Warm Up Find two ratios that are equivalent

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

Vocabulary Proportional Cross products Direct-proportional relationship Constant of proportionality

An equation that states that two ratios are equivalent is called a proportion. For example, the equation, or proportion, 2 =4 states that 3 6 2 4 the ratios and are equivalent. 3 6 Ratios that are equivalent are said to be proportional, or in proportion.

To find cross products, you multiply the numerator of one ratio by the denominator or another, then multiply the second numerator by the first denominator. In the proportion , the products a ∙ d and b ∙ c are called cross products. Proportion a∙ d = b ∙ c Cross Products One way to find whether two ratios are equivalent is to find their cross products. If the cross products are equal, the proportions are equivalent.

Class Example Tell whether the ratios are proportional. ? 4 6 = 15 10 6 ? 4 = 15 10 Find the cross products. ? 6 10 = 4 15 60 = 60 Since the cross products are equal, the ratios are proportional.

Individual Practice Tell whether the ratios are proportional. ? 5 = 10 5 ? 2 4 4=2 Find the cross products. 10 20 = 20 Since the cross products are equal, the ratios are proportional.

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

Partner Practice 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 is the height of the lighthouse? height of person length of shadow 1= x 3 36 1 1 3 Write a ratio comparing height of a person to shadow length. Set up the proportion. Let x represent the lighthouse height. 36 = 3 x Find the cross products. 36 = 3 x 3 3 Solve for x by dividing both sides of the equation by 3 12 = x What The height does this of the mean lighthouse for this situation? should be 12 meters.

In a direct-proportional relationship, as one amount increases, another amount increases at the same rate. The constant of proportionality is the value that relates the two amounts in a direct-proportional relationship. These ratios are directly proportional. What are the constants of proportionality? 1. 3 5 9 15 3 2. 10 20 12 24 2

Partner Practice 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? ? 15 quarts gasoline 4 parts gasoline = 1 part oil 5 quarts oil 4 ? 15 = 1 5 ? Set up equal ratios. Find the cross products. 4 5 = 1 15 20 15 The ratios are not equal. The mixture will not be correct.

Individual Practice 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? head length total length 1 5 1= x 5 20 Write a ratio comparing head length to total length. Set up the proportion. Let x represent the length of the cat's head. There's another way to solve fractions with a variable in the numerator. . . (20) 1 x = (20) 5 20 4=x Since x is divided by 20, multiply both sides of the equation by 20. The length of the cat's head should be 4 inches.