Identifying and Writing Proportions 4-2 Notes 16 Identifying and Writing Proportions 4-2

Vocabulary Equivalent ratios- ratios that name the same comparison. Proportion- an equation stating that two ratios are equivalent.

These ratios can be written as the fractions and . Since both simplify 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. These ratios can be written as the fractions and . Since both simplify to , they are equivalent. Equivalent ratios are ratios that name the same comparison. 6 4 21 14 3 2

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

If two ratios are equivalent, they are said to be proportional to each other, or in proportion.

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

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

Check It Out: Example 1A Solve on your own. Determine whether the ratios are proportional. 54 63 72 144

Check It Out: Example 1B Solve on your own. Determine whether the ratios are proportional. 135 75 9 4

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. 19 10 40 6 3 12 Cups of Water Cups of Rice Servings of Rice Write the ratios of rice to water for 12 servings and for 40 servings. 3 6 Ratio of rice to water, 12 servings: Write the ratio as a fraction. 10 19 Ratio of rice to water, 40 servings: Write the ratio as a fraction Write the ratios with a common denominator, such as 114. 3 6 3 · 19 6 · 19 57 114 10 19 10 · 6 19 · 6 60 114 = = = = 57 114 60 114 Since = , the two ratios are not proportional.

Check It Out: Example 2 Solve on your own. Use the data in the table to determine whether the ratios of beans to water are proportional for both servings of beans. 9 13 35 3 4 8 Cups of Water Cups of Beans Servings of Beans

You can find an equivalent ratio by multiplying or dividing both terms of a ratio by the same number.

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: 3 5 A. 3 5 3 · 2 5 · 2 6 10 = = Multiply both the numerator and denominator by any number such as 2. 3 5 6 10 = Write a proportion. B. 28 16 28 16 28 ÷ 4 16 ÷ 4 Divide both the numerator and denominator by any number such as 4. 7 4 = = 28 16 7 4 = Write a proportion.

Check It Out: Example 3 Find a ratio equivalent to each ratio. Then use the ratios to find a proportion. Possible Answers: 2 3 A. B. 16 12