Converting Units with Ratios

A conversion factor is a ratio worth 1

$12$ inches and $1$ foot are the same length. So this ratio equals exactly one:

$$\frac{12 \text{ inches}}{1 \text{ foot}} = 1$$

Multiplying by $1$ never changes an amount. It only changes how that amount is written — which is the whole trick behind every unit conversion.

$$5 \text{ feet} \times \frac{12 \text{ inches}}{1 \text{ foot}} = 60 \text{ inches}$$

Five feet and sixty inches are the same distance. Nothing grew.

Letting the unwanted unit cancel

Every conversion factor can be written two ways up:

$$\frac{12 \text{ inches}}{1 \text{ foot}} \qquad \text{or} \qquad \frac{1 \text{ foot}}{12 \text{ inches}}$$

Both equal $1$, so both are legal. Only one of them is useful, and the units say which.

Put the unit you want to lose on the opposite side from where it already is:

$$5 \;\cancel{\text{feet}} \times \frac{12 \text{ inches}}{1 \;\cancel{\text{foot}}} = 60 \text{ inches}$$

Feet appears once on top and once on the bottom, so it cancels the way a common factor does, and inches is what survives.

Turning the factor the wrong way up gives:

$$5 \text{ feet} \times \frac{1 \text{ foot}}{12 \text{ inches}} = \frac{5}{12} \frac{\text{feet}^2}{\text{inch}}$$

“Feet squared per inch” is not a length, and that is the signal. The units are the check.

Why this method is worth the extra writing

You can convert feet to inches by remembering to multiply by $12$. The trouble starts one step later: does converting inches back to feet multiply or divide? Does a speed in feet per second need the factor on the top or the bottom?

Setting the factor up so the units cancel answers all of those without a separate rule for each. The method is the same every time, and the units tell you when it is right.

That matters most when two conversions happen at once. Changing miles per hour into feet per second means converting a length and a time, in opposite directions, in one expression. A remembered rule about multiplying by $12$ has nothing to say about that. Cancelling units handles it in a single line.

Common conversion factors

From	To	Factor
feet	inches	$12$ inches per foot
yards	feet	$3$ feet per yard
miles	feet	$5280$ feet per mile
pounds	ounces	$16$ ounces per pound
hours	minutes	$60$ minutes per hour
minutes	seconds	$60$ seconds per minute
metres	centimetres	$100$ cm per metre
kilometres	metres	$1000$ m per kilometre

Chaining two conversions

Both factors go in the same line, and each unit cancels in turn:

$$2 \text{ hours} \times \frac{60 \text{ min}}{1 \text{ hour}} \times \frac{60 \text{ sec}}{1 \text{ min}} = 7200 \text{ seconds}$$

Hours cancels against the first factor, minutes cancels against the second, and seconds is left.

Worked examples

Common mistakes

Practice problems