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Understanding Ratios and Unit Rates in Mathematics

This text covers the concepts of ratios, unit rates, and density calculations through various examples and applications in mathematics, including scenarios involving buying items in different quantities and determining densities of substances. It also includes exercises for practice and examples for estimating unit rates and comparing costs based on unit prices.

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Understanding Ratios and Unit Rates in Mathematics

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  1. Week 16, Day Three HW # 56 - p. 230-231 # 17-29 odd Warm up Mr. Lynch bought some oranges and pears. After giving away 10 oranges, he had twice as many pears as oranges left. If he had 24 pears in the end, what was the ratio of oranges to pears in the beginning?

  2. Warm Up Response 24/2 = 12 He had 12 oranges in the end 12 + 10 = 22 He had 22 oranges in the beginning Oranges:Pears 22:24 11:12 The ratio of oranges to pears in the beginning was 11:12

  3. Homework Check p. 226-227 # 1-35 odd and # 36 21) No 23) no; she needs 3 ½ c of pancake mix to make 14 servings 25) – 27) Yes 29) No 5/6 31) Yes 33) No 11/6 35) 39/18 36) ---- 1) 2/5 3) 10/7 5) 15/1 7) No 9) Yes 11) Yes 13) 25/12 15) 12/1 17) 2/5 19) yes

  4. • Review for Mastery 5-1 • Review for Mastery 5-2 • Challenge 5-2

  5. A rate is a comparison of two quantities measured in different units. 90 3 Ratio: Read as “90 miles per 3 hours.” 90 miles 3 hours Rate:

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

  7. 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 minute 1 2 Write a rate. Multiply to find words per minute. 60 words 1 minute 30 words • 2 minute • 2 2 = 1 Geoff can type 60 words in one minute.

  8. 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. 45 words 1 minute 90 words ÷ 2 2 minutes ÷ 2 = Penelope can type 45 words in one minute.

  9. 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 the rate. Divide to find kilograms per 1 m3. 44,800 kg ÷ 5 5 m3 ÷ 5 8,960 kg 1 m3 Copper has a density of 8,960 kg/m3.

  10. 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 the rate. Multiply to find kilograms per 1 m3. 9650 kg • 2 0.5 m3 • 2 19,300 kg 1 m3 Gold has a density of 19,300 kg/m3.

  11. 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 the rate. Divide to find kilograms per 1 m3. 18,128 kg ÷ 4 4 m3 ÷ 4 4,532 kg 1 m3 The precious metal has a density of 4,532 kg/m3.

  12. Check It Out! Example 2B A piece of gemstone with a volume of 0.25 cubic meters weighs 3540 kilograms. What is the density of the gemstone? 3540 kg 0.25 m3 Write the rate. Multiply to find kilograms per 1 m3. 3540 kg • 4 0.25 m3 • 4 14,160 kg 1 m3 The gemstone has a density of 14,160 kg/m3.

  13. Additional Example 3A: Estimating Unit Rates Estimate each unit rate. 468 students to 91 computers  455 students 91 computers Choose a number close to 468 that is divisible by 91. 468 students 91 computers 5 students 1 computer Divide to find students per computer.  468 students to 91 computers is approximately 5 students per computer.

  14. Additional Example 3B: Estimating Unit Rates Estimate each unit rate. 313 feet in 8 seconds  320 feet 8 seconds Choose a number close to 313 that is divisible by 8. 313 feet 8 seconds 40 feet 1 second  Divide to find feet per second. 313 feet to 8 seconds is approximately 40 feet per second.

  15. Check It Out! Example 3A Estimate each unit rate. 583 soccer players to 85 soccer balls.  595 players 85 soccer balls Choose a number close to 583 that is divisible by 85. 583 players 85 soccer balls 7 players 1 soccer ball Divide to find players per soccer ball.  583 soccer players to 85 soccer balls is approximately 7 players per soccer ball.

  16. Check It Out! Example 3B Estimate each unit rate. 271 yards in 3 hours  270 yards 3 hours Choose a number close to 271 that is divisible by 3. 271 yards 3 hours 90 yards 1 hour  Divide to find yards per hour. 271 yards to 3 hours is approximately 90 yards per hour.

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

  18. Additional Example 4A: Finding Unit Prices to Compare Costs Pens can be purchased in a 5-pack for $1.95 or a 15-pack for $6.20. Which pack has the lower unit price? price for package number of pens $1.95 5 Divide the price by the number of pens. = = $0.39 price for package number of pens $6.20 15  = $0.41 The 5-pack for $1.95 has the lower unit price.

  19. Additional Example 4B: 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 jar has the lower unit price? price for jar number of ounces $2.19 15 Divide the price by the number of ounces.  = $0.15 price for jar number of ounces $2.78 20  = $0.14 The 20 oz jar for $2.78 has the lower unit price.

  20. Check It Out! Example 4A Golf balls can be purchased in a 3-pack for $4.95 or a 12-pack for $18.95. Which pack has the lower unit price? price for package number of balls $4.95 3  Divide the price by the number of balls. = $1.65  price for package number of balls $18.95 12 = $1.58 The 12-pack for $18.95 has the lower unit price.

  21. Check It Out! Example 4B John can buy a 24 oz bottle of ketchup for $2.19 or a 36 oz bottle for $3.79. Which bottle has the lower unit price? price for bottle number of ounces $2.19 24 Divide the price by the number of ounces.  = $0.09 price for bottle number of ounces $3.79 36  = $0.11 The 24 oz jar for $2.19 has the lower unit price.

  22. Lesson Quiz: Part I 12 1. Meka can make 6 bracelets per half hour. How many bracelets can she make per 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? ≈ 6.94 g/cm3 Estimate each unit rate. 3. $2.22 for 6 stamps 4. 8 heartbeats in 6 seconds $0.37 per stamp  1.3 beats/s

  23. Lesson Quiz: Part II Find each unit price. Then tell which has the lower unit price. 5. A half dozen carnations for $4.75 or a dozen a dozen for $9.24 6. 4 pens for $5.16 or a ten-pack for $12.90. They cost the same.

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