MATH 125 Section 3.2 - Pasadena City College | MATH125 Section 3.2 - Pasadena City College

Solve Percent Applications Learning Objectives By the end of this section, you will be able to: Translate and solve basic percent equations Solve percent applications Find percent increase and percent decrease Solve simple interest applications Solve applications with discount or mark-up Be Prepared! Before you get started, take this readiness quiz. 1. Convert 4.5% to a decimal. If you missed this problem, review Example 1.26. 2. Convert 0.6 to a percent. If you missed this problem, review Example 1.26. 3. Round 0.875 to the nearest hundredth. If you missed this problem, review Example 1.26. 4. Multiply (4.5)(2.38). If you missed this problem, review Example 1.26. 5. Solve 3.5 = 0.7n. If you missed this problem, review Example 1.26. 6. Subtract 50 − 37.45. If you missed this problem, review Example 1.26. Translate and Solve Basic Percent Equations We will solve percent equations using the methods we used to solve equations with fractions or decimals. Without the tools of algebra, the best method available to solve percent problems was by setting them up as proportions. Now as an algebra student, you can just translate English sentences into algebraic equations and then solve the equations. We can use any letter you like as a variable, but it is a good idea to choose a letter that will remind us of what you are looking for. We must be sure to change the given percent to a decimal when we put it in the equation. EXAMPLE 3.12 Translate and solve: What number is 35% of 90? Solution Translate into algebra. Let n = the number. Remember "of" means multiply, "is" means equals. Multiply. 31.5 is 35% of 90 TRY IT : : 3.23 Translate and solve: What number is 45% of 80? 312 Chapter 3 Math Models This OpenStax book is available for free at What number is 55% of 60? We must be very careful when we translate the words in the next example. The unknown quantity will not be isolated at first, like it was in Example 3.12. We will again use direct translation to write the equation. EXAMPLE 3.13 Translate and solve: 6.5% of what number is $1.17? Solution Translate. Let n = the number. Multiply. Divide both sides by 0.065 and simplify. 6.5% of $18 is $1.17 TRY IT : : 3.25 Translate and solve: 7.5% of what number is $1.95? TRY IT : : 3.26 Translate and solve: 8.5% of what number is $3.06? In the next example, we are looking for the percent. EXAMPLE 3.14 Translate and solve: 144 is what percent of 96? Solution Translate into algebra. Let p = the percent. Multiply. Divide by 96 and simplify. Convert to percent. 144 is 150% of 96 Note that we are asked to find percent, so we must have our final result in percent form. TRY IT : : 3.27 Translate and solve: 110 is what percent of 88? TRY IT : : 3.28 Translate and solve: 126 is what percent of 72? Chapter 3 Math Models 313Solve Applications of Percent Many applications of percent—such as tips, sales tax, discounts, and interest—occur in our daily lives. To solve these applications we’ll translate to a basic percent equation, just like those we solved in previous examples. Once we translate the sentence into a percent equation, we know how to solve it. We will restate the problem solving strategy we used earlier for easy reference. Now that we have the strategy to refer to, and have practiced solving basic percent equations, we are ready to solve percent applications. Be sure to ask yourself if your final answer makes sense—since many of the applications will involve everyday situations, you can rely on your own experience. EXAMPLE 3.15 Dezohn and his girlfriend enjoyed a nice dinner at a restaurant and his bill was $68.50. He wants to leave an 18% tip. If the tip will be 18% of the total bill, how much tip should he leave? Solution Step 1. Read the problem. Step 2. Identify what we are looking for. the amount of tip should Dezohn leave Step 3. Name what we are looking for. Choose a variable to represent it. Let t = amount of tip. Step 4. Translate into an equation. Write a sentence that gives the information to find it. Translate the sentence into an equation. Step 5. Solve the equation. Multiply. Step 6. Check. Does this make sense? Yes, 20% of $70 is $14. Step 7. Answer the question with a complete sentence. Dezohn should leave a tip of $12.33. Notice that we used t to represent the unknown tip. TRY IT : : 3.29 Cierra and her sister enjoyed a dinner in a restaurant and the bill was $81.50. If she wants to leave 18% of the total bill as her tip, how much should she leave? HOW TO : : USE A PROBLEM-SOLVING STRATEGY TO SOLVE AN APPLICATION. Read the problem. Make sure all the words and ideas are understood. Identify what we are looking for. Name what we are looking for. Choose a variable to represent that quantity. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation. Solve the equation using good algebra techniques. Check the answer in the problem and make sure it makes sense. Answer the question with a complete sentence. Step 1. Step 2. Step 3. Step 4. Step 5. Step 6. Step 7. 314 Chapter 3 Math Models This OpenStax book is available for free at Kimngoc had lunch at her favorite restaurant. She wants to leave 15% of the total bill as her tip. If her bill was $14.40, how much will she leave for the tip? EXAMPLE 3.16 The label on Masao’s breakfast cereal said that one serving of cereal provides 85 milligrams (mg) of potassium, which is 2% of the recommended daily amount. What is the total recommended daily amount of potassium? Solution Step 1. Read the problem. Step 2. Identify what we are looking for. the total amount of potassium that is recommended Step 3. Name what we are looking for. Choose a variable to represent it. Let a = total amount of potassium. Step 4. Translate. Write a sentence that gives the information to find it. Translate into an equation. Step 5. Solve the equation. Step 6. Check. Does this make sense? Yes, 2% is a small percent and 85 is a small part of 4,250. Step 7. Answer the question with a complete sentence. The amount of potassium that is recommended is 4,250 mg. TRY IT : : 3.31 One serving of wheat square cereal has seven grams of fiber, which is 28% of the recommended daily amount. What is the total recommended daily amount of fiber? TRY IT : : 3.32 One serving of rice cereal has 190 mg of sodium, which is 8% of the recommended daily amount. What is the total recommended daily amount of sodium? EXAMPLE 3.17 Mitzi received some gourmet brownies as a gift. The wrapper said each brownie was 480 calories, and had 240 calories of fat. What percent of the total calories in each brownie comes from fat? Solution Step 1. Read the problem. Step 2. Identify what we are looking for. the percent of the total calories from fat Step 3. Name what we are looking for. Chapter 3 Math Models 315Choose a variable to represent it. Let p = percent of fat. Step 4. Translate. Write a sentence that gives the information to find it. Translate into an equation. Step 5. Solve the equation. Divide by 480. Put in a percent form. Step 6. Check. Does this make sense? Yes, 240 is half of 480, so 50% makes sense. Step 7. Answer the question with a complete sentence. Of the total calories in each brownie, 50% is fat. TRY IT : : 3.33 Solve. Round to the nearest whole percent. Veronica is planning to make muffins from a mix. The package says each muffin will be 230 calories and 60 calories will be from fat. What percent of the total calories is from fat? TRY IT : : 3.34 Solve. Round to the nearest whole percent. The mix Ricardo plans to use to make brownies says that each brownie will be 190 calories, and 76 calories are from fat. What percent of the total calories are from fat? Find Percent Increase and Percent Decrease People in the media often talk about how much an amount has increased or decreased over a certain period of time. They usually express this increase or decrease as a percent. To find the percent increase, first we find the amount of increase, the difference of the new amount and the original amount. Then we find what percent the amount of increase is of the original amount. EXAMPLE 3.18 In 2011, the California governor proposed raising community college fees from $26 a unit to $36 a unit. Find the percent increase. (Round to the nearest tenth of a percent.) Solution Step 1. Read the problem. HOW TO : : FIND THE PERCENT INCREASE. Find the amount of increase. new amount − original amount = increase Find the percent increase. The increase is what percent of the original amount? Step 1. Step 2. 316 Chapter 3 Math Models This OpenStax book is available for free at Step 3. Name what we are looking for. Choose a variable to represent it. Let p = the percent. Step 4. Translate. Write a sentence that gives the information to find it. First find the amount of increase. new amount − original amount = increase 36 − 26 = 10 Find the percent. Increase is what percent of the original amount? Translate into an equation. Step 5. Solve the equation. Divide by 26. Change to percent form; round to the nearest tenth. Step 6. Check. Does this make sense? Yes, 38.4% is close to 1 3 , and 10 is close to 1 3 of 26. Step 7. Answer the question with a complete sentence. The new fees represent a 38.4% increase over the old fees. Notice that we rounded the division to the nearest thousandth in order to round the percent to the nearest tenth. TRY IT : : 3.35 Find the percent increase. (Round to the nearest tenth of a percent.) In 2011, the IRS increased the deductible mileage cost to 55.5 cents from 51 cents. TRY IT : : 3.36 Find the percent increase. In 1995, the standard bus fare in Chicago was $1.50. In 2008, the standard bus fare was $2.25. Finding the percent decrease is very similar to finding the percent increase, but now the amount of decrease is the difference of the original amount and the new amount. Then we find what percent the amount of decrease is of the original amount. EXAMPLE 3.19 HOW TO : : FIND THE PERCENT DECREASE. Find the amount of decrease. original amount − new amount = decrease Find the percent decrease. Decrease is what percent of the original amount?