## What you need to know

A unit is a standard measurement of a particular quantity. For example, metres and kilometres are both units for measuring distance, and seconds, minutes, and hours are all units for measuring time. You will be expected to be familiar with a bunch of different metric units (we’ll get into non-metric ones in a moment) but they’re all very common in day to day life, which helps. Including the ones mentioned above, you should also be familiar with units of money – pounds, pence – and mass – grams, kilograms, etc.

On top of this, you should be able to work with compound units. These are units made up of more than one standard unit (like the ones seen above). For example, one unit of speed is “metres per second” (written like $\text{m/s}$ or $\text{ms}^{-1}$), and one unit for density is “grams per cubic metre” (written like $\text{g/m}^3$ or $\text{gm}^{-3}$).

As mentioned, you come across these units a lot without even thinking about it, so they’ll usually end up sticking in your head whether you like it or not. The main thing to consider is that you will be expected to convert between units, namely scaling units up and down. Fortunately, if you can remember a few prefixes then it makes life a lot easier. Let’s consider units of length:

$1\text{ metre } = 1,000\textbf{ milli}\text{metres}$

$1\text{ metre} = 100\textbf{ centi}\text{metres}$

$1,000\text{ metres} = 1\textbf{ kilo}\text{metres}$

This means that to convert from metres into millimetres for example, all you need to do is multiply by 1,000 (this is effectively the scale factor). Knowing these will get you far as they apply to other units too, e.g. 1000 grams is a kilogram, 1 litre is 1000 millilitres, and so on. The exceptions to this in the metric system are money – we don’t tend to say there are 100 centipounds in £1, although maybe we should – and time, although I’m sure you all know there are 60 seconds in a minute and 60 minutes in an hour. When it comes to compound units, simply convert each part of it individually.

Example: A car is driving at 50 km per hour. What is the speed of the car in metres per second?

We know that 1km is 1,000m. This implies that 50km is $50 \times 1000 = 50,000$ m. So, the speed of the car is 50,000 metres per hour.

Next, we know that there are 60 seconds in a minute and 60 minutes in an hour, so there is a total of $60 \times 60 = 3,600$ seconds in 1 hour. In other words, it will take this particular car 3,600 seconds to travel 50,000 metres, so the number of metres they travel every second will be $50,000\div 3,600 = 13.9$ (1 dp). Thus, the speed of the car is 13.9 metres per second.

Sometimes it can be a little bit confusing working out whether to multiply or divide by the scale factor during parts of the question. It helps just to think if you’re answer makes sense, e.g. if I had multiplied by 3,600 in that last question, then my answer for the speed of the car would be 180 million metres per second, which clearly is not right. Watch out for these indicators if you’re unsure.

Imperial Units & Conversion Graphs

Imperial units are messy and tricky to remember, like the fact that there are 1,760 yards in a mile. I don’t know who thought that would be a good idea. Fortunately, you don’t have to remember them. That said, you do still have to convert using imperial units, including between metric and imperial units, but you’ll always be given the conversion rate in the question.

Example: A baby is born weighing 4.2kg. What is the mass of the baby in pounds to 3sf? Use the conversion rate $1\text{ pound } = 0.4536\text{ kg}$.

We have to convert a value from kilograms into pounds. That conversion rate tells us that to get from pounds to kilograms, we would need to multiply by 0.4536. Therefore, in order for us to convert kilograms to pounds we must do the opposite and divide by 0.4536.

$\text{Mass of baby } = 4.2 \div 0.4536 = 9.26\text{ pounds (3 sf)}$

We can ask the question: does this answer make sense? Yes, it does, because if 1 pound is equivalent to 0.4536 kg, then the mass of the baby when measured in pounds should be a bigger number than if it were measured in kg, and indeed our answer is bigger than the starting value.

Another way you might be expected to convert between units, typically metric and imperial, is by using a conversion graph. It’ll be a straight-line graph with one unit on the x-axis and another on the y-axis, and the line will be our guide for converting between the two units. Let’s have a look.

Example: Use the conversion graph below to convert 4 miles into kilometres.

To do this, we find 4 miles on the y-axis, draw a horizontal line from that point until it hits the blue line, and then draw a vertical line down the x-axis to find the corresponding value in kilometres.

Using the red line we’ve drawn, we can conclude that according to our conversion graph, 4 miles is equivalent to 6.4km.

Try to be as accurate as you can, as sometimes the line you draw might be in between two squares. This is okay, you just might need an extra decimal place for accuracy.

## Example Questions

We can either take of the 3% at the start at the end (it won’t make a difference); we’ll do it first. A 3% decrease means she will have 97% of her money left over, so

$\text{Money left } = \dfrac{97}{100} \times 500 = \pounds 485$

Now, if £1 is worth $1.56 dollars, this means to convert from pounds to dollars we would need to multiply by 1.56. So, we get $\text{Total holiday spend } = 485 \times 1.56$=$ 756.60.

The conversion rate is between metres and feet, so first we need to convert 2.3 km into metres. We know that 1 km is 1,000 m, so

$2.3\text{ km} = 2.3 \times 1,000 = 2,300\text{ m}$.

Then, if 1 foot is equal to 0.3048 metres, that means we would have to multiply a distance in feet by 0.3048 in order to get the equivalent distance in metres. So, to go from metres to feet, we will have to divide by 0.3048. This means Diana’s walk in feet is

$2,300 \div 0.3048 = 7,546\text{ feet (to nearest whole number)}$.

We have two conversions so will have to firstly draw up from 4 on the x-axis to the line and then across, and secondly draw across from 5 on the y-axis to the line and then down. This will look like:

Looking at the red line first, we can see that it falls about half way between 2.2 litres and 2.4 litres, so to 1 decimal place we can say that 4 pints is equal to 2.3 litres.

Looking at the green line second, we can see that it falls in between 8.6 pints and 8.8 pints but ultimately, it’s closest to 8.8 pints, so to 1 decimal place we can say that 5 litres is equal to 8.8 pints.

## Conversions Teaching Resources

You may be a GCSE Maths tutor in Birmingham or a Maths teacher in York, where ever you are teaching GCSE Maths to whichever students, the conversion questions and revision notes on this page will be of use. From a detailed specification of what students need to know to conversion worksheets and answers. For more fantastic GCSE Maths resources visit the GCSE Maths revision page.