Solving Proportions and Percent Equations

Transcription

1 Lesson 3.2 Objectives Use cross-products to solve proportions. Use proportions and percent equations to solve percent problems. Solving Proportions and Percent Equations The scale drawing of an office building has a scale of 1 inch to 4 feet. This means that one inch on the drawing is the same as four feet in the actual office building. The scale is written as 1" : 4'. A ratio is a fraction that compares two quantities. Thus, at c you can also write the scale as 1 ".. However, it is 4' better to keep the 4 ratio fe in the same units. Because 4 feet equals 48 inches, this ratio can be written in one of three ways: 1 1" : 48" 1 : scale ratio fraction The length of the office building on the drawing is 12 inches. How can you find the full-scale length? '-0" '-6" 2'-3" 2'-3" 3'-6" 4'-6" 2'-3" '-3" '-6" " REV. No 1 REV. No 2 FOUNDATION AND FIRST FLOOR FRAMING PLAN REVISIONS 23 NOV DEC A ELECTRONICS OFFICES AND SALES GRAHAM HOLCOMB ARCHITECT COMMISSION 5907 DATE NOV. 1, 1996 SCALE 1" = 4' DRAWING SI 148 Chapter 3 Solving Equations

2 Proportions A proportion is an equation with a ratio on each side of the equal sign. The two middle values in the ratios of a proportion are the means. The first and the last values in the ratios of a proportion are the extremes. Multiplying the means and then the extremes is an example of cross-multiplying. Activity Cross-Multiplication 1 Find the product of the means. Then find the product of the extremes. a. 24; 24; b. 60; 60; c. 40; 40 a b c What do you notice about the results of the products of the means and the product of the extremes in parts a, b, and c? The products are equal. 3 Use your observation to solve the proportion below d Find the product of the means. Find the product of the extremes. 10d; 60 4 Write an equation. Solve for d. 10d 5 60; d Check your answer by substituting 6 for d in the proportion. Find the product of the means. Find the product of the extremes. ; 60; 60 Cross-multiplying is a method used to test whether two ratios are equal. For example, you can use cross-multiplication to show the two fractions, 3 6 and 4 8, are equal. Find the product of the means: Find the product of the extremes: The cross-products are equal. Thus, the ratios are equal. This leads to an important property for proportions. Proportion Property In a true proportion, the cross-products are equal. 3.2 Solving Proportions and Percent Equations 149

3 Example 1 Solving a Proportion Reread the opening paragraph of this lesson. Find the full-scale length of the office building. Solution 1 The proportion s2 represents this situation, where s is the full-scale length of the office h building. the la Cross multiply to find the value of s. 1 1 Given 4 8 s2 1 s Proportion Property s 576 Identity Property of Multiplication Thus, the actual length of the office building is 576 inches or 48 feet. Ongoing Assessment A scale drawing for a rectangular tabletop measures 3 inches wide and 4 inches long. You want to make a table that is 5 feet wide. How long will it be? Show your work. Express your answer in feet and inches. 6 feet 8 inches Percent Equations Percent means hundredths of a quantity. For example, 8% is just another way to write the ratio 8:100 or You can use proportions to change fractions or decimals to percents. mals Example 2 Changing Fractions to Percents Change 5 8 to a percent. Solution Write a proportion and solve. 5 8 x 100 x represents a % 100 Thus, 5 8 is equal to 62.5%. 8x 500 x 62.5 Proportion Property Although proportions solve percent problems, it is usually easier to use a percent equation. One way to write a percent equation is a% of b c. In this form, a is the percent, b is the base, and c is the percentage. 150 Chapter 3 Solving Equations

4 Example 3 Using Percent to Find Sales Shannon earns 6% of her sales. If Shannon earns $150, what are her sales? Solution Use a percent equation. The percent is 6, and the percentage is 150. Let b represent the base (Shannon s sales). Write a percent equation b b Multiply each side by b 2,500 Shannon has sales of $2,500. Problem Solving Using the Four-Step Plan see margin Write and solve a proportion for this problem. Jodie has a map with scale 1 cm 500 km. If the distance between two cities on this map is 1.75 cm, what is the actual distance between these cities? Step 1 Understand the Problem Which distance between the two cities is given? Which distance do you have to find? What given information relates the two types of distances? Why is a proportion an appropriate model? Step 2 Develop a Plan Problem-solving strategy: Use an equation. Write the given scale in the form of a ratio, map distance actual distance. Write a word equation in the form of a proportion to show correspondence between map distance and actual distance. Step 3 Carry Out the Plan Identify a variable for the unknown quantity. Write a proportion following the pattern of your word equation. Include units so that you can see the proper unit for the result. Cross multiply to solve the proportion. Step 4 Check the Results Determine if you have equivalent ratios: What is the ratio between the given map distance for the two cities and your resulting actual distance? Is this the same as the ratio in the given scale? 3.2 Solving Proportions and Percent Equations 151

5 Lesson Assessment Think and Discuss 1. How are ratios and proportions related? Give an example. 2. Explain the meaning of the Proportion Property. 3. Explain how to use a proportion to change a fraction to a percent. 4. Explain how to use a percent equation to find what percent one number is of another. Practice and Problem Solving Solve each proportion. r r m 9 m t a 16 t a 8 a 9. 3 a m 7 m n x n 12 x c 25 c 10 Set up a proportion and solve. Round answers to the nearest tenth. see margin for proportions 14. What is 25% of 120? What percent of 10 is 6? 60% is what percent of 256? 25% 17. Find 12.5% of is what percent of 9? 66.7% is what percent of 6? 150% Set up a percent equation and solve. Round answers to the nearest tenth. see margin for equations 20. What percent of 300 is 13.5? 21. What is 300% of $20? 4.5%. $ What is 0.5% of $120? 23. What is 15% of $225? $0.60. $ $2.50 is what percent of $10? 25. What percent of 12 is 2.4? 25%. 20% 26. What percent of 84 is 63? 27. What is 22% of $275.00? 75%. $ $58.50 is what percent % of what number of $450? 13%. is 81.9? 1, Chapter 3 Solving Equations

6 For each problem, set up a proportion or percent equation and solve. see margin for equations 30. Keesha bought a sweater for $50. If the sales tax rate is 8.5%, find the total amount Keesha paid the cashier. $ Sami borrowed $1,200 for one year. If Sami paid $90 simple interest, what was the interest rate? 7.5% 32. Tom pays 6.2% of his wages for social security. If Tom paid $186 in social security taxes, what was his income? $3, Lian sells cosmetics at a department store. She earns $275 per week, plus a 9% commission on all her sales. How much will Lian earn in one week if her sales total $1,875 for the week? $ Rhonda buys her plumbing supplies at wholesale. She buys pipe for $4/ft and sells it for $5/ft. What percent profit is she making? 20% Mixed Review 35. Simplify the expression ( )( ) Water is stored in a rectangular container that is 8 inches by 12 inches by 4 inches. The water is transferred to a spherical container with a radius of 10 inches. Is the spherical container large enough to hold the water? By how much is it too small or too large? The spherical tank is about 3,804 cubic inches larger. 4 in. r 10 in. 12 in. 8 in. 37. Evaluate 8c 2d 4 when c is 5 and d is Solve the equation mx b for x. Show your steps. What properties did you use? ; Division Property of Equality 3.2 Solving Proportions and Percent Equations 153