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Ratios and unit rates

Step 8 · Ratios and negative numbers topics

A ratio compares two quantities. A unit rate tells you how much of one quantity goes with 1 of the other.

What a ratio is

A ratio compares two quantities by saying how much of one there is for a given amount of the other. If a recipe uses 2 cups of flour for every 3 cups of oats, the ratio of flour to oats is 2 to 3. You can write a ratio three ways:

2 to 3
2 : 3
$\frac{2}{3}$

Order matters. The ratio of oats to flour is 3 to 2, which is a different ratio.

Equivalent ratios

Double the recipe and you use 4 cups of flour and 6 cups of oats. The amounts changed, but the mixture is the same, because 4 to 6 is the same comparison as 2 to 3. Ratios you get by multiplying or dividing both parts by the same number are called equivalent ratios. A table keeps them organized:

Flour (cups)	Oats (cups)
2	3
4	6
6	9
10	15

To write a ratio in simplest form, divide both parts by the largest number that goes into both. The ratio 4 : 6 simplifies to 2 : 3.

Rates and unit rates

A rate is a ratio that compares quantities with different units, such as miles and hours, or dollars and pounds. A unit rate says how much of the first quantity goes with exactly 1 of the second. To find a unit rate, divide.

Example: Speed

A car travels 150 miles in 3 hours. What is its speed in miles per hour?

Write the rate: 150 miles in 3 hours.
Divide the first quantity by the second: $150 \div 3 = 50$ .
Attach the units: 50 miles for every 1 hour.

Answer50 miles per hour

Example: Better buy

A 12-pack of juice boxes costs $4.20. An 8-pack costs $3.04. Which is the better buy?

Find the price of one box in the 12-pack: $4.20 \div 12 = 0.35$ , so $0.35 per box.
Find the price of one box in the 8-pack: $3.04 \div 8 = 0.38$ , so $0.38 per box.
Compare the unit prices: $0.35 is less than $0.38.

AnswerThe 12-pack is the better buy.

Try it yourself

Work each one on paper first, then open the answer to check.

A class has 12 girls and 15 boys. Write the ratio of girls to boys in simplest form.
Show answer
4 : 5. Both 12 and 15 divide by 3.
Fill in the missing number: 3 : 5 = 12 : ?
Show answer
20, because the first part was multiplied by 4, so the second part is too: $5 \times 4 = 20$ .
A printer prints 96 pages in 4 minutes. What is the unit rate?
Show answer
24 pages per minute, because $96 \div 4 = 24$ .
Six oranges cost $2.70. What does one orange cost?
Show answer
$0.45, because $2.70 \div 6 = 0.45$ .
Runner A covers 400 meters in 80 seconds. Runner B covers 300 meters in 50 seconds. Who is faster?
Show answer
Runner B. A runs $400 \div 80 = 5$ meters per second and B runs $300 \div 50 = 6$ meters per second.