# Dividing by 9 – Secrets of the Mathematical Ninja

Dividing by 9 has always been an awkward one for the mathematical ninja - it ought to be a simple operation, but for some reason it's never stuck.

However, there IS an easy way that involves not much more than adding up and (possibly) using your fingers to track a carry.

Allow me to show you how. Suppose you want to divide 23068 by 9.

You say the first number (2) - this is both the (provisional) first digit of the answer and the running total so far.

Add it to the next digit (2+3): 5 is the provisional next digit, so our answer so far is 25.

Repeat: (5 + 0) 5 is the provisional next digit; our answer so far is 255.

Then we hit a problem: 5 + 6 is 11, which isn't a digit. Instead, we need to add one to our provisional answer (now it's 256) and add the digits of 11 together (to make 2) as our new running total, and next digit. So far, we have 2562.

The last digit is when we get into decimal territory - and it works just the same. Our running total becomes 10 which is too large; we carry one onto the answer (2563) and the first digit after the dot will be (1+0 = 1). In fact, because we're dealing with ninths, EVERY digit after the dot will be a 1, and our answer is 2563.1111…

Although handling the carry is a bit of a brainful at first, with a bit of practise, you can get very good at this - yet another way for the budding mathematical ninja to wow his or her classmates with seemingly impossible arithmetic skills. ## 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.

### 10 comments on “Dividing by 9 – Secrets of the Mathematical Ninja”

• ##### Quinn

Great little article! One thing that should be mentioned is that you don’t “use up” your tens. You have to keep a running total the whole way.
51348/9
5
56
569
569(13)=5703 (the 13 is from 9+4)
5703(21)=5705.1 (the 21 is from 13+8)

• ##### Colin

Good point! I’ll mention that when I finally revise it, which might not be for a while.

• ##### Nathan

I think the article is correct. The 10s get absorbed into digit sums as you go along so the running total is never more than 9 when you move to the next digit.

51348/9
5
56
569
569(13) (the 13 is from 9+4)
=570[1+3] (1 and 3 are the digits of 13)
=5704(4) (…so the new running total is just 4)
5704(12) (the 12 is from 8+4)
=5705[1+2] (1 and 2 are the digits of 12)
=5705.3(3) (…so the new running total is just 3)
5705.333r

• ##### Mathsnooob

So when you divide a number that is divisible by 9 e.g 558 / 9 you get

5
5 + 5 = 10 and 1 + 0 = 1 So
61 (+5)
8 + new running total (1) = 9 so
61.9 (+0)

And you just keep going until you hit zeroes (which won’t happen) so you’re left with .99999r

Is that how it’s supposed to look or should we end up with an actual integer?

Thanks

• ##### Colin

Good spot! It’s not obvious, but it turns out that $61.\dot{9}$ is an integer – it’s $61 \frac{9}{9}$, or 62.

• ##### Mathsnooob

Yeh, that’s awesome. There always seems to be a recurring decimal which I’d never noticed and wasn’t intuitive for me.

Great site, btw. These tricks are way more fun than using a mobile phone.

• ##### Colin

Thanks!

• ##### Colin

I get 26.9999…, which is the correct answer 😉

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