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5-3 Identifying and Writing Proportions Preview Warm Up California Standards Lesson Presentation Holt CA Course 1 5-3 Identifying and Writing Proportions Warm Up Find the unit rate. Evaluate. 1. 18 miles in 3 hours 3(15 – 6) + (18 – 12)2 6 mi/h 2. 6 apples for $3.30 A. 36 $0.55 per apple 3. 3 cans for $0.87 B. 45 $0.29 per can C. 63 4. 5 CDs for $43 $8.60 per CD D. 75 Holt CA Course 1 5-3 Identifying and Writing Proportions California Standards Number Sense (NS1.2) - Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations ( a , a to b, a:b ). b Holt CA Course 1 5-3 Identifying and Writing Proportions Objective: You will learn how to find equivalent ratios and identify proportions. Holt CA Course 1 5-3 Identifying and Writing Proportions Vocabulary equivalent ratios proportion Holt CA Course 1 5-3 Identifying and Writing Proportions Students are measuring the width w and the length l of their heads. The ratio of l to w is 10 inches to 6 inches for Jean and 25 centimeters to 15 centimeters for Pat. Holt CA Course 1 5-3 Identifying and Writing Proportions These ratios can be written as the 10 Fractions 6 and 25 . Since both ratios 15 simplify to 5, they are equivalent. 3 Equivalent ratios are ratios that name the same comparison. Holt CA Course 1 5-3 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 10 6 and 25 are equivalent. 15 10 = 25 6 15 Reading Math If two ratios are equivalent, they are said to be proportional to each other, or in proportion. Holt CA Course 1 5-3 Identifying and Writing Proportions Example 1: Comparing Ratios in Simplest Forms Determine whether the ratios are proportional. 24 , 72 51 128 24 ÷ 3 8 Simplify 24 = 51 51 ÷ 3 17 72 ÷ 8 9 72 = Simplify 128 ÷ 8 16 128 Since 8 ≠ 9 , the ratios are not proportional. 17 16 Holt CA Course 1 5-3 Identifying and Writing Proportions Example 2: Comparing Ratios in Simplest Forms Determine whether the ratios are proportional. 150 , 90 105 63 150 ÷ 15 10 Simplify 150 = 105 105 ÷ 15 7 90 ÷ 9 10 90 = Simplify 63 ÷ 9 7 63 10 10 Since = , the ratios are proportional. 7 7 Holt CA Course 1 5-3 Identifying and Writing Proportions Check It Out! Example 3 Determine whether the ratios are proportional. 54 , 72 63 144 54 ÷ 9 6 Simplify 54 = 63 63 ÷ 9 7 72 ÷ 72 1 = Simplify 72 144 ÷ 72 2 144 6 1 Since ≠ , the ratios are not proportional. 7 2 Holt CA Course 1 5-3 Identifying and Writing Proportions Check It Out! Example 4 Determine whether the ratios are proportional. 135 , 9 75 4 135 ÷ 15 9 Simplify 135 75 ÷ 15 = 75 5 9 9 is already in simplest form. 4 4 9 9 Since ≠ , the ratios are not proportional. 5 4 Holt CA Course 1 5-3 Identifying and Writing Proportions Example 5: Comparing Ratios Using a Common Denominator Directions for making 12 servings of rice call for 3 cups of rice and 6 cups of Servings Cups of Cups of water. For 40 servings, the directions of Rice Rice Water call for 10 cups of rice and 19 cups of 12 3 6 water. Determine whether the ratios of 40 10 19 rice to water are proportional for both servings of rice. Write the ratios of rice to water for 12 servings and for 40 servings. 3 Ratio of rice to water, 12 servings: 6 Write the ratio as a fraction. 10 Ratio of rice to water, 40 servings: Write the ratio as a fraction. 19 Write the ratios with a 3 = 3 · 19 = 57 10 = 10 · 6 = 60 common denominator, 6 6 · 19 114 19 19 · 6 114 such as 114. 57 60 Since ≠ , the two ratios are not proportional. 114 114 Holt CA Course 1 5-3 Identifying and Writing Proportions Check It Out! Example 6 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 Beans Cups of Water 8 4 3 35 13 9 Write the ratios of beans to water for 8 servings and for 35 servings. 4 Ratio of beans to water, 8 servings: Write the ratio as a fraction. 3 13 Ratio of beans to water, 35 servings: Write the ratio as a fraction. 9 Write the ratios with a 4 = 4·3 = 12 13 common denominator, 3 3·3 9 9 such as 9. Since 12 ≠ 13 , the two ratios are not proportional. 9 9 Holt CA Course 1 5-3 Identifying and Writing Proportions You can find an equivalent ratio by multiplying or dividing the numerator and the denominator of a ratio by the same number. Holt CA Course 1 5-3 Identifying and Writing Proportions Example 7: Finding Equivalent Ratios and Writing Proportions Find a ratio equivalent to each ratio. Then use the ratios to find a proportion. A. 3 Multiply both the 5 numerator and 3 3·2 = 6 = denominator by any 5 5 · 2 10 number, such as 2. 3 = 6 5 10 Write a proportion. B. 28 16 Divide both the 28 = 28 ÷ 4 = 7 numerator and denominator by any 16 16 ÷ 4 4 number, such as 4. 28 = 7 16 4 Write a proportion. Holt CA Course 1 5-3 Identifying and Writing Proportions Check It Out! Example 8 Find a ratio equivalent to each ratio. Then use the ratios to find a proportion. Possible Answers: A. 2 Multiply both the 3 numerator and 2 2·3 = 6 = denominator by any 3 3·3 9 number, such as 3. 2 = 6 3 9 Write a proportion. B. 16 12 Divide both the 16 = 16 ÷ 4 = 4 numerator and 12 12 ÷ 4 3 denominator by any number, such as 4. 16 = 4 12 3 Write a proportion. Holt CA Course 1 5-3 Identifying and Writing Proportions On-Line Tutoring Holt CA Course 1 5-3 Identifying and Writing Proportions Lesson Quiz: Part I Determine whether the ratios are proportional. 9 12 12 10 1. 30 , 40 2. 21 , 15 3 , 3 ; proportional 4 , 2 ; not proportional 10 10 7 3 Find a ratio equivalent to each ratio. Then use the ratios to write a proportion. 3 15 ; 3 = 15 4. 3 9 ; 3 = 9 3. 8 40 8 40 7 21 7 21 Holt CA Course 1 5-3 Identifying and Writing Proportions Lesson Quiz: Part II 5. In one pre-school, there are 5 children for every 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 pre-schools. 5 = 20 1 4 Holt CA Course 1