1. Evaluating Algebraic Expressions 5-3 Proportions AF4.2 Solve multistep problems involving rate, average speed, distance, and time or a direct variation. California Standards

2. Evaluating Algebraic Expressions 5-3 Proportions 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. 4646 2323 = 4646 2323

3. Evaluating Algebraic Expressions 5-3 Proportions Proportion In the proportion, the products a ∙ d and b ∙ c are called cross products. cdcd abab = a∙ d = b ∙ cCross Products One way to find whether two ratios are equivalent is to find their cross products. cdcd abab =

4. Evaluating Algebraic Expressions 5-3 Proportions

5. Evaluating Algebraic Expressions 5-3 Proportions Tell whether the ratios are proportional. 4 10 6 15 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 = ? 6  10 = 4  15 ?

6. Evaluating Algebraic Expressions 5-3 Proportions 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? 4 parts gasoline 1 part oil = ? 15 quarts gasoline 5 quarts oil 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 15 5 4141 = ? 4  5 = 1  15 ?

7. Evaluating Algebraic Expressions 5-3 Proportions 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. 2424 5 10 = ? 20 = 20 2424 5 10 = ? 5  4 = 2  10 ?

8. Evaluating Algebraic Expressions 5-3 Proportions 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 3 parts tea 1 part sugar = ? 12 tablespoons tea 4 tablespoons sugar The ratios are equal. The mixture will be correct. Set up equal ratios. Find the cross products. 12 = 12 12 4 3131 = ? 3  4 = 1  12 ?

9. Evaluating Algebraic Expressions 5-3 Proportions 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 1313 height of person length of shadow Since x is divided by 36, multiply both sides of the equation by 36. 12 = x 1313 = x 36 (36) = (36) 1313 x 36 The length of the lighthouse’s shadow should be 12 meters.

10. Evaluating Algebraic Expressions 5-3 Proportions 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 1515 head length total length Since x is divided by 20, multiply both sides of the equation by 20. 4 = x 1515 = x 20 (20) = (20) 1515 x 20 The length of the cat's head should be 4 inches.

11. Evaluating Algebraic Expressions 5-3 Proportions 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 = 5w Find the cross products. Divide both sides by 5. 5w55w5 220 5 = Additional Example 3: Using Cross Products to Solve Proportions = weight 1 length 2 weight 2 length 1 55 5 = w4w4 44 = w Simplify. Set up a proportion using the information. Let w represent Marco’s weight. Marco weighs 44 lb.

12. Evaluating Algebraic Expressions 5-3 Proportions 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 = 4w Find the cross products. Divide both sides by 4. 4w44w4 192 4 = Check It Out! Example 3 = weight 1 length 2 weight 2 length 1 32 4 = w6w6 48 = w Simplify. Set up a proportion using the information. Let w represent Kaylee’s weight. Kaylee weighs 48 lbs.

13. Evaluating Algebraic Expressions 5-3 Proportions 30 ∙ 225 = 45xFind the cross products. Divide both sides by 45. 45x 45 6750 45 = 30 45 = x 225 150 = xSimplify. 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.

14. Evaluating Algebraic Expressions 5-3 Proportions 21 ∙ 160 = 24mFind the cross products. Divide both sides by 24. 24m 24 3360 24 = 21 24 = m 160 140 = mSimplify. 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.