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Lesson 4.2.1Lesson 4.2.1

Ratios and RatesRatios and Rates

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Lesson

4.2.1Ratios and RatesRatios and Rates

California Standard:Measurement and Geometry 3.2Use measures expressed as rates (e.g., speed, density) and measures expressed as products (e.g., person-days) to solve problems; check the units of the solutions; 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

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Ratios and RatesRatios and RatesLesson

4.2.1

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.

Imagine buying apples for $2 per pound — the cost will increase by $2 for every pound you buy.

Best apples$2 per pound

A rate tells you how much one thing changes when something else changes by a certain amount.

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Ratios and RatesRatios and Rates

Ratios are Used to Compare Two Numbers

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4.2.1

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.

The ratio 5 : 6 could also be expressed as “5 to 6”

or as the fraction .5

6

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Ratios and RatesRatios and Rates

Example 1

Solution follows…

Lesson

4.2.1

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.

This could also be written “4 to 3” or . 4

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Ratios and RatesRatios and Rates

Ratios Compare Quantities With Different Units

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4.2.1

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.

For example, if you travel 60 miles in 3 hours

you would be traveling at a rate of .60 miles

3 hours

60 miles

3 hoursSo = , or 20 miles per hour.

20 miles

1 hour

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Ratios and RatesRatios and Rates

Example 2

Solution follows…

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4.2.1

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:

55 steps

1 minute

110 steps

2 minutes= = 55 steps per minute

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Ratios and RatesRatios and Rates

Numerator ÷ Denominator Gives a Unit Rate

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4.2.1

Dividing the numerator by the denominator of a rate gives the unit rate.

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

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Ratios and RatesRatios and Rates

Example 3

Solution follows…

Lesson

4.2.1

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.

= (54 ÷ 3) miles per hour = 18 miles per hour.54 miles

3 hours

This is a unit rate because the denominator is now

1 (it’s equivalent to mi/h).18

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Ratios and RatesRatios and Rates

Example 4

Solution follows…

Lesson

4.2.1

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.

The rate is revolutions per minute. 420

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(420 ÷ 7) revolutions per minute = 60 revolutions per minute.

This is a unit rate because 60 revolutions per minute has a

denominator of 1 (60 = ).60

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Ratios and RatesRatios and Rates

Guided Practice

Solution follows…

Lesson

4.2.1

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

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Ratios and RatesRatios and Rates

Use “Unit Rates” to Find the Better Buy

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4.2.1

Stores often sell different sizes of the same thing, such as clothes detergent or fruit juice.

But this isn’t always the case, so it’s useful to be able to work out which is the better buy.

You can do this by finding the price for a single unit of each product. The units can be ounces, liters, meters, or whatever is most sensible.

A bigger size is often a better buy — meaning that you get more product for the same amount of money.

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Ratios and RatesRatios and Rates

Example 5

Solution follows…

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4.2.1

A store sells two sizes of cereal. Which is the better buy?

Solution

CEREAL

CEREAL

$3.20 for 16 ounce box$4.32 for 24 ounce box

16 ounce box:

Unit rate = (3.20 ÷ 16) dollars per ounce = $0.20 per ounce

3.20 dollars

16 ouncesRate is .

The 24 ounce box is the better buy — the price per ounce is lower.

24 ounce box:

Unit rate = (4.32 ÷ 24) dollars per ounce = $0.18 per ounce

4.32 dollars

24 ouncesRate is .

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Ratios and RatesRatios and Rates

Guided Practice

Solution follows…

Lesson

4.2.1

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.

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Ratios and RatesRatios and Rates

Independent Practice

Solution follows…

Lesson

4.2.1

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$40 per item

$0.75 per pen

40 pages per day

$6 per hour

12.5 miles per hour

60 miles per hour

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Ratios and RatesRatios and Rates

Independent Practice

Solution follows…

Lesson

4.2.1

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

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Ratios and RatesRatios and RatesLesson

4.2.1

Round UpRound Up

Rates compare one thing to another and always have units.

In the next Lesson you’ll see how rate is related to the slope of a graph.

A unit rate is a rate that has a denominator of one.