Equivalent ratios and simplest form

If you are making orange squash and you mix one part orange to four parts water, then the ratio of orange to water will be \(1:4\) (\({1}\) to \({4}\)).

  • If you use \({1}\) litre of orange, you will use \({4}\) litres of water \((1:4)\).
  • If you use \({2}\) litres of orange, you will use \({8}\) litres of water \((2:8)\).
  • If you use \({10}\) litres of orange, you will use \({40}\) litres of water \((10:40)\).

These ratios are all equivalent:

\[1:4~=~2:8~=~10:40\]

Both sides of the ratio can be multiplied or divided by the same number to give an equivalent ratio.

Question

Write the ratio \(40:28\) in its simplest form.

To work this out, look for a number that will divide into \({40}\) and \({28}\).

\({2}\) divides into both numbers, so \(40:28\) can be written as \(20:14\).

You can divide these by \({2}\), so the simplified ratio is \(10:7\).

No number divides into \({10}\) and \({7}\) exactly, so \(10:7\) is the simplest form of the ratio.