5.1 Units in calculations

Suppose you walk a distance of 100 metres in a time of 50 seconds. Your average speed is given by dividing the distance by the time taken; in this case, 100 divided by 50 gives 2. But 2 what? Again, the answer needs to be quoted with units in order to be meaningful. Fortunately, the units of the answer can be found very easily; we divided the distance by the time taken and we can treat the units in a similar fashion. The units of the answer are metres divided by seconds, frequently written as m/s and said as ‘metres per second’. Rather than treating the numbers and units separately, it is better to include units as part of the calculation itself. So, in the example we have just been considering:

average speed = = 2 , more usually written as 2 m/s.

In the same way, if we multiply two lengths (measured in metres) together to give an area (see Section 6), it is obvious from the calculation that the units of the answer are m m, more usually written as m² and said as ‘metres squared’ or ‘square metres’. For example:

4 m 3 m = 12 m m = 12 m²

Finally, suppose that we want to find out what fraction of a particular garden fertiliser is nitrogen, and that we know a 3 000 gram bag of the fertiliser contains 210 grams of nitrogen. To find the fraction of the total mass that is nitrogen, we need to divide 210 grams by 3 000 grams:

Notice that in the first step above we cancelled the unit of grams, since this was the same on the top and the bottom of the fraction. Units can be cancelled in the same way as numbers, and in this case the correct final answer is a fraction with no units. (i.e. 7%) of the mass of the fertiliser is nitrogen.