Math Core

You already know how to find a unit rate like 12 miles in 3 hours: divide to get 4 miles per hour. But real life is messier. What if you walk $\dfrac{1}{2}$ mile in $\dfrac{1}{6}$ of an hour? The same idea still works. You just have to divide fractions.

A rate "per 1"

A unit rate tells you how much of one quantity goes with 1 of another. To find it, divide the first quantity by the second.

\text{unit rate} = \frac{\text{amount of the first quantity}}{\text{amount of the second quantity}}

When both amounts are fractions, you get a fraction on top of a fraction, called a complex fraction. It looks strange, but the fraction bar just means "divide."

\frac{\frac{1}{2}}{\frac{1}{6}} = \frac{1}{2} \div \frac{1}{6}

Definition

Complex fraction

A complex fraction is a fraction whose numerator, denominator or both are also fractions. To simplify it, divide the numerator by the denominator: multiply the top by the reciprocal of the bottom.

Why dividing makes sense

Think about the walk: $\dfrac{1}{2}$ mile in $\dfrac{1}{6}$ hour. There are 6 sixths in a whole hour. If you keep walking at the same pace for 6 of those sixths, you walk 6 times as far:

6 \times \frac{1}{2} = 3 \text{ miles in 1 hour.}

Dividing gives the same thing, faster:

\frac{1}{2} \div \frac{1}{6} = \frac{1}{2} \times \frac{6}{1} = \frac{6}{2} = 3.

Your speed is 3 miles per hour.

Unit rates with fractions

To find "A per B," compute $A \div B$ , even when $A$ and $B$ are fractions or mixed numbers.

Change mixed numbers to fractions first.
Multiply by the reciprocal of the second number.
Say the answer with its units: "miles per hour," "dollars per pound."

Worked example: A walking speed

Jada walks $\dfrac{2}{3}$ mile in $\dfrac{1}{4}$ hour. What is her speed in miles per hour?

Miles per hour means miles $\div$ hours:

\frac{2}{3} \div \frac{1}{4} = \frac{2}{3} \times \frac{4}{1} = \frac{8}{3} = 2\frac{2}{3}.

Jada walks $2\dfrac{2}{3}$ miles per hour.

Check: a quarter hour fits into an hour 4 times, and $4 \times \dfrac{2}{3} = \dfrac{8}{3}$ . It matches.

Worked example: A recipe rate

A recipe uses $\dfrac{3}{4}$ cup of flour for $\dfrac{2}{3}$ of a batch. How much flour is that per whole batch?

Cups per batch means cups $\div$ batches:

\frac{3}{4} \div \frac{2}{3} = \frac{3}{4} \times \frac{3}{2} = \frac{9}{8} = 1\frac{1}{8}.

A whole batch needs $1\dfrac{1}{8}$ cups of flour. That makes sense: a whole batch is a little more than $\dfrac{2}{3}$ of a batch, so it needs a little more than $\dfrac{3}{4}$ cup.

Two unit rates for every situation

Every rate can be flipped. If Maya mows $1\dfrac{1}{4}$ acres in $\dfrac{3}{4}$ hour, you can ask "acres per hour" or "hours per acre." Both are useful.

Worked example: Mixed numbers, both ways

Maya mows $1\dfrac{1}{4}$ acres in $\dfrac{3}{4}$ hour.

How many acres does she mow per hour?
How many hours does she need per acre?

First write $1\dfrac{1}{4} = \dfrac{5}{4}$ .

Acres per hour: $\dfrac{5}{4} \div \dfrac{3}{4} = \dfrac{5}{4} \times \dfrac{4}{3} = \dfrac{20}{12} = \dfrac{5}{3} = 1\dfrac{2}{3}$ acres per hour.
Hours per acre: $\dfrac{3}{4} \div \dfrac{5}{4} = \dfrac{3}{4} \times \dfrac{4}{5} = \dfrac{12}{20} = \dfrac{3}{5}$ hour per acre.

Notice that $\dfrac{5}{3}$ and $\dfrac{3}{5}$ are reciprocals. The two unit rates always are.

Common mistake

Watch the order. "Miles per hour" is miles $\div$ hours, not hours $\div$ miles. The word right after "per" goes on the bottom. If you divide in the wrong order, you get the reciprocal of the rate you wanted.

Comparing deals

Unit rates let you compare things fairly, even when the amounts are different.

Worked example: Which cheese is a better buy?

Store A sells $\dfrac{2}{3}$ pound of cheese for $3. Store B sells $\dfrac{3}{4}$ pound for $3.60. Which is cheaper per pound?

Store A: $3 \div \dfrac{2}{3} = 3 \times \dfrac{3}{2} = \dfrac{9}{2} = 4.50$ , so $4.50 per pound.
Store B: $3.60 \div \dfrac{3}{4} = 3.60 \times \dfrac{4}{3} = \dfrac{14.40}{3} = 4.80$ , so $4.80 per pound.

Store A is the better buy by 30 cents per pound.

Tip

Check your answer with a quick estimate. In the cheese example, $\dfrac{2}{3}$ pound costs $3, so a full pound should cost a bit more than $3, but less than double. $4.50 fits.