5.2 Percent: Converting Between Fractions, Decimals, and Percents

Transcription

1 5.2 Percent: Converting Between Fractions, Decimals, and Percents The concept of percent permeates most common uses of mathematics in everyday life. We pay taes based on percents, many people earn income as a percent sales commission, investments and banks compare alternatives using percent, and sports team records are represented as a percent. A percent is a ratio compared with. When we pay 7.25% sales ta, we pay $7.25 ta for every $ in sales price. As a ratio, this can be written as: $7.25 ta $ price = Since the denominator of a percent is, we can easily convert percents to decimals by dividing by. Recall this involves moving the decimal point two place values to the left. Similarly, we can convert percents to reduced fractions. Given the percent 25%, we convert to fraction: 25% = = 25 4 = 1 4 Eample 1 Convert each percent to the indicated quantity. a. 28% to a fraction b. 19.8% to a decimal c. 3.5% to a fraction d. 325% to a decimal Solution a. Writing the percent as a fraction and simplifying: 28% = 28 = = 7 25 b. Writing the percent as a ratio and dividing: 19.8% = 19.8 = c. Writing the percent as a fraction, eliminating decimals, and simplifying: 3.5% = = 35 0 = =

2 d. Writing the percent as a ratio and dividing: 325% = 325 = 3.25 Since percents are used so frequently to compare different ratios, we often want to convert ratios (or fractions) to percents. Suppose we want to convert the fraction 7 8 to a percent. Recall that a percent is a ratio with a denominator of. Thus we want to find the value of such that is equivalent to the original fraction 7. We solve the proportion: 8 = = = 175 = = 87.5% The fraction 7 8 is equivalent to 87.5%. Eample 2 Convert each fraction or mied number to a percent. a. 3 4 b c. 2 3 d

3 Solution a. Solving the proportion: = 3 4 = 3 4 = 75% Thus 3 4 = 75%. b. First write as the fraction 9. Solving the proportion: 2 = 9 2 = 9 2 = 450% Thus = 450%. c. Solving the proportion: = = = 200 Thus 2 3 = %. = = % 374

4 d. First write as the fraction Solving the proportion: = = = 850 Thus = %. = = % Converting a decimal to a percent is etremely easy. Given the decimal 0.824, we are looking for the ratio which is equivalent to Setting up the equation: = = = 82.4% Thus = 82.4%. Note that, in effect, we are merely multiplying the decimal by to convert it to a percent. Eample 3 Convert each decimal to a percent. a b c d

5 Solution a. Solving the equation: = = = 56.9% Thus = 56.9%. b. Solving the equation: = 0.75 = 0.75 = 75% Thus 0.75 = 75%. c. Solving the equation: = = = 167.1% Thus = 167.1%. d. Solving the equation: = = = 0.84% Thus = 0.84%. The remainder of eamples in this section are devoted to applications of percents. 376

6 Eample 4 Solution Mr. Jones pays $12,480 income ta for $46,520 in taable income. Find his income ta rate, epressed as a percent rounded to the nearest hundredth. income ta His ta rate is given as the ratio of. So, we want to find the ratio taable income $12, 480 which is equivalent to. Setting up and solving the proportion: $46,520 12, 480 = 46, = = = ! 26.83% Mr. Jones income ta rate is approimately 26.83%. In the previous eample note that, in solving the proportion, we did not use the LCM of and. Instead, we used a convenient number which we knew would clear fractions in the equation. This is common practice in solving percent proportion applications, since the numbers are often fairly large and finding the LCM can be time consuming. Eample 5 Solution At one point of the season, the San Francisco Giants had won 26 games and lost 14 games. Epress their win to games played ratio as a percent. First note that they have played = 40 games. Their win to games played ratio is 26 40, which we want to epress as a percent. Setting up and solving the proportion: = = = 130 = = 65% The Giants winning percent is 65%. For baseball fans, note that this is equivalent to 0.650, which is how these winning percents are represented in the media. 377

7 Eample 6 Solution A car salesperson earns $ commission on the sale of a car with a purchase price of $26,500. If her commission rate is a percent based on the purchase price, find the salesperson s commission rate epressed as a percent. commission Her commission rate is given as the ratio. We want to find the purchase price $ ratio which is equivalent to. Setting up and solving the $26,500 proportion: = = = = = 2.5% The salesperson s commission rate is 2.5%. In the net section we will solve a variety of additional percent problems. Terminology percent Eercise Set 5.2 Convert each percent to a fraction. Be sure to reduce all answers % 2. 75% 3. 48% 4. 56% 5. 90% 6. 24% % % % % % % % % % % 378

8 Convert each percent to a decimal % % % % % % % % % % % % Convert each fraction or mied number to a percent Convert each decimal to a percent

9 Solve each of the following percent applications. 59. A small business pays $13,860 ta for $93,500 in taable income. Find their ta rate, epressed as a percent rounded to the nearest hundredth. 60. A small business pays $22,691 ta for $121,640 in taable income. Find their ta rate, epressed as a percent rounded to the nearest hundredth. 61. You pay $56.25 sales ta on a purchase of $1250. Find the sales ta rate, epressed as a percent. 62. You pay $55.90 sales ta on a purchase of $860. Find the sales ta rate, epressed as a percent. 63. A salesperson earns $1,467 commission on sales of $32,600. If the commission is based on the sales, find the salesperson s commission rate epressed as a percent. 64. A salesperson earns $2,250 commission on sales of $125,000. If the commission is based on the sales, find the salesperson s commission rate epressed as a percent. 65. A company has 23 full-time employees and 17 part-time employees. What percent of its employees are full-time employees? 66. A small hot dog stand makes 125 jumbo dogs and 75 chili dogs for a baseball game. What percent of its hot dogs made are chili dogs? 67. During one week, Frank has cleaning orders for 275 shirts and 125 pants. What percent of his cleaning orders are pants? 68. During one day, Todd makes 18 business-related phone calls and 22 personal phone calls. What percent of his phone calls are personal calls? 69. During a playoff game, Michael Jordan made 19 shots out of 32 shot attempts. Epress his shots made as a percent of shot attempts. 70. During a playoff game, Kobe Bryant made 15 shots out of 25 shot attempts. Epress his shots made as a percent of shot attempts. 71. At one point of the season, the Chicago Cubs had won 32 games and lost 38 games. Epress their win to games played ratio as a percent, rounded to the tenths place. 72. At the end of a season, the Atlanta Braves had won 102 games and lost 60 games. Epress their win to games played ratio as a percent, rounded to the tenths place. 73. An eperimental asthma drug is tested on 429 people, and 391 show improvement. What percent of the tested people showed improvement? Round your answer to the nearest tenth of a percent. 74. An eperimental chemotherapy drug is tested on 1200 people, and 965 show decrease in cancer levels. What percent of the tested people showed decrease in cancer levels? Round your answer to the nearest tenth of a percent. 75. A large computer manufacturer reduces prices of its computers from $1400 to $1190. Epress the ratio of amount of price decrease to original price as a percent. 76. A cellular phone company reduces prices of its phones from $120 to $ Epress the ratio of amount of price decrease to original price as a percent. 380

10 77. An investment earns $380 profit on an investment of $2,000. Epress the profit to investment ratio as a percent. 78. An investment earns $900 profit on an investment of $5,000. Epress the profit to investment ratio as a percent. 381