3rd week of the first six week 5-1 Ratios and Proportions Course 3 3rd week of the first six week 7th pre a/p

Learning Objective: to find equivalent ratios to create proportions.

Vocabulary ratio equivalent ratio proportion

A ratio is a comparison of two quantities by division A ratio is a comparison of two quantities by division. In one rectangle, the ratio of shaded squares to unshaded squares is 7:5. In the other rectangle, the ratio is 28:20. Both rectangles have equivalent shaded areas. Ratios that make the same comparison are equivalent ratios. 7:5 28:20

Ratios can be written in several ways Ratios can be written in several ways. 7 to 5, 7:5, and name the same ratio. Reading Math 7 5

Additional Example 1: Finding Equivalent Ratios Find two ratios that are equivalent to each given ratio. Multiply or divide the numerator and denominator by the same nonzero number. = 9 27 = 9 • 2 27 • 2 18 54 A. 9 27 = = 9 ÷ 9 27 ÷ 9 1 3 Two ratios equivalent to are and . 9 27 18 54 1 3 = 64 • 2 24 • 2 64 24 = 128 48 Two ratios equivalent to are and . 64 24 128 48 8 3 B. 64 24 = = 64 ÷ 8 24 ÷ 8 8 3

Check It Out: Example 1 Find two ratios that are equivalent to each given ratio. Multiply or divide the numerator and denominator by the same nonzero number. = 8 16 = 8 • 2 16 • 2 16 32 A. 8 16 = = 8 ÷ 4 16 ÷ 4 2 4 Two ratios equivalent to are and . 8 16 32 2 4 32 16 = = 32 • 2 16 • 2 64 32 Two ratios equivalent to are and . 32 16 64 4 2 B. 32 16 = = 32 ÷ 8 16 ÷ 8 4 2

Ratios that are equivalent are said to be proportional, or in proportion. Equivalent ratios are identical when they are written in simplest form.

Additional Example 2: Determining Whether Two Ratios are in Proportion Simplify to tell whether the ratios form a proportion. Since , the ratios are in proportion. 1 9 = 3 27 A. and 2 18 3 27 = = 3 ÷ 3 27 ÷ 3 1 9 2 18 = = 2 ÷ 2 18 ÷ 2 1 9 12 15 B. and 27 36 12 15 = = 12 ÷ 3 15 ÷ 3 4 5 Since , the ratios are not in proportion. 4 5  3 27 36 = = 27 ÷ 9 36 ÷ 9 3 4

Simplify to tell whether the ratios form a proportion. Check It Out: Example 2 Simplify to tell whether the ratios form a proportion. Since , the ratios are in proportion. 1 5 = 3 15 A. and 9 45 3 15 = = 3 ÷ 3 15 ÷ 3 1 5 9 45 = = 9 ÷ 9 45 ÷ 9 1 5 14 49 = = 14 ÷ 7 49 ÷ 7 2 7 Since , the ratios are not in proportion. 2 7  4 9 14 49 B. and 16 36 16 36 = = 16 ÷ 4 36 ÷ 4 4 9

Additional Example 3: Earth Science Application At 4°C, four cubic feet of silver has the same mass as 42 cubic feet of water. At 4°C, would 210 cubic feet of water have the same mass as 20 cubic feet of silver? Since , 210 cubic feet of water would have the same mass at 4°C as 20 cubic feet of silver. 2 21 = 4 42 ? = 20 210 4 ÷ 2 42 ÷ 2 ? = 20 ÷ 10 210 ÷ 10 Divide. 2 21 =

Check It Out: Example 3 At 4°C, two cubic feet of silver has the same mass as 21 cubic feet of water. At 4°C, would 105 cubic feet of water have the same mass as 10 cubic feet of silver? Since , 105 cubic feet of water would have the same mass at 4°C as 10 cubic feet of silver. 2 21 = 2 21 ? = 10 105 ? = 10 ÷ 5 105 ÷ 5 2 21 Divide. 2 21 =

Demonstration of Learning page 1 Find two ratios that are equivalent to each given ratio. 4 15 1. 8 21 2. Simplify to tell whether the ratios form a proportion. 16 10 3. 32 20 and 36 24 4. 28 18 and

Demonstration of learning page 2 5. Kate poured 8 oz of juice from a 64 oz bottle. Brian poured 16 oz of juice from a 128 oz bottle. What ratio of juice is missing from each bottle? Are the ratios proportional?