# Dividing by 63

At a recent MathsJam, @brownmaths -- who really should have known better -- showed up with a calculator. Dear oh dear.

His excuse was that it was in his teaching satchel, and he sometimes needed it to work out trigonometric functions (the Mathematical Ninja rolled his eyes, but I said fair enough) or "If there's a test and it's out of 63, and I need to work out the percentages..."

"Divide by 7, then by 9" was the obvious response from the gathering, but I wondered if there was a nicer way.

63 is reasonably close to 62.5, which is 500/8. That suggests that multiplying the raw scores by 16 and dividing by 10 should give you a pretty good approximation for the percentage score. (Since $63\times16=1008$, the answer will be 0.8% too high - which you can adjust for.)

For example, it's a tough test, and someone has managed 19/63. $19 \times 16 = 304$, so that's about 30% (which makes sense -- 21/63 is a third). You could then take off 0.8 for every hundred -- subtracting 2.4 gives 301.6, which you turn into 30.16%. In fact, it's $30.\dot 15873\dot 0$, so not bad.

Let's say a better student has 58/63. That's harder to multiply by 16, but not that hard -- double it, it's 116, and $116 \times 8$ is 928 (I did 888 + 40). Now to lose 0.8 for every hundred, which is 7.2, or -- if you feel like showing off, 7.4, as you have nine-and-a-quarter hundreds. That makes it 92.06% -- and again, it's bang on to two decimal places.

* Edited 2015-03-26 for clarity.
* Edited 2016-06-05 to fix LaTeX ## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

### 8 comments on “Dividing by 63”

• ##### Claudio Meller

I think that :
That suggests that multiplying the raw scores by 16 and dividing by 10…
is
That suggests that multiplying the raw scores by 16 and dividing by 1000

• ##### Colin

I’ve folded the percentage calculation into this step, although I could certainly make that clearer. That’s where the extra factor of 100 comes from.

• ##### gmsc

If you know your multiples of 19, you can use leapfrog division ( http://headinside.blogspot.com/2013/02/leapfrog-division.html ) to work this out.

58/63 = 174/189, so leapfrog division will treat this as starting with 174/19.

174 / 19 = 9 remainder 3
39 / 19 = 2 remainder 1
12 / 19 = 0 remainder 12
120 / 19 = 6 remainder 6
66 / 19 = 3 remainder 9
93 / 19 = 4 remainder 17
174 / 19 = we’ve already done this, so we’ll start repeating here.

The answer then, is 0.920634 repeating.

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