5. An equation that states that two ratios are equivalent is called a proportion . For example, the equation, or proportion, states that the ratios and are equivalent. Ratios that are equivalent are said to be proportional , or in proportion . 4 6 2 3 = 4 6 2 3
6. Proportion a∙ d = b ∙ c Cross Products One way to find whether two ratios are equivalent is to find their cross products. In the proportion , the products a ∙ d and b ∙ c are called cross products . c d a b = c d a b =
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8. Tell whether the ratios are proportional. Since the cross products are equal, the ratios are proportional. Additional Example 1A: Using Cross Products to Identify Proportions Find the cross products. 60 = 60 4 10 6 15 = ? 4 10 6 15 = ? 6 10 = 4 15 ?
9. A mixture of fuel for a certain small engine should be 4 parts gasoline to 1 part oil. If you combine 5 quarts of oil with 15 quarts of gasoline, will the mixture be correct? The ratios are not equal. The mixture will not be correct. Set up equal ratios. Find the cross products. Additional Example 1B: Using Cross Products to Identify Proportions 20 15 4 parts gasoline 1 part oil = ? 15 quarts gasoline 5 quarts oil 15 5 4 1 = ? 4 5 = 1 15 ?
10. Tell whether the ratios are proportional. Check It Out! Example 1A Since the cross products are equal, the ratios are proportional. Find the cross products. 20 = 20 2 4 5 10 = ? 2 4 5 10 = ? 5 4 = 2 10 ?
11. A mixture for a certain brand of tea should be 3 parts tea to 1 part sugar. If you combine 4 tablespoons of sugar with 12 tablespoons of tea, will the mixture be correct? Check It Out! Example 1B The ratios are equal. The mixture will be correct. Set up equal ratios. Find the cross products. 12 = 12 3 parts tea 1 part sugar = ? 12 tablespoons tea 4 tablespoons sugar 12 4 3 1 = ? 3 4 = 1 12 ?
12. The ratio of the length of the actual height of a person to the length of the shadow cast by the person is 1:3. At the same time, a lighthouse casts a shadow that is 36 meters long. What should the length of its shadow be? Write a ratio comparing height of a person to shadow length. Set up the proportion. Let x represent the shadow length. Additional Example 2: Using Properties of Equality to Solve Proportions 1 3 height of person length of shadow Since x is divided by 36, multiply both sides of the equation by 36. 12 = x The length of the lighthouse’s shadow should be 12 meters. 1 3 = x 36 (36) = (36) 1 3 x 36
13. For most cats, the ratio of the length of their head to their total body length is 1:5. If a cat is 20 inches in length, what should the total length of their head be? Write a ratio comparing head length to total length. Set up the proportion. Let x represent the length of the cat's head. Check It Out! Example 2 1 5 head length total length Since x is divided by 20, multiply both sides of the equation by 20. 4 = x The length of the cat's head should be 4 inches. 1 5 = x 20 (20) = (20) 1 5 x 20
14. Allyson weighs 55 pounds and sits on a seesaw 4 feet away from it center. If Marco sits on the seesaw 5 feet away from the center and the seesaw is balanced, how much does Marco weigh? 55 ∙ 4 = 5 w Find the cross products. Divide both sides by 5. Additional Example 3: Using Cross Products to Solve Proportions 44 = w Simplify. Set up a proportion using the information. Let w represent Marco’s weight. Marco weighs 44 lb. 5 w 5 220 5 = = weight 1 length 2 weight 2 length 1 55 5 = w 4
15. Austin weighs 32 pounds and sits on a seesaw 6 feet away from it center. If Kaylee sits on the seesaw 4 feet away from the center and the seesaw is balanced, how much does Kaylee weigh? 32 ∙ 6 = 4 w Find the cross products. Divide both sides by 4. Check It Out! Example 3 48 = w Simplify. Set up a proportion using the information. Let w represent Kaylee’s weight. Kaylee weighs 48 lbs. 4 w 4 192 4 = = weight 1 length 2 weight 2 length 1 32 4 = w 6
16. 30 ∙ 225 = 45 x Find the cross products. Divide both sides by 45. 150 = x Simplify. It will take 150 minutes to complete the job. Nate has already spent 30 minutes, so it will take him 150 – 30 = 120 more minutes to finish the job. Nate has 225 envelopes to prepare for mailing. He takes 30 minutes to prepare 45 envelopes. If he continues at the same rate, how many more minutes until he has completed the job? Additional Example 4: Business Application Set up the proportion. Let x represent the number of minutes it takes to complete the job. 45 x 45 6750 45 = 30 45 = x 225
17. 21 ∙ 160 = 24 m Find the cross products. Divide both sides by 24. 140 = m Simplify. It will take 140 minutes to complete the job. Nemo has already spent 21 minutes, so it will take him 140 – 21 = 119 more minutes to finish the job. Nemo has to make 160 muffins for the bake sale. He takes 21 minutes to make 24 muffins. If he continues at the same rate, how many more minutes until he has completed the job? Check It Out! Example 4 Set up the proportion. Let m represent the number of minutes it takes to complete the job. 24 m 24 3360 24 = 21 24 = m 160
18. Lesson Quiz: Part I Tell whether the ratios are proportional. 1. 2. 3. The ratio of violins to violas in an orchestra is 5:3. t The orchestra has 9 viola players. How many t violinists are in the orchestra? yes no 15 0.5 ft 4. Two weights are balanced on a fulcrum. If a 6 lb weight is positioned 1.5 ft from the fulcrum, at what distance from the fulcrum must an 18 lb weight be placed to keep the weights balanced? 48 42 = ? 16 14 40 15 = ? 3 4
19. Lesson Quiz: Part II 5. An elevator travels 342 feet as it goes from the lobby of an office building to the top floor. It takes 7 seconds to travel the first 133 feet. If the elevator travels at the same rate, how much longer does it take to reach the top floor? 11 s
