# Ask Uncle Colin: Dividing by decimals

Dear Uncle Colin, A question came up asking me to work out $4.5 + 1.5k = 18-3k$. Obviously, I can rearrange that to get $4.5k = 13.5$ - but how can I divide by 4.5 without a calculator? - Calculating A Number Turning On Ratio

Hi CANTOR, and thank for the question! There are several ways to approach this.

My favoured one uses the “fix the ugly thing” strategy: you’ve got an unpleasant decimal; how about multiplying both sides by the same thing to make it not a decimal? Naively, you might pick 10, getting $45k = 135$; but oh, look! there’s a factor of 5 there, so $9k = 27$ and $k=3$.

More straightforwardly, you might have picked 2, getting $9k = 27$ directly.

A second method is to turn everything into fractions: $\frac{9}{2} k = \frac{27}{2}$, so $k = \frac{27}{2} \div \frac{9}{2} = 3$. (The halves, in a sense, cancel.)

Option 3 is a bit piratical: you can just count up in 4.5s until you get to 13.5 (you can estimate that it’s about 3, because 3 × 4 = 12 and 3 × 5 =15) and - lo and behold, the 4.5 times table goes 4.5, 9, 13.5. This is not advised as a general rule, because if your answer isn’t a whole number, you run into problems.

Long division is possible, but not especially useful for this method (you need to work out your 4.5 times table to do it, and if you don’t spot that the answer is 3 at that stage, you’re doing it wrong.)

I hope that helps!

Uncle Colin