4 3 Identifying and Writing Proportions Vocabulary equivalent

4 -3 Identifying and Writing Proportions Vocabulary equivalent ratios proportion

4 -3 Identifying and Writing Proportions Students in Mr. Howell’s math class are measuring the width w and the length l of their faces. The ratio of l to w is 6 inches to 4 inches for Jean and 21 centimeters to 14 centimeters for Pat.

4 -3 Identifying and Writing Proportions These ratios can be written as the 6 4 21 14. fractions and Since both simplify to 32 , they are equivalent. Equivalent ratios are ratios that name the same comparison.

4 -3 Identifying and Writing Proportions An equation stating that two ratios are equivalent is called a proportion. The equation, or proportion, below states that the ratios 6 and 21 are equivalent. 4 Reading Math 14 6 = 21 14 4 Read the proportion 6 = 21 by saying “six is to four 4 14 as twenty-one is to fourteen. ”

4 -3 Identifying and Writing Proportions If two ratios are equivalent, they are said to be proportional to each other, or in proportion.

4 -3 Identifying and Writing Proportions Additional Example 1 A: Comparing Ratios in Simplest Forms Determine whether the ratios are proportional. 24 , 72 51 128 8 24 ÷ 3 = 17 51 ÷ 3 Simplify 24. 51 72 ÷ 8 9 = 16 128 ÷ 8 Simplify 72. 128 Since 9 8 = , the ratios are not proportional. 17 16

4 -3 Identifying and Writing Proportions Additional Example 1 B: Comparing Ratios in Simplest Forms Determine whether the ratios are proportional. 150 , 90 105 63 150 ÷ 15 10 = 7 105 ÷ 15 90 ÷ 9 10 = 7 63 ÷ 9 Since Simplify 150. 105 Simplify 90. 63 10 10 = , the ratios are proportional. 7 7

4 -3 Identifying and Writing Proportions Check It Out: Example 1 A Determine whether the ratios are proportional. 18 , 54 36 108 Since 1 18 ÷ 18 = 2 36 ÷ 18 and 54 ÷ 54 1, = 108 ÷ 54 2 the ratios are proportional.

4 -3 Identifying and Writing Proportions Check It Out: Example 1 B Determine whether the ratios are proportional. 54, 72 63 144 Since and 6 54 ÷ 9 = 7 63 ÷ 9 72 ÷ 72 = 1 2 144 ÷ 72 , the ratios are not proportional.

4 -3 Identifying and Writing Proportions Additional Example 2: Comparing Ratios Using a Common Denominator Directions for making 12 servings of rice call for 3 cups of rice and 6 cups of water. For 40 servings, the directions call for 10 cups of rice and 19 cups of water. Determine whether the ratios of rice to water are proportional for both servings of rice. Servings Cups of of Rice Water 12 3 6 40 10 19 Write the ratios of rice to water for 12 servings and for 40 servings. 3 Ratio of rice to water, 12 servings: 6 10 Ratio of rice to water, 40 servings: 19 3 = 3 · 19 = 57 6 6 · 19 114 Since 10 = 10 · 6 = 60 19 19 · 6 114 Write the ratio as a fraction Write the ratios with a common denominator, such as 114. 57 60 = , the two ratios are not proportional. 114

4 -3 Identifying and Writing Proportions Check It Out: Example 2 Use the data in the table to determine whether the ratios of beans to water are proportional for both servings of beans. Servings of Beans Cups of Water 8 4 3 35 13 9 Since 36 = 39 , the two ratios are not proportional. 27 27

4 -3 Identifying and Writing Proportions You can find an equivalent ratio by multiplying or dividing both terms of a ratio by the same number.

4 -3 Identifying and Writing Proportions Additional Example 3: Finding Equivalent Ratios and Writing Proportions Find a ratio equivalent to each ratio. Then use the ratios to find a proportion. Possible Answers: A. 3 5 3 = 3 · 2 =6 5 5 · 2 10 3 = 6 10 5 B. 28 16 28 = 28 ÷ 4 7 = 16 16 ÷ 4 4 28 = 7 16 4 Multiply both the numerator and denominator by any number such as 2. Write a proportion. Divide both the numerator and denominator by any number such as 4. Write a proportion.

4 -3 Identifying and Writing Proportions Check It Out: Example 3 Find a ratio equivalent to each ratio. Then use the ratios to write a proportion. A. 16 12 Possible answer: 16 ÷ 4 4 ; 16 = 4 = 12 ÷ 4 3 12 3 B. 108 40 Possible answer: 108 ÷ 4 = 27 108 = 27 ; 40 ÷ 4 10 40 10

4 -3 Identifying and Writing Proportions Check It Out: Example 3 C. 169 52 169 ÷ 13 = 13 ; 169 = 13 Possible answer: 52 ÷ 13 4 52 4

4 -3 Identifying and Writing Proportions Lesson Quiz: Part I Determine whether the ratios are proportional. 1. 9 , 12 30 40 3 , 3 ; proportional 10 10 2. 12 , 10 21 15 4 , 2 ; not proportional 7 3 Find a ratio equivalent to each ratio. Then use the ratios to write a proportion. 3. 3 8 15 ; 3 = 15 40 40 8 4. 3 7 9 ; 3 = 9 21 21 7

4 -3 Identifying and Writing Proportions Lesson Quiz: Part II 5. In one pre-school, there are 5 children for every one teacher. In another pre-school, there are 20 children for every 4 teachers. Determine whether the ratios of children to teachers are proportional in both preschools. 5 = 20 4 1