You have converted units with ratios before. In algebra and science you often convert rates, like miles per hour into feet per second, where two units change at once. One tool handles all of it: multiplying by unit fractions.
Why it works
Since $12$ inches and $1$ foot are the same length, the fraction $\frac{12\text{ in}}{1\text{ ft}}$ equals $1$. Multiplying by $1$ never changes an amount; it only changes how the amount is written. Units also behave like factors: a unit on top cancels the same unit on the bottom, just as $\frac{3}{3} = 1$. So you choose each unit fraction so that the unit you want to remove cancels.
Key idea
Multiply by unit fractions equal to $1$, placing each old unit opposite where it already sits so it cancels. Check that only the units you want are left.
Worked examples
Worked example: Two units at once
Convert $60$ miles per hour to feet per second.
- Write the rate as a fraction: $\frac{60\text{ mi}}{1\text{ h}}$.
- Miles are on top, so use $\frac{5280\text{ ft}}{1\text{ mi}}$ (miles on the bottom). Hours are on the bottom, so use $\frac{1\text{ h}}{3600\text{ s}}$ (hours on top).
- $\frac{60\text{ mi}}{1\text{ h}} \cdot \frac{5280\text{ ft}}{1\text{ mi}} \cdot \frac{1\text{ h}}{3600\text{ s}}$. Miles and hours cancel.
- $\frac{60 \cdot 5280}{3600} = \frac{316800}{3600} = 88$.
Answer: $88$ feet per second.
Worked example: Square units
Convert $3$ square yards to square feet.
- A square yard is a square $3$ ft by $3$ ft, so it holds $3 \cdot 3 = 9$ square feet.
- With unit fractions: $3\text{ yd}^2 \cdot \frac{3\text{ ft}}{1\text{ yd}} \cdot \frac{3\text{ ft}}{1\text{ yd}}$. Both yards cancel.
- $3 \cdot 9 = 27$.
Answer: $27$ square feet.
Common mistake
Multiplying by a conversion fraction upside down, such as $\frac{1\text{ mi}}{5280\text{ ft}}$ when you start in miles. The units then do not cancel: you would get $\frac{\text{mi}^2}{\text{ft}}$. Check two things: the units left over are the ones you want, and the size makes sense (a smaller unit needs a bigger number).