Unit Rates: Finding the Price of One

A rate compares different units

A ratio can compare anything. A rate compares two quantities measured in different units:

$$150 \text{ miles in } 3 \text{ hours} \qquad 4 \text{ apples for } 3 \text{ dollars}$$

The word per usually signals a rate: miles per hour, dollars per apple, words per minute.

A unit rate is a rate for exactly one of the second quantity.

Finding one by dividing

Divide until the second amount is $1$.

$$\frac{150 \text{ miles}}{3 \text{ hours}} = \frac{50 \text{ miles}}{1 \text{ hour}} = 50 \text{ mph}$$

That is scaling a ratio down, the way any equivalent ratio is found — but scaled to a second part of exactly $1$.

Miles	Hours
$150$	$3$
$100$	$2$
$50$	$1$

The last row is the unit rate.

Why one is the useful number

Two packages of rice sit on a shelf:

• $5$ pounds for $8$ dollars
• $12$ pounds for $18$ dollars

Which is cheaper? Nothing in those numbers answers that, because the two packages are different sizes and different prices at the same time.

Reducing both to a price per pound removes the size difference:

$$\frac{8}{5} = 1.60 \text{ per pound} \qquad \frac{18}{12} = 1.50 \text{ per pound}$$

The larger bag is cheaper per pound. Once both are measured against the same $1$, the comparison is a single subtraction.

This is what unit rates are for. Speeds, prices, fuel economy and wages are all quoted “per one” for exactly this reason — so that two of them can be set side by side without any further work.

Rate or ratio?

Comparison	Units	Type
$3$ girls to $2$ boys	same (people)	ratio
$150$ miles in $3$ hours	different	rate
$4$ apples for $3$ dollars	different	rate

Every rate is a ratio. The extra word marks the case where the two quantities are measured in different things, which is when “per” starts to mean something.

Two directions, two rates

Every rate can be turned around, and both versions are correct.

$$\frac{4 \text{ apples}}{3 \text{ dollars}} = 1.33 \text{ apples per dollar}$$ $$\frac{3 \text{ dollars}}{4 \text{ apples}} = 0.75 \text{ dollars per apple}$$

Which one you want depends on the question. “How much does one apple cost?” needs dollars per apple. “How many apples can I get for a dollar?” needs the other.

The unit you want one of goes on the bottom.

Worked examples

Common mistakes

Practice problems