Understanding Unit Rates with Fractions

A unit rate tells you how much of one quantity corresponds to exactly one unit of another quantity. You encounter unit rates all the time, like speed (miles per hour) or price (dollars per pound).

Sometimes, the quantities given in a problem are fractions. To find the unit rate, the process is exactly the same as with whole numbers: you divide the first quantity by the second quantity.

How to Find a Unit Rate with Fractions

When you divide a fraction by a fraction, you are evaluating a complex fraction. The rule for dividing fractions is simple: multiply by the reciprocal (often remembered as "keep-change-flip").

If you have a ratio of $\frac{a}{b}$ to $\frac{c}{d}$, the unit rate is:

$\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Let's apply this to some real-world examples.

Example 1: Calculating Speed

Problem: You walk $\frac{1}{2}$ mile in $\frac{2}{3}$ hour. What is your speed in miles per hour?

Solution: We want to find the miles per hour, so we divide the miles by the hours.

1. Set up the division: $\frac{1}{2} \div \frac{2}{3}$

2. Rewrite as multiplication by flipping the second fraction: $\frac{1}{2} \times \frac{3}{2}$

3. Multiply straight across (numerators together, denominators together): $\frac{1 \times 3}{2 \times 2} = \frac{3}{4}$

Answer: Your speed is $\frac{3}{4}$ miles per hour.

Example 2: Recipe Ratios

Problem: A recipe uses $\frac{3}{4}$ cup of sugar for $\frac{2}{3}$ cup of flour. How much sugar is needed per cup of flour?

Solution: We need to find sugar per flour, so we divide the amount of sugar by the amount of flour.

1. Set up the division: $\frac{3}{4} \div \frac{2}{3}$

2. Multiply by the reciprocal: $\frac{3}{4} \times \frac{3}{2}$

3. Multiply straight across: $\frac{3 \times 3}{4 \times 2} = \frac{9}{8}$

4. Convert the improper fraction to a mixed number (optional but helpful for recipes): $\frac{9}{8} = 1 \frac{1}{8}$

Answer: You need $1 \frac{1}{8}$ cups of sugar per cup of flour.