7-4 Solving Proportions Warm Up Problem of the Day Lesson Presentation Pre-Algebra

7-4 Solving Proportions Warm Up Pre-Algebra 7-4 Solving Proportions 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

Unequal masses will not balance on a fulcrum if they are an equal distance from it; one side will go up and the other side will go down. Unequal masses will balance when the following proportions is true: mass 2 length 1 mass 1 length 2 = Mass 1 Mass 2 Fulcrum Length 1 Length 2

One way to find whether ratios, such as those above, are equal is to find a common denominator. The ratios are equal if their numerators are equal after the fractions have been rewritten with a common denominator. 72 96 6 8 = 72 96 9 12 = 9 12 6 8 =

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

Additional Example 1A: Using Cross Products to Identify Proportions Tell whether the ratios are proportional. 4 10 6 15 = ? A. 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? 4 parts gasoline 1 part oil = ? 15 quarts gasoline 5 quarts oil 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.

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

Try This: 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 ratios. Find the cross products. 3 • 4 = 12 1 • 12 = 12 12 = 12 The ratios are equal. The mixture will be correct.

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

Additional Example 2: Solving Proportions Solve the proportion. 5 6 p 12 = 6p = 12 • 5 Find the cross products. 6p = 60 Solve. ; the proportion checks. 5 6 10 12 = p = 10

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

Additional Example 3: Physical Science Application Allyson weighs 55 lbs and sits on a seesaw 4 ft away from its center. If Marco sits 5 ft away from the center and the seesaw is balanced, how much does Marco weigh? mass 1 length 2 = mass 2 length 1 Set up the proportion. Let x represent Marco’s weight. x 4 55 5 = 55 • 4 = 5x Find the cross products. 220 = 5x Multiply. 5x 5 220 = Solve. Divide both sides by 5. 44 = x Marco weighs 44 lb.

Try This: Example 3 Robert weighs 90 lbs and sits on a seesaw 5 ft away from its center. If Sharon sits 6 ft away from the center and the seesaw is balanced, how much does Sharon weigh? mass 1 length 2 = mass 2 length 1 Set up the proportion. Let x represent Sharon’s weight. x 5 90 6 = 90 • 5 = 6x Find the cross products. 450 = 6x Multiply. 6x 6 450 = Solve. Divide both sides by 5. 75 = x Sharon weighs 75 lb.

= = = = Lesson Quiz Tell whether each pair of ratios is proportional. 48 42 = ? 16 14 20 15 = ? 3 4 1. yes 2. no Solve each proportion. 45 18 n 12 = n 24 6 9 = 3. n = 30 4. n = 16 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