Unit Rates with Fractions

A rate compares two quantities measured in different units, like miles per hour or dollars per pound. Rates with fractions happen whenever one or both of those quantities is a fraction instead of a whole number, for example traveling \(\frac{3}{4}\) of a mile in \(\frac{1}{2}\) of an hour. These often appear as complex fractions, where a fraction sits on top of another fraction, such as \(\dfrac{3/4}{1/2}\).

The goal is almost always the same: turn the rate into a unit rate, meaning the second quantity is reduced to 1. That tells you exactly how much of the first quantity corresponds to a single unit of the second, such as "miles per one hour."

Any rate of the form "\(a\) per \(b\)" can be written as the fraction \(\dfrac{a}{b}\), which really means \(a \div b\). When \(a\) and \(b\) are themselves fractions, you get a complex fraction. To simplify it, use the rule for dividing fractions: multiply by the reciprocal of the bottom fraction.

\( \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} \)

To find a unit rate with fractions, follow the same steps every time:

1. Write the rate as a fraction, with the quantity you want "per" in the denominator.
2. Rewrite the fraction as a division problem between the two fractions.
3. Flip the second fraction (the divisor) to get its reciprocal, and change division to multiplication.
4. Cancel common factors where possible, then multiply the numerators and denominators.
5. Simplify the result to lowest terms or a decimal, so it reads as "amount per 1 unit."

A recipe uses \(\frac{2}{3}\) cup of flour for every \(\frac{1}{4}\) batch of cookies. How much flour is needed per whole batch?

Set up the rate: \(\dfrac{2/3}{1/4}\). Rewrite as division and multiply by the reciprocal:

\( \dfrac{2}{3} \div \dfrac{1}{4} = \dfrac{2}{3} \times \dfrac{4}{1} = \dfrac{8}{3} \)

So the unit rate is \(\dfrac{8}{3}\) cups, or \(2\frac{2}{3}\) cups of flour per batch.

A car uses \(1\frac{1}{2}\) gallons of gas to travel \(\frac{3}{5}\) of a trip. What is the unit rate of gallons per whole trip?

Convert the mixed number first: \(1\frac{1}{2} = \dfrac{3}{2}\). The rate becomes \(\dfrac{3/2}{3/5}\).

\( \dfrac{3}{2} \div \dfrac{3}{5} = \dfrac{3}{2} \times \dfrac{5}{3} = \dfrac{15}{6} = \dfrac{5}{2} = 2.5 \)

The car uses 2.5 gallons per whole trip. Notice how cross-simplifying the 3's before multiplying kept the numbers small and made the final simplification quicker.

Rates with fractions connect closely to ratios and to proportions, since a unit rate is really a special ratio where the second term equals 1. Once you can confidently divide fractions and simplify complex fractions, comparing prices, speeds, or mixture strengths that involve fractional amounts becomes far more manageable.

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