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**Objectives Write and simplify ratios.**

Use proportions to solve problems.

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**A ratio compares two numbers by division**

A ratio compares two numbers by division. The ratio of two numbers a and b can be written as a to b, a:b, or , where b ≠ 0. For example, the ratios 1 to 2, 1:2, and all represent the same comparison.

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**Example 1: Writing Ratios**

Write a ratio expressing the slope of l. Substitute the given values. Simplify.

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Example 2: Using Ratios The ratio of the side lengths of a triangle is 4:7:5, and its perimeter is 96 cm. What is the length of the shortest side? Let the side lengths be 4x, 7x, and 5x. Then 4x + 7x + 5x = 96 . After like terms are combined, 16x = 96. So x = 6. The length of the shortest side is 4x = 4(6) = 24 cm.

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Check It Out! Example 2 The ratio of the angle measures in a triangle is 1:6:13. What is the measure of each angle? x + y + z = 180° x + 6x + 13x = 180° 20x = 180° x = 9° y = 6x z = 13x y = 6(9°) z = 13(9°) y = 54° z = 117°

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**A proportion is an equation stating that two ratios**

are equal. In the proportion , the values a and d are the extremes. The values b and c are the means. When the proportion is written as a:b = c:d, the extremes are in the first and last positions. The means are in the two middle positions.

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**In Algebra 1 you learned the Cross Products Property**

In Algebra 1 you learned the Cross Products Property. The product of the extremes ad and the product of the means bc are called the cross products.

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**Example 3A: Solving Proportions**

Solve the proportion. 7(72) = x(56) Cross Products Property 504 = 56x Simplify. x = 9 Divide both sides by 56.

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**Example 3B: Solving Proportions**

Solve the proportion. (z – 4)2 = 5(20) Cross Products Property (z – 4)2 = 100 Simplify. (z – 4) = 10 Find the square root of both sides. Rewrite as two eqns. (z – 4) = 10 or (z – 4) = –10 z = 14 or z = –6 Add 4 to both sides.

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Check It Out! Example 3a Solve the proportion. 3(56) = 8(x) Cross Products Property 168 = 8x Simplify. x = 21 Divide both sides by 8.

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Check It Out! Example 3b Solve the proportion. 2y(4y) = 9(8) Cross Products Property 8y2 = 72 Simplify. y2 = 9 Divide both sides by 8. y = 3 Find the square root of both sides. y = 3 or y = –3 Rewrite as two equations.

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Check It Out! Example 3d Solve the proportion. (x + 3)2 = 4(9) Cross Products Property (x + 3)2 = 36 Simplify. (x + 3) = 6 Find the square root of both sides. (x + 3) = 6 or (x + 3) = –6 Rewrite as two eqns. x = 3 or x = –9 Subtract 3 from both sides.

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**The following table shows equivalent forms of the Cross Products Property.**

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**Example 4: Using Properties of Proportions**

Given that 18c = 24d, find the ratio of d to c in simplest form. 18c = 24d Divide both sides by 24c. Simplify.

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**Understand the Problem**

Example 5: Problem-Solving Application Marta is making a scale drawing of her bedroom. Her rectangular room is 12 feet wide and 15 feet long. On the scale drawing, the width of her room is 5 inches. What is the length? 1 Understand the Problem The answer will be the length of the room on the scale drawing.

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Example 5 Continued 2 Make a Plan Let x be the length of the room on the scale drawing. Write a proportion that compares the ratios of the width to the length.

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Example 5 Continued Solve 3 5(15) = x(12.5) Cross Products Property 75 = 12.5x Simplify. x = 6 Divide both sides by 12.5. The length of the room on the scale drawing is 6 inches.

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Lesson Quiz 1. The ratio of the angle measures in a triangle is 1:5:6. What is the measure of each angle? Solve each proportion. 4. Given that 14a = 35b, find the ratio of a to b in simplest form. 5. An apartment building is 90 ft tall and 55 ft wide. If a scale model of this building is 11 in. wide, how tall is the scale model of the building? 15°, 75°, 90° 3 7 or –7 18 in.

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Homework Pg 8-15, 17-19, 22-29

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