4-1 Ratios, Rates, and Unit Rates
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1 Warm Up Problem of the Day Lesson Presentation Lesson Quizzes
2 Warm Up Divide. Round answers to the nearest tenth
3 Learn to work with rates and ratios.
4 Vocabulary rate unit rate unit price
5 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.
6 Unit rates are rates in which the second quantity is 1. The ratio 90 3 can be simplified by dividing: 90 3 = 30 1 unit rate: 30 miles, 1 hour or 30 mi/h
7 Example 1: Finding Unit Rates A. Geoff can type 30 words in half a minute. How many words can he type in 1 minute? 30 words minute 1 2 Write a rate. 30 words minute 2 = 60 words 1 minute Multiply to find words per minute. Geoff can type 60 words in one minute.
8 Check It Out B. Penelope can type 90 words in 2 minutes. How many words can she type in 1 minute? 90 words 2 minutes 90 words 2 2 minutes 2 = 45 words 1 minute Write a rate. Divide to find words per minute. Penelope can type 45 words in one minute.
9 Example 2: Chemistry Application A. 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 8,960 kg 1 m 3 Divide to find kilograms per 1 m 3. Copper has a density of 8,960 kg/m 3.
10 Example 2: Chemistry Application B. 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 kg 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.
11 Check It Out B. 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.
12 Check It Out C. 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 kg 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.
13 Example 3 : 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 Find the ratio of distance to time. Substitute 356 for d and 2 hours for t. = 178 mi/h Divide to find the unit rate. The driver's average speed is 178 mi/h.
14 Example 3: 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 _ = 178t Substitute 500 for d and 178 for r Divide both sides by t Simplify. Yes; at an average speed of 178 mi/h, the race will take about 2.8 hours.
15 Helpful Hint The formula r = as shown below. d t is equivalent to d= rt, r = d t r t = d t t rt = d
16 Check It Out 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? r = d t Find the ratio of distance to time. = 14 mi 2 h Substitute 14 for d and 2 hours for t. = 7 mi/h Divide to find the unit rate. The cyclist's average speed is 7 mi/h.
17 Use the formula d = rt. Check It Out 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. _ 70 = 7t Substitute 70 for d and 7 for r. 7 7 Divide both sides by = t Simplify. No; at an average speed of 7 mi/h, the race will take about 10 hours.
18 Unit price is a unit rate used to compare price per item.
19 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 number of ounces = $2.19 $ Divide the price by the number of ounces. price for jar number of ounces = $ $0.14 The better buy is the 20-oz jar for $2.78.
20 Check It Out Golf balls can be purchased in a 3-pack for $4.95 or a 12-pack for $ Which is the better buy? price for package number of balls = $4.95 $ Divide the price by the number of balls. price for package number of balls = $ $1.58 The better buy is the 12-pack for $18.95.
21 Lesson Quiz
22 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 cm 3. What is the approximate density of a penny? 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
23 Lesson Quiz: Part II Determine the better buy. 5. A half dozen carnations for $4.75 or a dozen for $9.24 a dozen
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