4-1 Ratios & Proportions

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

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.

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.

Practice 15 cows to 25 sheep 24 cars to 18 trucks 30 Knives to 27 spoons

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

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

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

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

Notes Ratios that make the same comparison are equivalent ratios. To check whether two ratios are equivalent, you can write both in simplest form.

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

Practice

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: ,

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

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.

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

Example ÷ 5 3 7 = 8 40 ÷ 5

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.

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

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

Solving a Proportion Using Cross Products Use the cross products to create an equation. Solve the equation for the variable using the inverse operation.

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

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 = ¼

Example 3: Solve the Proportion 2𝑥+1 4 = 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

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