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Lesson 4.2.1

Ratios and Rates. Lesson 4.2.1. Lesson 4.2.1. Ratios and Rates. California Standard: Measurement and Geometry 3.2

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Lesson 4.2.1

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  1. Ratios and Rates Lesson 4.2.1

  2. Lesson 4.2.1 Ratios and Rates California Standard: Measurement and Geometry 3.2 Use measures expressed as rates (e.g., speed, density)and measures expressed as products (e.g., person-days) to solve problems; check the units of thesolutions; and use dimensional analysis to check the reasonableness of the answer. What it means for you: You’ll learn what rates are and how you can use them to compare things — such as which size product is better value. • Key words: • rate • ratio • fraction • denominator • unit rate

  3. Best apples$2 per pound Lesson 4.2.1 Ratios and Rates Rates are used a lot in daily life. You often hear people talk about speed in miles per hour, or the cost of groceries in dollars per pound. A rate tells you how much one thing changes when something else changes by a certain amount. Imagine buying apples for $2 per pound — the cost will increase by $2 for every pound you buy.

  4. The ratio 5 : 6 could also be expressed as “5 to 6” or as the fraction . 5 6 Lesson 4.2.1 Ratios and Rates Ratios are Used to Compare Two Numbers You might remember ratios from grade 6. Ratios compare two numbers, and don’t have any units. For example, the ratio of boys to girls in a class might be 5 : 6. There are three ways of expressing a ratio.

  5. This could also be written “4 to 3” or . 4 3 Lesson 4.2.1 Ratios and Rates Example 1 There are four nuts between three squirrels. What is the ratio of nuts to squirrels? Solution There are 4 nuts to 3 squirrels so the ratio of nuts to squirrels is 4 : 3. Solution follows…

  6. For example, if you travel 60 miles in 3 hours you would be traveling at a rate of . 60 miles 60 miles 20 miles 3 hours 3 hours 1 hour So = , or 20 miles per hour. Lesson 4.2.1 Ratios and Rates Ratios Compare Quantities With Different Units A rate is a special kind of ratio, because it compares two quantities that have different units. You’d normally write this as a unit rate. That’s one with a denominator of 1.

  7. 110 steps 55 steps = = 55 steps per minute 2 minutes 1 minute Lesson 4.2.1 Ratios and Rates Example 2 John takes 110 steps in 2 minutes. What is his unit rate in steps per minute? Solution 110 steps in 2 minutes means a rate of: Solution follows…

  8. So, if it costs 2 dollars for 3 apples, the unit rate is the price per apple, which is = 2 dollars ÷ 3 apples = 0.67 dollars per apple 2 dollars 3 apples Lesson 4.2.1 Ratios and Rates Numerator ÷ Denominator Gives a Unit Rate Dividing the numerator by the denominator of a rate gives the unit rate.

  9. 54 miles = (54 ÷ 3) miles per hour = 18 miles per hour. 3 hours This is a unit rate because the denominator is now 1 (it’s equivalent to mi/h). 18 1 Lesson 4.2.1 Ratios and Rates Example 3 A car goes 54 miles in 3 hours. Write this as a unit rate in miles per hour. Solution Divide the top by the bottom of the rate. Solution follows…

  10. 420 The rate is revolutions per minute. 7 This is a unit rate because 60 revolutions per minute has a denominator of 1 (60 = ). 60 1 Lesson 4.2.1 Ratios and Rates Example 4 If a wheel spins 420 times in 7 minutes, what is its unit rate in revolutions per minute? Solution Divide the top by the bottom of the rate. (420 ÷ 7) revolutions per minute =60 revolutions per minute. Solution follows…

  11. Lesson 4.2.1 Ratios and Rates Guided Practice In Exercises 1–3, find the unit rates. 1. $3.60 for 3 pounds of tomatoes. 2. $25 for 500 cell phone minutes. 3. 648 words typed in 8 minutes. 4. Joaquin buys 2 meters of fabric, which costs him $9.50. What was the price per meter? 5. Mischa buys a $19.98 ticket for unlimited rides at a fairground. She goes on six rides. How much did she pay per ride? $1.20 per pound of tomatoes $0.05 per minute 81 words per minute $4.75 per meter $3.33 per ride Solution follows…

  12. Lesson 4.2.1 Ratios and Rates Use “Unit Rates” to Find the Better Buy Stores often sell different sizes of the same thing, such as clothes detergent or fruit juice. A bigger size is often a better buy — meaning that you get more product for the same amount of money. But this isn’t always the case, so it’s useful to be able to work out which is thebetter buy. You can do this by finding the price for a single unitof each product. The units can be ounces, liters, meters, or whatever is most sensible.

  13. CEREAL $3.20 for 16 ounce box $4.32 for 24 ounce box CEREAL 4.32 dollars 3.20 dollars Rate is . Rate is . 16 ounces 24 ounces Lesson 4.2.1 Ratios and Rates Example 5 A store sells two sizes of cereal. Which is the better buy? Solution 16 ounce box: Unit rate = (3.20 ÷ 16) dollars per ounce = $0.20 per ounce 24 ounce box: Unit rate = (4.32 ÷ 24) dollars per ounce = $0.18 per ounce The 24 ounce box is the better buy — the price per ounce is lower. Solution follows…

  14. Lesson 4.2.1 Ratios and Rates Guided Practice 6. Determine which phone company offers the better deal:Phone Company A: $40 for 800 minutes.Phone Company B: $26 for 650 minutes. 7. Determine which is the better deal on carrots: $1.20 for 2 lb or $2.30 for 5 lb. Unit rate from Company A = $40 ÷ 800 minutes = 5 ¢ per minuteUnit rate from Company B = $26 ÷ 650 minutes = 4 ¢ per minute So, phone Company B offers the best deal. Unit rate deal 1 = $1.20 ÷ 2 lb = 60 ¢ per lbUnit rate deal 2 = $2.30 ÷ 5 lb = 46 ¢ per lbSo, deal 2 — $2.30 for 5 lb is best. Solution follows…

  15. Lesson 4.2.1 Ratios and Rates Independent Practice In Exercises 1–6, write each as a unit rate. 1. $4.50 for 6 pens 2. 100 miles in 8 h 3. 200 pages in 5 days 4. 120 miles in 2 h 5. $400 for 10 items 6. $36 in 6 hours $0.75 per pen 12.5 miles per hour 40 pages per day 60 miles per hour $40 per item $6 per hour Solution follows…

  16. Lesson 4.2.1 Ratios and Rates Independent Practice 7. Peanuts are either $1.70 per pound or $8 for 5 pounds. Which is the better buy? 8. Lemons sell for $4.50 for 6, or $10.50 for 15. Which is the better buy? 9. “$40 for 500 pins or $60 for 800 pins.” Which is the better buy? $8 for 5 pounds $10.50 for 15 $60 for 800 pins Solution follows…

  17. Lesson 4.2.1 Ratios and Rates Round Up Rates compare one thing to another and always have units. A unit rate is a rate that has a denominator of one. In the next Lesson you’ll see how rate is related to the slope of a graph.

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