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Warm Up

This lesson presentation covers working with rates and ratios, including vocabulary, unit rates, and practical applications in chemistry and auto racing. The presentation includes examples and quizzes to reinforce learning.

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Warm Up

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  1. Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

  2. 420 18 73 21 430 18 380 16 Warm Up Divide. Round answers to the nearest tenth. 1.2. 3.4. 23.3 3.5 23.9 23.8

  3. 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. Learn to work with rates and ratios.

  5. Vocabulary rate unit rate unit price

  6. A rate is a comparison of two quantities that have different units. 90 3 Ratio: Read as “90 miles per 3 hours.” 90 miles 3 hours Rate:

  7. 90 3 The ratio can be simplified by dividing: Unit rates are rates in which the second quantity is 1. 90 3 30 1 = 30 miles, 1 hour unit rate: or 30 mi/h

  8. 30 words minute 1 2 30 words • 2 minute • 2 1 2 Additional Example 1: Finding Unit Rates Geoff can type 30 words in half a minute. How many words can he type in 1 minute? Write a rate. Multiply to find words per minute. 60 words 1 minute = Geoff can type 60 words in one minute.

  9. 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. Divide to find words per minute. 90 words ÷ 2 2 minutes ÷ 2 45 words 1 minute = Penelope can type 45 words in one minute.

  10. Additional Example 2A: Chemistry Application Five cubic meters of copper has a mass of 44,800 kilograms. What is the density of copper? 44,800 kg 5 m3 Write a rate. 44,800 kg ÷ 5 5 m3 ÷ 5 Divide to find kilograms per 1 m3. 8,960 kg 1 m3 Copper has a density of 8,960 kg/m3.

  11. Additional Example 2B: 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 m3 Write a rate. 9650 kg • 2 0.5 m3 • 2 Multiply to find kilograms per 1 m3. 19,300 kg 1 m3 Gold has a density of 19,300 kg/m3.

  12. Check It Out: Example 2A 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 m3 Write a rate. 18,128 kg ÷ 4 4 m3 ÷ 4 Divide to find kilograms per 1 m3. 4,532 kg 1 m3 Precious metal has a density of 4,532 kg/m3.

  13. Check It Out: Example 2B 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 m3 Write a rate. 3540 kg • 4 0.25 m3 • 4 Multiply to find kilograms per 1 m3. 14,160 kg 1 m3 The gem stone has a density of 14,160 kg/m3.

  14. d t r = 356 mi 2 h = Additional Example 3A: 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? Find the ratio of distance to time. Substitute 356 for d and 2 hours for t. Divide to find the unit rate. = 178 mi/h The driver's average speed is 178 mi/h.

  15. Additional Example 3B: 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. Substitute 500 for d and 178 for r. 500 = 178t _ ___178 178 Divide both sides by 178. 2.8 ≈ t Simplify. Yes; at an average speed of 178 mi/h, the race will take about 2.8 hours.

  16. Helpful Hint The formula r = is equivalent to d= rt, as shown below. r = r ▪ t = ▪ t rt = d d t dt dt

  17. d t r = 14 mi 2 h = Check It Out: Example 3A 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? Find the ratio of distance to time. Substitute 14 for d and 2 hours for t. Divide to find the unit rate. = 7 mi/h The cyclist's average speed is 7 mi/h.

  18. Check It Out: Example 3B 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. Substitute 70 for d and 7 for r. 70 = 7t _ ___7 7 Divide both sides by 7. 10 = t Simplify. No; at an average speed of 7 mi/h, the race will take about 10 hours.

  19. Unit price is a unit rate used to compare price per item.

  20. price for jar number of ounces price for jar number of ounces 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? Divide the price by the number of ounces. $2.19 15 =  $0.15 $2.78 20 =  $0.14 The better buy is the 20-oz jar for $2.78.

  21. 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? Divide the price by the number of balls. price for package number of balls $4.95 3 =  $1.65 price for package number of balls $18.95 12  = $1.58 The better buy is the 12-pack for $18.95.

  22. Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

  23. Lesson Quiz: Part I 1. Meka can make 6 bracelets per half hour. How many bracelets can she make in 1 hour? 2. A penny has a mass of 2.5 g and a volume of approximately 0.360 cm3. What is the approximate density of a penny? 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? ≈ 6.94 g/cm3 12 53 mi/h

  24. Lesson Quiz: Part II Determine the better buy. 5. A half dozen carnations for $4.75 or a dozen for $9.24 a dozen

  25. 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

  26. 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

  27. 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

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