Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

Find two ratios that are equivalent to each given ratio. Warm Up Find two ratios that are equivalent to each given ratio. Possible answers: 3 5 9 15 6 10 , 10 12 5 6 , 20 24 1. 2. 45 30 3 2 , 90 60 8 9 16 18 , 24 27 3. 4.

Problem of the Day Replace each • with a digit from 1 to 7 to write a proportion. Use each digit once. The digits 2 and 3 are already shown. •• • 23 = 14 7 23 56 = Possible answer:

Learn to solve proportions.

Vocabulary cross product

Additional Example 1A: Using Cross Products to Identify Proportions Tell whether the ratios are proportional. 4 10 6 15 = ? 4 10 6 15 60 Find cross products. 60 60 = 60 Since the cross products are equal, the ratios are proportional.

Additional Example 1B: Using Cross Products to Identify 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? Set up equal ratios. 4 parts gasoline 1 part oil = ? 15 quarts gasoline 5 quarts oil Find the cross products. 4 • 5 = 20 1 • 15 = 15 20 ≠ 15 The cross products are not equal. The mixture will not be correct.

= Check It Out: Example 1A Tell whether the ratios are proportional. 2 4 5 10 = ? 2 4 5 10 20 Find cross products. 20 20 = 20 Since the cross products are equal, the ratios are proportional.

= Check It Out: Example 1B 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? 3 parts tea 1 part sugar = ? 12 tablespoons tea 4 tablespoons sugar Set up equal ratios. Find the cross products. 3 • 4 = 12 1 • 12 = 12 12 = 12 The cross products are equal. The mixture will be correct.

Additional Example 2: Using Properties of Equality to Solve Proportions The ratio of boys to girls in a soccer league is 5:4. If there are 52 boys in the league, how many girls are there? 5 4 girls boys = Write a ratio comparing girls to boys. 5 4 = x 52 Set Up the proportion. Let x represent the number of girls. (52) (52)‏ 5 4 = x 52 Since x is divided by 52, multiply both sides of the equation by 56. 65 = x There are 65 girls in the league.

Check It Out: Example 2 The ratio of cats to dogs at a kennel is 3:2. If there are 48 dogs at the kennel, how many cats are there? 3 2 Cats Dogs = Write a ratio comparing cats to dogs. 3 2 = x 48 Set Up the proportion. Let x represent the number of cats. (48) (48)‏ 3 2 = x 48 Since x is divided by 48, multiply both sides of the equation by 48. 72 = x There are 72 cats at the kennel.

Additional Example 3: Using Cross Products to Solve Proportions A banana slug travels 4.5 inches in 120 minutes. At this rate of speed, how long would it take the slug to travel 12 inches? At a constant rate of speed ratios of distance to time are equivalent. distance 1 time 1 = distance 2 time 2 Set up a proportion that compares distance to time. 4.5 in 120 in = 12 in s Let s represent the time the takes to travel 12 in. 4.5 ▪ s = 120 ▪ 12 Find the cross products.

Additional Example 3 Continued A banana slug travels 4.5 inches in 120 minutes. At this rate of speed, how long would it take the slug to travel 12 inches? 4.5s = 1440 Multiply. 4.5 4.5 Divide both sides by 25. s = 320 Simplify. The slug will travel 12 inches in 320 min. or 5 h 20 min.

Check It Out: Example 3 Continued A cable car travels 15 miles in 80 minutes. At this rate of speed, how long would it take the cable car to travel 45 miles? 15s = 3600 Multiply. 15 15 Divide both sides by 15. s = 240 Simplify. The cable car will travel 15 miles in 240 min. or 4 hours.

Additional Example 4: Business Application 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? Let x represent the number of minutes it takes to complete the job. 30 45 = x 225 Set up the proportion. 30 ∙ 225 = 45x Find the cross products. 45x 45 6750 = 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.

Check It Out: Example 4 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? Let m represent the number of minutes it takes to complete the job. 21 24 = m 160 Set up the proportion. 21 ∙ 160 = 24m Find the cross products. 24m 24 3360 = 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.

Lesson Quiz for Student Response Systems Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems 19 19

= = = = Lesson Quiz Tell whether the ratios are proportional. yes 1. 48 42 = ? 16 14 yes 40 15 = ? 3 4 1. 2. no Solve each proportion. 45 18 n 12 = n 24 6 9 = n = 30 4. 3. n = 16 5. An elevator travels 342 feet as it goes from the lobby of a 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 reach the top floor? 11s

Lesson Quiz for Student Response Systems 1. Identify the ratios that are proportional. A. B. C. D. 21 21

Lesson Quiz for Student Response Systems 2. Solve the given proportion. A. p = 8 B. p = 9 C. p = 10 D. p = 11 22 22

Lesson Quiz for Student Response Systems 3. A 225 kg weight is positioned 5 m from a fulcrum. If a 300 kg weight is placed at the opposite end of the balance, how far from the fulcrum should it be positioned? A. 5.75 m B. 4.5 m C. 4.25 m D. 3.75 m 23 23