Solving Proportions: Ratios and Cross Products

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

2. Solving Proportions 7.4 Pre-Algebra

3. Learn to solve proportions.

4. Vocabulary cross product

5. Cross Products

6. Helpful Hint The cross product represents the numerator of the fraction when a common denominator is found by multiplying the denominators.

7. 6 15 6 15 4 10 4 10 ? = Example: Using Cross Products to Identify Proportions Tell whether the ratios are proportional. A. 60 Find cross products. 60 60 = 60 Since the cross products are equal, the ratios are proportional.

8. 4 parts gasoline 1 part oil ? 15 quarts gasoline 5 quarts oil = Example: 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 ratios. Find the cross products. 4 • 5 = 20 1 • 15 = 15 20 ≠ 15 The ratios are not equal. The mixture will not be correct.

9. 5 10 5 10 2 4 2 4 ? = Try This Tell whether the ratios are proportional. A. 20 Find cross products. 20 20 = 20 Since the cross products are equal, the ratios are proportional.

10. 48 42 20 15 16 14 3 4 ? ? = = Lesson Quiz Tell whether each pair of ratios is proportional. yes 1. 2. no

11. 3 parts tea 1 part sugar ? 12 tablespoons tea 4 tablespoons sugar = Try This 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? Set up ratios. Find the cross products. 3 • 4 = 12 1 • 12 = 12 12 = 12 The ratios are equal. The mixture will be correct.

12. Solving with Cross-Products When you do not know one of the four numbers in a proportion, set the cross products equal to each other and solve.

13. 5 6 p 12 = 10 12 5 6 = ; the proportion checks. Example: Solving Proportions Solve the proportion. 6p = 12 • 5 Find the cross products. Solve. 6p = 60 p = 10

14. 2 3 14 g = 14 21 2 3 = ; the proportion checks. Try This Solve the proportion. 14 • 3 = 2g Find the cross products. Solve. 42 = 2g 21 = g

15. n 12 6 9 45 18 n 24 = = Lesson Quiz Solve each proportion. 3. n = 30 4. n = 16

16. pounds length pounds length = 220 5 55 5 5x 5 x 4 = = Example: Physical Science Application Allyson weighs 55 lbs and sits on a seesaw 5 ft away from its center. If Marco sits 4 ft away from the center and the seesaw is balanced, how much does Marco weigh? Set up the proportion. Let x represent Marco’s weight. 55 • 4 = 5x Find the cross products. 220 = 5x Multiply. Solve. Divide both sides by 5. 44 = x Marco weighs 44 lb.

17. pounds length pounds length = 450 6 90 6 6x 6 x 5 = = Try This Robert weighs 90 lbs and sits on a seesaw 6 ft away from its center. If Sharon sits 5 ft away from the center and the seesaw is balanced, how much does Sharon weigh? Set up the proportion. Let x represent Sharon’s weight. 90 • 5 = 6x Find the cross products. 450 = 6x Multiply. Solve. Divide both sides by 5. 75 = x Sharon weighs 75 lb.

18. Lesson Quiz 5. Two weights are balanced on a fulcrum. If a 6lb 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? 0.5 ft