Besides coming up with the equations to solve in word problems, another task is often to convert units of measurement. For example, what is the area of a rectangle 3 feet long and 10 centimetres wide. The area of a rectangle is the two dimensions multiplied together. The problem here is that one dimension is in feet and the other in centimetres. The units of area for this problem should be either feet-squared (ft²) or centimetres-squared (cm²).

So you look up the conversion between feet and centimetres and see that there are 30.48 centimetres in a foot. What do I do with this number?

So if the area is to be expressed in cm², you need to convert the 3 feet to centimetres. Do I multiply or divide the 3 feet by 30.48? One way to determine this is to think of the relative sizes of the units. A centimetre is smaller than a foot, so you would expect a dimension in feet to be a bigger number when expressed in centimetres. That tells you to multiply and you would get 3 ft × 30.48 = 91.44 cm and that is correct.

Conversely, if you wanted to convert the centimetres to feet, since feet are larger than centimetres, you would divide the centimetres dimension: 10 cm ÷ 30.48 = 0.33 ft (rounded to 2 decimal places).

Another way to determine what to do is to look at the units themselves. Units in numbers behave just like variables in that they can be added, subtracted, multiplied, divided, and cancelled using the same rules as with variables. For example, if I walked 2 km, stopped, then walked 500 m, how far did I walk? Well you just can’t add the numbers together as 502 would be in what units? 2 km cannot be directly added to 500 m just as 2x + 500y can’t be added because the variables are different. But if I convert one of the units to the other so that the units are the same, then I can add the numbers. 500 m is 0.5 km so: 2 km + 0.5 km = 2.5 km. This works for the same reason that 2x + 0.5x = 2.5x works.

Now let’s look at the ft/cm conversions by looking at the units. The conversion number 30.48 has units as well. There are 30.48 cm per ft or 30.48 cm/ft. The units part can be seen as a fraction with cm in the numerator and ft in the denominator. So if we want to convert ft to cm, we want to form an expression so that the “ft” part cancels just like a variable would:

${3}\hspace{0.33em}\rlap{-}{\mathrm{ft}}\hspace{0.33em}\times\hspace{0.33em}{30}{.}{48}\frac{\mathrm{cm}}{\rlap{-}{\mathrm{ft}}}\hspace{0.33em}{=}\hspace{0.33em}{91}{.}{48}\hspace{0.33em}{\mathrm{cm}}$

If we want to convert the cm to ft, then we would want to cancel the cm part. The conversion number can also be interpreted as being 1 ft per 30.48 cm or 1 ft/30.48 cm. Multiplying by this fraction, effectively divides the 10 cm by 30.48 and the “cm” units cancel:

${10}\hspace{0.33em}\rlap{-}{\mathrm{c}}\rlap{-}{\mathrm{m}}\hspace{0.33em}\times\hspace{0.33em}\frac{{1}\hspace{0.33em}{\mathrm{ft}}}{{30}{.}{48}\hspace{0.33em}\rlap{-}{\mathrm{c}}\rlap{-}{\mathrm{m}}}\hspace{0.33em}{=}\hspace{0.33em}{0}{.}{33}\hspace{0.33em}{\mathrm{ft}}$

So now, the area of the rectangle can be found:

91.44 cm × 10 cm = 914.4 cm × cm = 914.4 cm²

or

3 ft × 0.33 ft = 0.99 ft × ft = 0.99 ft²

This is why area units are “squared”.