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Warm Up Solve each equation. Check your answer. 1. 6x = 36 2. 6 Simplify. 4. 2.3 – 3.6 / 4 – 1.7 5. -0.4 + 1.3 * 4 0.5 – 5.1 / 3 6 48 3.6 -0.3 -4

Learning Targets Vocabulary ratio proportion rate cross products Write and use ratios, rates, and unit rates. Write and solve proportions. Vocabulary ratio proportion rate cross products Unit Price unit rate

A ratio is a comparison of two quantities by division A ratio is a comparison of two quantities by division. The ratio of a to b can be written a:b or , where b ≠ 0. Ratios that name the same comparison are said to be equivalent. A statement that two ratios are equivalent, such as , is called a proportion.

A rate is a ratio of two quantities with different units, such as Rates are usually written as unit rates. A unit rate is a rate with a second quantity of 1 unit, such as or 17 mi/gal. You can convert any rate to a unit rate. Unit price is a unit rate used to compare price per item.

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Example 1: Finding Unit Rates Raulf Laue of Germany flipped a pancake 416 times in 120 seconds to set the world record. Find the unit rate. Round your answer to the nearest hundredth. Write a proportion to find an equivalent ratio with a second quantity of 1. Divide 416 by 120 on the left side to find x. The unit rate is about 3.47 flips/s.

On Your Own Cory earns $52.50 in 7 hours. Find the unit rate.

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Additional Example 1: Finding Unit Prices to Compare Costs Jamie can buy a 15-oz jar of peanut butter for $2.19 or a 20-oz jar for $2.78. Which is the better buy? Divide the price by the number of ounces. price for jar number of ounces $2.19 15 =  $0.15 price for jar number of ounces $2.78 20 =  $0.14 The better buy is the 20-oz jar for $2.78.

On Your Own Golf balls can be purchased in a 3-pack for $4.95 or a 12-pack for $18.95. Which is the better buy? price for package number of balls = price for package number of balls = The better buy is the

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Example 2: Converting Rates Serena ran a race at a rate of 10 kilometers per hour. What was her speed in kilometers per minute? Round your answer to the nearest hundredth. To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity. The rate is about 0.17 kilometer per minute.

A cyclist travels 56 miles in 4 hours A cyclist travels 56 miles in 4 hours. What is the cyclist’s speed in feet per second? Round your answer to the nearest tenth, and show that your answer is reasonable. Step 1 Convert the speed to feet per hour. The speed is 73,920 feet per hour.

Step 2 Convert the speed to feet per minute. To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity. The speed is 1232 feet per minute. Step 3 Convert the speed to feet per second. To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity. The speed is approximately 20.5 feet per second.

On Your Own An engineer opens a valve that drains 60 gallons of water per minute from a tank. How many quarts were drained per second?

Cross Products Property In the proportion , the products a • d and b • c are called cross products. You can solve a proportion for a missing value by using the Cross Products property. Cross Products Property WORDS NUMBERS ALGEBRA If and b ≠ 0 and d ≠ 0 then ad = bc. In a proportion, cross products are equal. 2 • 6 = 3 • 4

Example 3: Solving Proportions Solve each proportion. A. B. Use cross products. Use cross products. 6(7) = 2(y – 3) 42 = 2y – 6 3(m) = 5(9) 3m = 45 Add 6 to both sides. +6 +6 48 = 2y m = 15 Divide both sides by 3. 24 = y Divide both sides by 2.

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On your Own Solve each proportion. A. B.

Example 5: Application A contractor has a blueprint for a house drawn to the scale 1 in: 3 ft. A wall on the blueprint is 6.5 inches long. How long is the actual wall? blueprint 1 in. actual 3 ft. Write the scale as a fraction. Let x be the actual length. x • 1 = 3(6.5) Use the cross products to solve. x = 19.5 The actual length of the wall is 19.5 feet.

On Your Own A scale model of a human heart is 16 ft. long. The scale is 32:1. How many inches long is the actual heart it represents? model 32 in. actual 1 in. Write the scale as a fraction. Let x be the actual length. Convert 16 ft to inches. Use the cross products to solve. Since x is multiplied by 32, divide both sides by 32 to undo the multiplication.