Today I am going to improve your minds Nah, this is prompted by a discussion with a colleague yesterday, during the course of which I was utterly shocked to discover that he was ignorant of what I consider to be a basic and fundamental fact in accounting and mathematics. I expressed my disgust with the intellect of the youth of today to my mother last night, and was equally shocked and horrified that she had no idea what I was talking about either.

For the past two weeks I have been working at a very nice client in the city centre. The accountant there is a very nice man, friendly and pleasant to talk to, but the drawback to the job so far this year has been his total failure to provide me with a final set of accounts to review. Seeing as I am being employed for two weeks to review a set of accounts, this has been somewhat hampering my work. By dint of a bit of wheedling, I have managed to acquire several provisional versions of the accounts and review the numbers which I don’t think are going to change, but the more significant and important numbers have been eluding me.

That is, until Wednesday evening, literally as I was just about to walk out of the door, the client finally presents me with a set of accounts I was dead chuffed and went home a happy little auditor, looking forward to a nice relaxing Thursday spent adding them up. When, Thursday dawned, however, my good mood swiftly vanished. The accounts didn’t balance

To an accountant, having a set of accounts which don’t balance is kind of akin to a mechanic having a car they can’t mend or a doctor having a disease they can’t cure. At first you set about the problem in a cheerful sort of way, confident that with intelligence and hard work you will be able to figure out what the problem is and set it right. But, as the hours tick by and you fail to solve it, you become increasingly frustrated and start banging your head against the table until either you accidentally knock yourself out with your stapler, or it’s time to go home.

These particular accounts had a difference of £27,000. I remarked to my colleague that £27k was an excellent sort of difference to have on a set of accounts. He looked at me blankly, unsure whether this was sarcasm or genuine ecstasy, so I repeated the point in an enthusiastic manner. He continued to look at me blankly, but I wasn’t having him on. Anyone have any ideas why £27k is an infinitely better difference than £26k, £28k, or even £29k?!

I hope this is actually self evident, but if not I’ll give you the clue that £27k is divisible by nine…

No?

Okay I’m the only one sad enough to find this exciting, but this is actually a useful piece of information, honest; not just if you are a professional accountant, but even if you’re just trying to sort out your own bank balance or something like that.

If you’re adding something up and the difference between what it is supposed to be and what it actually appears to be is a multiple of nine, then you are in luck, because an error caused by the transposition of two digits is always divisible by nine.

To clarify what I mean by transposition, supposed you are trying to add 100 to 123. You expect the answer to be 223, however when you type it into your calculator you get 232. The difference between what you wanted and what you got is 9, and the reason is that you actually added 100 to 132, having accidentally transposed (or swapped around) two of the digits in 123. Obviously that’s a pretty silly example, but the same holds true in any circumstance.

Why? I’m going to do a proof

Consider the set {a, b e N ¦ a, b < 10}

Now assume x, y e N such that x, y differ by a single transposition of adjacent digits.

x = … a b …
y = ... b a …

WLOG assume a > b

Then a ~ a x 10^p and b ~ b x 10^p-1 where p e N is the place-holder of b in the sequence
Hence ab ~ (a x 10^p)+ (b x 10^p-1)
= 10^p-1 x (10a + b)

Similarly, ba = 10^p-1 x (10b + a)

ab – ba = 10^p-1 x (9a – 9b)

9 | 10^p-1 x (9a – 9b)

QED

Did that make you cry?

What I’m essentially saying in English, is choose two positive whole numbers less than ten and call them a and b. Let’s take 5 and 7 for the sake of argument. Now imagine two bigger numbers x and y which have 5 and 7 as some of their digits. x can be 1752 and y can be 1572. I want a to be bigger than b, so I’m defining a to be 7 and b to be 5.

Now if a is the third digit in x = 1752, a is actually equivalent to 700. That is to say, a = 7 x (10 x 2) where 2 = p. Similarly, b is actually 50, so b = 5 x (10 x 1) where 1 = p – 1.

So, ab = (7 x 10 x 2) + (5 x 10 x 1)
= 10 x (7 x 10 + 5)

ba = 10 x (10 x 5 + 7)

ab – ba = 10 x (9 x 7 – 9 x 5)
= 18

Clearly, nine is a factor of eighteen.

Result~!

If this fact were properly taught in Britain’s accountancy colleges today, the number of auditors committing work-related suicide might half overnight. If you ever find yourself with a numerical difference, before you re-do a day’s work, try dividing it by nine. If your problem turns out to be purely transposition, you can potentially solve it in five minutes Sadly, whilst I found the client’s transposition fairly swiftly, his accounts proved to be riddled with other, terminal mistakes, and so I have had a very, very frustrating day

Tags: accountancy, transposition errors

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