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4 -1 Ratios, Rates, and Unit Rates Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

4 -1 Ratios, Rates, and Unit Rates Warm Up Divide. Round answers to the nearest tenth. 1. 420 23. 3 18 2. 73 3. 5 21 3. 380 23. 8 16 4. 430 23. 9 18

4 -1 Ratios, Rates, and Unit Rates Problem of the Day There are 3 bags of flour for every 2 bags of sugar in a freight truck. A bag of flour weighs 60 pounds, and a bag of sugar weighs 80 pounds. Which part of the truck’s cargo is heavier, the flour or the sugar? flour

4 -1 Ratios, Rates, and Unit Rates Learn to work with rates and ratios.

4 -1 Ratios, Rates, and Unit Rates Vocabulary rate unit price

4 -1 Ratios, Rates, and Unit Rates A rate is a comparison of two quantities that have different units. Ratio: 90 3 Rate: 90 miles 3 hours Read as “ 90 miles per 3 hours. ”

4 -1 Ratios, Rates, and Unit Rates Unit rates are rates in which the second quantity is 1. The ratio 90 can be simplified by dividing: 3 90 = 30 3 1 unit rate: 30 miles, or 30 mi/h 1 hour

4 -1 Ratios, Rates, and Unit Rates Additional Example 1: Finding Unit Rates Geoff can type 30 words in half a minute. How many words can he type in 1 minute? 30 words 1 2 minute Write a rate. 30 words • 2 = 60 words 1 1 minute 2 minute • 2 Multiply to find words per minute. Geoff can type 60 words in one minute.

4 -1 Ratios, Rates, and Unit Rates Check It Out: Example 1 Penelope can type 90 words in 2 minutes. How many words can she type in 1 minute? 90 words 2 minutes Write a rate. 90 words ÷ 2 = 45 words 2 minutes ÷ 2 1 minute Divide to find words per minute. Penelope can type 45 words in one minute.

4 -1 Ratios, Rates, and Unit Rates Additional Example 2 A: Chemistry Application Five cubic meters of copper has a mass of 44, 800 kilograms. What is the density of copper? 44, 800 kg 5 m 3 Write a rate. 44, 800 kg ÷ 5 5 m 3 ÷ 5 Divide to find kilograms per 1 m 3. 8, 960 kg 1 m 3 Copper has a density of 8, 960 kg/m 3.

4 -1 Ratios, Rates, and Unit Rates Additional Example 2 B: Chemistry Application A piece of gold with a volume of 0. 5 cubic meters weighs 9650 kilograms. What is the density of gold? 9650 kg 0. 5 m 3 Write a rate. 9650 kg • 2 0. 5 m 3 • 2 Multiply to find kilograms per 1 m 3. 19, 300 kg 1 m 3 Gold has a density of 19, 300 kg/m 3.

4 -1 Ratios, Rates, and Unit Rates Check It Out: Example 2 A Four cubic meters of precious metal has a mass of 18, 128 kilograms. What is the density of the precious metal? 18, 128 kg 4 m 3 Write a rate. 18, 128 kg ÷ 4 4 m 3 ÷ 4 Divide to find kilograms per 1 m 3. 4, 532 kg 1 m 3 Precious metal has a density of 4, 532 kg/m 3.

4 -1 Ratios, Rates, and Unit Rates Check It Out: Example 2 B A piece of gem stone with a volume of 0. 25 cubic meters weighs 3540 kilograms. What is the density of the gem stone? 3540 kg 0. 25 m 3 Write a rate. 3540 kg • 4 0. 25 m 3 • 4 Multiply to find kilograms per 1 m 3. 14, 160 kg 1 m 3 The gem stone has a density of 14, 160 kg/m 3.

4 -1 Ratios, Rates, and Unit Rates Additional Example 3 A: Application A driver is competing in a 500 -mile auto race. In the first 2 hours of the race, the driver travels 356 miles. What is the driver's average speed? r= d t 356 mi = 2 h = 178 mi/h Find the ratio of distance to time. Substitute 356 for d and 2 hours for t. Divide to find the unit rate. The driver's average speed is 178 mi/h.

4 -1 Ratios, Rates, and Unit Rates Additional Example 3 B: Application A driver is competing in a 500 -mile auto race. The driver estimates that he will finish the race in less than 3 hours. If the driver keeps traveling at the same average speed, is his estimate reasonable? Explain. Determine how long the trip will take. Use the formula d = rt. 500 = 178 t _ ___ 178 2. 8 ≈ t Substitute 500 for d and 178 for r. Divide both sides by 178. Simplify. Yes; at an average speed of 178 mi/h, the race will take about 2. 8 hours.

4 -1 Ratios, Rates, and Unit Rates Helpful Hint The formula r = d is equivalent to d= rt, t as shown below. r= d t d r▪t= ▪ t t rt = d

4 -1 Ratios, Rates, and Unit Rates Check It Out: Example 3 A A cyclist is competing in a 70 -mile bike race. In the first 2 hours of the race, the cyclist travels 14 miles. What is the cyclist's average speed? d r= Find the ratio of distance to time. t 14 mi = 2 h = 7 mi/h Substitute 14 for d and 2 hours for t. Divide to find the unit rate. The cyclist's average speed is 7 mi/h.

4 -1 Ratios, Rates, and Unit Rates Check It Out: Example 3 B A cyclist is competing in a 70 -mile bike race. The cyclist estimates that he will finish the race in less than 8 hours. If the cyclist keeps traveling at the same average speed, is the estimate reasonable? Explain. Determine how long the trip will take. Use the formula d = rt. 70 7 t _ = ___ 7 7 10 = t Substitute 70 for d and 7 for r. Divide both sides by 7. Simplify. No; at an average speed of 7 mi/h, the race will take about 10 hours.

4 -1 Ratios, Rates, and Unit Rates Unit price is a unit rate used to compare price per item.

4 -1 Ratios, Rates, and Unit Rates Additional Example 4: 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? price for jar = $2. 19 $0. 15 number of ounces 15 Divide the price by the number of ounces. price for jar = $2. 78 $0. 14 number of ounces 20 The better buy is the 20 -oz jar for $2. 78.

4 -1 Ratios, Rates, and Unit Rates Check It Out: Example 4 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 = $4. 95 $1. 65 Divide the price by the number of balls 3 of balls. price for package = $18. 95 $1. 58 number of balls 12 The better buy is the 12 -pack for $18. 95.

4 -1 Ratios, Rates, and Unit Rates Lesson Quizzes Standard Lesson Quiz for Student Response Systems

4 -1 Ratios, Rates, and Unit Rates Lesson Quiz: Part I 1. Meka can make 6 bracelets per half hour. How many bracelets can she make in 1 hour? ≈ 6. 94 g/cm 3 2. A penny has a mass of 2. 5 g and a volume of approximately 0. 360 cm 3. What is the approximate density of a penny? 12 3. Melissa is driving to her grandmother's house. In the first 3 hours of the drive, she travels 159 miles. What is Melissa's average speed? 53 mi/h

4 -1 Ratios, Rates, and Unit Rates Lesson Quiz: Part II Determine the better buy. 5. A half dozen carnations for $4. 75 or a dozen for $9. 24 a dozen

4 -1 Ratios, Rates, and Unit Rates Lesson Quiz for Student Response Systems 1. John can walk 16 miles in 4 hours. How many miles can he walk in one hour? A. 16 miles B. 8 miles C. 4 miles D. 2 miles

4 -1 Ratios, Rates, and Unit Rates Lesson Quiz for Student Response Systems 2. Estimate the unit rate. 272 sailors in 17 ships A. 12 sailors per ship B. 14 sailors per ship C. 16 sailors per ship D. 18 sailors per ship

4 -1 Ratios, Rates, and Unit Rates Lesson Quiz for Student Response Systems 3. Which of the following would be a better buy than purchasing 4 mangoes for $16? A. 2 mangoes for $10 B. half a dozen mangoes for $25 C. 8 mangoes for $ 28 D. one dozen mangoes for $54