The true mathematical ninja gets immense satisfaction from one thing above all others: showing off.

And so, when you can eyeball a messy fraction and say ‘that’s about…’ and get it right to two decimal places or so… well, you earn the baffled respect of everyone around you, and gain a reputation as ‘being good at maths.’

Having a reputation for being good at maths is one of the most powerful things you can have if you want to actually be good at maths.

So, how can you see a fraction such as - say - $\frac{7}{16}$ and be able to say, confidently, “that’s about 0.42, or a bit higher, let’s say 0.44”? (It’s 0.4375). Let me explain.

The numbers 96, 98, 99, 102 and 104 are Very Useful Numbers when it comes to ninja maths, because they’re composite numbers near 100. If you can ‘cancel up’ a fraction to one of those numbers, you can easily make a solid guess at the decimal value, and adjust it as needed.

That’s exactly what I did with $\frac{7}{16}$. I know that 96 is $16 \times 6$, so $\frac{7}{16} = \frac{42}{96}$. As a first guess, $\frac{42}{100} = 0.42$ isn’t too shabby a guess. However, since I know that 96 is 4% below 100, my guess is also going to be about 4% low, so I added on 4% of 0.4 - again, not perfectly precise, but good enough - to get 0.436. Knowing that would still be on the low side, I went for 0.44*.

If you know the factors of the composite numbers near 100, you can do this with nearly any simple fraction. I’ll assume you know the obvious ones like 2, 3, 4, 5 and 10, but you can easily make up fractions with bottoms of:

  • 6 - multiply by 16 to get 96
  • 7 - multiply by 14 to get 98
  • 8 - multiply by 12 to get 96
  • 9 - multiply by 11 to get 99
  • 11 - multiply by 9 to get 99
  • 12 - multiply by 8 to get 96
  • 13 - multiply by 8 to get 104
  • 14 - multiply by 7 to get 98
  • 15 - multiply by 7 to get 105
  • 16 - multiply by 6 to get 96
  • 17 - multiply by 6 to get 102

It turns out that 18 is the first number you can’t get within 5% of 100 (but you can get it very close to 200 if you multiply by 11 and apply the same kind of technique).

So there you go: to get a good decimal representation of a nasty fraction, simply make it up to a nicer one near 100 and adjust by however many percent off you are.

* I’d have done better to work out that 4% of .42 is .0168 and add that on, making 0.4368, but I’m not quite that good yet.