Ratio, Proportion, and Other Applied Problems Section 7.6 Ratio, Proportion, and Other Applied Problems

Ratio and Proportion A ratio is a comparison of two quantities. A proportion is an equation that states that two ratios are equal. The Proportion Equation If then ad = bc for all real numbers a, b, c, and d, where b ≠ 0 and d ≠ 0. This is sometimes called cross multiplying. 2

Example If ¾ inch on a map represents an actual distance of 20 miles, how long is the distance represented by inches on the same map. Initial measurement on map 2nd measurement on map Initial distance 2nd distance 5 2 Continued 3

Example (cont) 2 1 inches on the map represents an actual distance of 110 miles. 4

Example A B Triangles A and B are similar. Find the length of side y. 17 14 n A 23 x y B 1. Understand the problem. Similar triangles have the same shape, but different sizes. The corresponding sides of the triangles are proportional. 2. Write an equation. Base of Triangle A Short side of Triangle A Base of Triangle B Short side of Triangle B Continued 5

Example (cont) A B 3. Solve and state the answer. Cross multiply. Divide each side by 14. 17 14 n A The length of the base of Triangle B is 23 x B 6

distance = rate × time (d = rt). Example Terry drove his car to Cleveland while Kathy drove her car to Columbus. Terry drove 360 kilometers while Kathy drove 280 kilometers. Terry drove 20 kilometers per hour faster than Kathy on his trip. What was the average speed in kilometers per hour for each driver? 1. Understand the problem. Distance problems can be solved using the formula distance = rate × time (d = rt). Let r = the rate of Kathy’s car. Let r + 20 = the rate of Terry’s car. The time, t, for each driver was the same. Continued 7

Example (cont) d Make a table. 2. Write an equation. 280 Kathy r + 20 360 Terry d 2. Write an equation. Since the time for each driver was the same, we can set the times equal to each other. Continued 8

Example (cont) 3. Solve and state the answer. Cross multiply. Distribute to remove parentheses. Subtract 280r from each side. Divide each side by 80. Kathy’s rate was 70 kilometers per hour. Terry’s rate was 90 kilometers per hour. 9

Example At Key Bank it takes a computer 4 hours to process and print monthly statements. When a second computer is used and the two computers work together, the statements can be printed in 3 hours. How long would it take the second computer by itself to process and print the monthly statements? 1. Understand the problem. If it takes the first computer 4 hours to print the statements, then in 1 hour, the computer would finish of the job. If it takes both computers working together 3 hours to print the statements, then in 1 hour, both computers would finish of the job. Continued 10

Example (cont) Let x = the number of hours it takes the second computer to finish the job. In one hour, the second computer would finish of the job. 2. Write an equation. Amount of work done by first computer Amount of work done by second computer 1 (one whole task completed) + = + = Continued 11

Example (cont) 3. Solve and state the answer. Multiply each term by 12x. Simplify. Subtract 3x from each side. When working alone, it would take the second computer 12 hours to print the statements. 12