FHSST Biology Info: How to Change Units

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How to Change Units-- the "Multiply by 1" Technique

Firstly you obviously need some relationship between the two units that you wish to convert between. Let us demonstrate with a simple example. We will consider the case of converting millimetres (mm) to metres (m)—the SI unit of length. We know that there are 1000 mm in 1 m which we can write as

$\begin{array}{c}1000\text{ mm}=1\text{ m}\end{array}$.

Now multiplying both sides by

$\frac{1}{1000\text{ mm}}$

we get

$\begin{array}{c}\frac{1}{1000\text{ mm}}1000\text{ mm}=\frac{1}{1000\text{ mm}}1\text{ m}\end{array}$,

which simply gives us

$\begin{array}{c}1=\frac{1\text{ m}}{1000\text{ mm}}\end{array}$.

This is the conversion ratio from millimetres to metres. You can derive any conversion ratio in this way from a known relationship between two units. Let's use the conversion ratio we have just derived in an example:

Question: Express 3800 mm in metres.

Answer:

$\begin{array}{ccc}3800\text{ mm}& =& 3800\text{ mm}\times 1\\ & =& 3800\text{ mm}\times \frac{1\text{ m}}{1000\text{ mm}}\\ & =& 3.8\text{ m}\end{array}$.

Note that we wrote every unit in each step of the calculation. By writing them in and cancelling them properly, we can check that we have the right units when we are finished. We started with mm and multiplied by $\frac{\text{m}}{\text{mm}}$

This cancelled the mm leaving us with just m—the SI unit we wanted to end up with! If we wished to do the reverse and convert metres to millimetres, then we would need a conversion ratio with millimetres on the top and metres on the bottom.

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