## What you need to know

There are a few unit conversions that you should know by heart. They are listed below.

Length:

$1\text{cm}=10\text{mm}$

$1\text{m}=1,000\text{mm}$

$1\text{m}=100\text{cm}$

$1\text{km}=1,000\text{m}$

Mass:

$1\text{kg}=1,000\text{g}$

$1\text{ tonne}=1,000\text{kg}$

Area:

$1\text{cm}^2=100\text{mm}^2$

$1\text{m}^2=10,000\text{cm}^2$

Volume:

$1\text{cm}^3=1\text{ml}$

$1\text{ litre}=1,000\text{cm}^3=1,000\text{ml}$

$1\text{cm}^3=1,000\text{mm}^3$

$1\text{m}^3=1,000,000\text{cm}^3$

NOTE: For the area and volume conversions, you can save yourself a bunch of memorising with the following observation. We know that

$1\text{m}=100\text{cm}$

Now, if we’re discussing area then ‘$\text{m}$’ becomes ‘$\text{m}^2$’ and ‘$\text{cm}$’ becomes ‘$\text{cm}^2$’. In other words, all the units are squared. As a consequence of this, we must also square the numbers. Doing this, we get

$1\text{m}^2=100^2\text{cm}^2=10,000\text{cm}^2$

This works for any conversion, e.g. it would also work if you had to convert between $\text{km}^2$ and $\text{m}^2$.

Furthermore, it also works with volume! In the volume case, all the units are cubed, so we must also cube the numbers. Again, taking $1\text{m}=100\text{cm}$, we get

$1\text{m}^3=100^3\text{cm}^3=1,000,000\text{cm}^3$

Example: A bath contains 230 litres of water. Find the volume of the bath in cubic metres.

Firstly, we will use the conversion rate to turn 230 litres into centimetres cubed.

$230\text{litres} \equiv 230 \times 1,000=230,000\text{cm}^3$

Next, we have to convert 230,000 centimetres cubed to metres cubed. Observing that

$1\text{m}^3=100^3\text{cm}^3=1,000,000\text{cm}^3$

We see that we’ll have to divide by 1,000,000 to get our answer. So, we get

$230,000\div 1,000,000=0.23\text{m}^3$

***

Often people get mixed up on whether to multiply or divide by the number in question to get your answer. The easiest way to avoid tripping up here is to look at your answer and ask: does this make sense? If I had accidentally multiplied by 1,000,000 in this last example, I would get

$230,000\text{cm}^3=230,000,000,000\text{m}^3$

Now, this is a lot of metres cubed. Does it make sense to have so many metres cubed in bath? Not really. So, we must’ve got the calculation wrong.

***

Metric to Imperial

Metric units are modern units that typically work in terms of multiples of 10, 100, or 1,000. All the units listed above are metric units. Imperial units are the old-fashioned ones that don’t work in tens.

Imperial units include: miles, feet, pounds (for weight, not money), stone, ounces,  and acres.

If you ever have to convert between metric and imperial, the conversion will be given in the question.

Example: Using the conversion rate below. Convert:

a) 12.5 miles to km

b) 128.8 km to miles

Conversion rate: 1 mile = 1.61km

a) We want to convert miles to km. The result, in kilometres, should be bigger than 12.5. This tells us we should multiply by 1.61:

$12.5\text{ miles}\equiv 12.5\times 1.61=20.125\text{km}$

b) We want to convert km to miles. The result, in miles, should be smaller than 128.8. This tells us we should divide by 1.61:

$128.8\text{ km}\equiv 128.8\div 1.61=80\text{ miles}$

## Example Questions

1 tonne is equal to 1,000kg, so we get

$8.5\text{ tonnes}\equiv 8.5\times 1,000=8,500\text{kg}$

First, we will convert the side-length into metres. There are 100cm in 1m, so we get

$41\text{cm}\equiv 41\div 100=0.41\text{m}$

Then, we find the area of the square by multiplying the side-length by itself:

$0.41\times 0.41=0.1681\text{m}^2$

Note: it’s also possible to find the area in centimetres squared and then convert that result to metres squared.

Firstly, we will convert 2,016g in oz by dividing by 28:

$2,016\div 28=72\text{ oz}$

Next, we will convert this value to lbs by dividing by 16:

$72\div 16 = 4.5\text{ lbs}$