In Grade 6 you found unit rates like "120 miles in 3 hours is 40 miles per hour." A unit rate answers one question: how much for 1? Now the amounts can be fractions. Walking $\frac{1}{2}$ mile in $\frac{1}{4}$ hour is a rate too, and it has a unit rate.
Idea clave
To find "A per 1 B," divide: $\text{A} \div \text{B}$. The quantity after the word "per" goes in the divisor. With fractions, divide the way you learned in dividing fractions: multiply by the reciprocal.
Por qué funciona
In $\frac{1}{4}$ hour you walk $\frac{1}{2}$ mile. An hour holds 4 quarter hours, so in 1 hour you walk 4 times as far: $4 \times \frac{1}{2} = 2$ miles. Multiplying both amounts by 4 keeps the rate the same and turns $\frac{1}{4}$ hour into 1 hour. That is exactly what $\frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times 4 = 2$ does.
Worked examples
Ejemplo resuelto: A walking speed
Ana walks $\frac{1}{2}$ mile in $\frac{1}{4}$ hour. What is her speed in miles per hour?
- "Miles per hour" means miles for 1 hour, so hours go in the divisor: $\frac{1}{2} \div \frac{1}{4}$.
- Multiply by the reciprocal of $\frac{1}{4}$: $\frac{1}{2} \times \frac{4}{1} = \frac{4}{2} = 2$.
- Check: at 2 miles per hour, a quarter hour covers a quarter of 2 miles, which is $\frac{1}{2}$ mile ✓.
Respuesta: $2$ miles per hour
Ejemplo resuelto: A recipe rate
A recipe uses $\frac{3}{4}$ cup of sugar for every $\frac{1}{2}$ cup of flour. How much sugar is that per cup of flour?
- Sugar per 1 cup of flour: $\frac{3}{4} \div \frac{1}{2}$.
- $\frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = 1\frac{1}{2}$.
- So each cup of flour goes with $1\frac{1}{2}$ cups of sugar. That makes sense: 1 cup of flour is two half cups, so it needs twice $\frac{3}{4}$.
Respuesta: $1\frac{1}{2}$ cups of sugar per cup of flour
Ejemplo resuelto: Mixed numbers
A bike rider goes $2\frac{1}{2}$ miles in $\frac{3}{4}$ hour. What is the speed in miles per hour?
- Write the mixed number as a fraction: $2\frac{1}{2} = \frac{5}{2}$.
- Divide by the hours: $$\frac{5}{2} \div \frac{3}{4} = \frac{5}{2} \times \frac{4}{3} = \frac{20}{6} = \frac{10}{3}$$
- $\frac{10}{3} = 3\frac{1}{3}$. Estimate to check: in a bit less than 1 hour the rider went $2\frac{1}{2}$ miles, so in a full hour a bit more than that ✓.
Respuesta: $3\frac{1}{3}$ miles per hour
Error común
Dividing the other way: $\frac{1}{4} \div \frac{1}{2} = \frac{1}{2}$. That is a real rate, but it is hours per mile (half an hour for each mile), not miles per hour. Before you divide, say the units out loud: "miles per hour" means miles $\div$ hours. A quick check: the two unit rates are reciprocals, $2$ and $\frac{1}{2}$.
You may see a rate written as a fraction of fractions, like $\dfrac{\frac{1}{2}}{\frac{1}{4}}$. It means the same division: $\frac{1}{2} \div \frac{1}{4}$.