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4-1 Ratios & Proportions

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**7 5 Notes A ratio is a comparison of two quantities.**

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

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**Example 1: Writing Ratios in Simplest Form**

Write the ratio 15 bikes to 9 skateboards in simplest form. bikes skateboards 15 9 Write the ratio as a fraction. = 15 ÷ 3 9 ÷ 3 5 3 Simplify. = = 5 3 The ratio of bikes to skateboards is , 5:3, or 5 to 3.

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**The ratio of shirts to jeans is , 8:3, or 8 to 3.**

Check It Out! Example 2 Write the ratio 24 shirts to 9 jeans in simplest form. shirts jeans 24 9 Write the ratio as a fraction. = 24 ÷ 3 9 ÷ 3 8 3 = = Simplify. 8 3 The ratio of shirts to jeans is , 8:3, or 8 to 3.

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**Practice 15 cows to 25 sheep 24 cars to 18 trucks**

30 Knives to 27 spoons

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When simplifying ratios based on measurements, write the quantities with the same units, if possible.

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**Example 3: Writing Ratios Based on Measurement**

Write the ratio 3 yards to 12 feet in simplest form. First convert yards to feet. 3 yards = 3 ● 3 feet There are 3 feet in each yard. = 9 feet Multiply. Now write the ratio. = 3 yards 12 feet 9 feet 12 feet 3 4 = 9 ÷ 3 12 ÷ 3 = Simplify. The ratio is , 3:4, or 3 to 4. 3 4

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Check It Out! Example 3 Write the ratio 36 inches to 4 feet in simplest form. First convert feet to inches. 4 feet = 4 ● 12 inches There are 12 inches in each foot. = 48 inches Multiply. Now write the ratio. = 36 inches 4 feet 36 inches 48 inches 3 4 = 36 ÷ 12 48 ÷ 12 = Simplify. The ratio is , 3:4, or 3 to 4. 3 4

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Practice 4 feet to 24 inches 3 yards to 12 feet 2 yards to 20 inches

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**Notes Ratios that make the same comparison are equivalent ratios.**

To check whether two ratios are equivalent, you can write both in simplest form.

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**Example 4: Determining Whether Two Ratios Are Equivalent**

Simplify to tell whether the ratios are equivalent. Since , the ratios are equivalent. 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 equivalent. 4 5 3 27 36 = = 27 ÷ 9 36 ÷ 9 3 4

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Practice

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Lesson Quiz: Part I Write each ratio in simplest form. 1. 22 tigers to 44 lions 2. 5 feet to 14 inches 1 2 30 7 Find a ratio that is equivalent to each given ratio. 4 15 3. 8 30 12 45 Possible answer: , 7 21 4. 1 3 14 42 Possible answer: ,

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Lesson Quiz: Part II Simplify to tell whether the ratios are equivalent. 16 10 5. 32 20 8 5 = ; yes and 36 24 6. 28 18 3 2 14 9 ; no and 7. Kate poured 8 oz of juice from a 64 oz bottle. Brian poured 16 oz of juice from a 128 oz bottle. Are the ratios of poured juice to starting amount of juice equivalent? 8 64 16 128 and ; yes, both equal 1 8

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Vocabulary A proportion is an equation stating that two ratios are equal. To prove that two ratios form a proportion, you must prove that they are equivalent. To do this, you must demonstrate that the relationship between numerators is the same as the relationship between denominators.

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**Examples: Do the ratios form a proportion?**

Yes, these two ratios DO form a proportion, because the same relationship exists in both the numerators and denominators. 7 21 , 10 30 x 3 ÷ 4 8 2 No, these ratios do NOT form a proportion, because the ratios are not equal. , 9 3 ÷ 3

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Example ÷ 5 3 7 = 8 40 ÷ 5

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Cross Products When you have a proportion (two equal ratios), then you have equivalent cross products. Find the cross product by multiplying the denominator of each ratio by the numerator of the other ratio.

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**Example: Do the ratios form a proportion? Check using cross products.**

4 3 , 12 9 These two ratios DO form a proportion because their cross products are the same. 12 x 3 = 36 9 x 4 = 36

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Example 2 5 2 , 8 3 No, these two ratios DO NOT form a proportion, because their cross products are different. 8 x 2 = 16 3 x 5 = 15

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**Solving a Proportion Using Cross Products**

Use the cross products to create an equation. Solve the equation for the variable using the inverse operation.

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**Example 1: Solve the Proportion**

Start with the variable. 20 k = 17 68 Simplify. Now we have an equation. To get the k by itself, divide both sides by 68. 68k 17(20) = 68k = 340 68 68 k = 5

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**Example 2: Solve the Proportion**

2𝑥 3 = 5 30 Start with the variable. Simplify. Now we have an equation. Solve for x. 2x(30) 5(3) = 60x = 15 60 60 x =

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**Example 3: Solve the Proportion**

2𝑥 = 5 3 Start with the variable. Simplify. Now we have an equation. Solve for x. (2x +1)3 5(4) = 6x + 3 = 20 x = 17 6

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**Example 4: Solve the Proportion**

3 4 = 𝑥+2 𝑥 Cross Multiply. Simplify. Now we have an equation with variables on both sides. Solve for x. 3x 4(x+2) = 3x = 4x + 8 x = -8

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