what percent of 70? % of 80 is what number? % of 6 is what number? % of what number is 0? 0. 0% of what number is 1? 03