Math Core

A 6-pack of water costs $3 and a 24-pack costs $10. Which is the better deal? It is hard to tell until you find the price of one bottle in each pack. That price for one is called a unit rate, and it makes comparing easy.

Rates

When a ratio compares two quantities with different units, like miles and hours or dollars and pounds, it is called a rate.

150 miles in 3 hours
$12 for 4 pounds
90 words in 2 minutes

Unit rates

A rate tells you the most when the second quantity is 1. "150 miles in 3 hours" is the same speed as "50 miles in 1 hour."

Definition

Unit rate

A unit rate is a rate that tells how much of the first quantity there is for 1 of the second quantity. To find it, divide:

\text{unit rate} = \frac{\text{first quantity}}{\text{second quantity}}

Unit rates often use the word per, which means "for each one." For example, 50 miles per hour, or $3 per pound.

Worked example: Finding unit rates

Find each unit rate.

A car travels 150 miles in 3 hours.
Four pounds of apples cost $12.
A typist types 90 words in 2 minutes.

Solutions.

$150 \div 3 = 50$ , so the car travels 50 miles per hour.
$12 \div 4 = 3$ , so the apples cost $3 per pound.
$90 \div 2 = 45$ , so the typist types 45 words per minute.

A ratio table shows why this works. Dividing both rows by the same number keeps the rate the same:

Hours	3	1
Miles	150	50

Using a unit rate

Once you know the amount for 1, you can find the amount for any number by multiplying.

Worked example: Scaling up

A faucet fills 28 liters of water in 4 minutes. How much water does it fill in 9 minutes?

First find the unit rate: $28 \div 4 = 7$ liters per minute.

Then multiply: $7 \times 9 = 63$ liters.

Comparing with unit rates

To decide which choice is faster, cheaper or better, find the unit rate for each one and compare.

Worked example: The better buy

A 6-pack of water costs $3. A 24-pack costs $10. Which pack has the lower price per bottle?

6-pack: $3 \div 6 = 0.50$ , so $0.50 per bottle.
24-pack: $10 \div 24 \approx 0.42$ , so about $0.42 per bottle.

The 24-pack is the better buy.

Common mistake

Divide in the right order. The unit rate "dollars per bottle" means dollars $\div$ bottles, not bottles $\div$ dollars. For the 6-pack, $6 \div 3 = 2$ tells you bottles per dollar. That is also a unit rate, but a bigger number of bottles per dollar is the better deal. Always say the units out loud so you know what your number means.

Graphing a unit rate

Here are the times and distances for a cyclist riding at 12 miles per hour:

Hours	1	2	3	4
Miles	12	24	36	48

Hours (x) and miles (y) for a cyclist riding 12 miles per hourOpen in grapher →

The unit rate is the $y$ -value when $x = 1$ : the point $(1, 12)$ . Each time you move 1 hour to the right, the distance goes up 12 miles.

Tip

Check a unit rate by estimating. If 7 notebooks cost $20.65, the price of each is a bit less than $21 \div 7 = 3$ dollars. The exact unit price, $2.95, fits that estimate.