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Bell Ringer Ratios and Proportions • A ratio is a comparison of a number “a” and a nonzero number “b” using division Example 1 Simplify Ratios Simplify the ratio. a. 60 cm : 200 cm 3 ft b. 18 in. SOLUTION a. 60 cm 60 cm : 200 cm can be written as the fraction . 200 cm 60 cm 60 ÷ 20 200 cm = 200 ÷ 20 3 = 10 Divide numerator and denominator by their greatest common factor, 20. Simplify.3 is read as “3 to 10.” 10 Example 1 b. Simplify Ratios 3 ft = 3 · 12 in. 18 in. 18 in. = 36 in. 18 in. = 36 ÷ 18 18 ÷ 18 = 2 1 Substitute 12 in. for 1 ft. Multiply. Divide numerator and denominator by their greatest common factor, 18. Simplify. 12 is read as “2 to 1.” Example 2 Use Ratios In the diagram, AB : BC is 4 : 1 and AC = 30. Find AB and BC. SOLUTION Let x = BC. Because the ratio of AB to BC is 4 to 1, you know that AB = 4x. AB + BC = AC 4x + x = 30 5x = 30 x=6 Segment Addition Postulate Substitute 4x for AB, x for BC, and 30 for A Add like terms. Divide each side by 5. Example 2 Use Ratios To find AB and BC, substitute 6 for x. AB = 4x = 4 · 6 = 24 BC = x = 6 ANSWER So, AB = 24 and BC = 6. Example 3 Use Ratios The perimeter of a rectangle is 80 feet. The ratio of the length to the width is 7 : 3. Find the length and the width of the rectangle. SOLUTION The ratio of length to width is 7 to 3. You can let the length l = 7x and the width w = 3x. 2l + 2w = P Formula for the perimeter of a rectangle 2(7x) + 2(3x) = 80 Substitute 7x for l, 3x for w, and 80 for 14x + 6x = 80 Multiply. 20x = 80 Add like terms. x=4 Divide each side by 20. Example 3 Use Ratios To find the length and width of the rectangle, substitute 4 for x. l = 7x = 7 · 4 = 28 ANSWER w = 3x = 3 · 4 = 12 The length is 28 feet, and the width is 12 feet. Now You Try 1. In the diagram, EF : FG is 2 : 1 and EG = 24. Find EF and FG. ANSWER EF = 16; FG = 8 2. The perimeter of a rectangle is 84 feet. The ratio of the length to the width is 4 : 3. Find the length and the width of the rectangle. ANSWER length, 24 ft; width, 18 ft An equation that states that two ratios are equal is called a proportion Example 4 Solve a Proportion y+2 5 = Solve the proportion . 3 6 SOLUTION y+2 5 = 3 6 5 · 6 = 3(y + 2) 30 = 3y + 6 30 – 6 = 3y + 6 – 6 24 = 3y 24 3y = 3 3 8=y Write original proportion. Cross product property Multiply and use distributive property Subtract 6 from each side. Simplify. Divide each side by 3. Simplify. Now You Try Solve the proportion. 3 6 = 3. x 8 ANSWER 4 15 = 4. 5 y 3 ANSWER 9 m + 2 14 5. = 5 10 ANSWER 5 Complete #s 2-44 even only