Before doing a calculation, ask yourself what you are calculating

By MUNGAI KIHANYA

The Sunday Nation

Nairobi,

05 June 2016

Haron Wachira came across a puzzle on the internet and he asked me to unravel it for him. Here it is: suppose you have Sh500 and you spent Sh200; your balance will be Sh300. Then some time later you spend Sh150; you are now left with Sh150. From this you use Sh90 and now you have Sh60 remaining. Finally you spend the Sh60 and you now have nothing.

Now, we do the accounts: the total of all the expenses is Sh200 + Sh150 + Sh90 + Sh60 = Sh500. However, when we add up all the balances we get: Sh300 + Sh150 + Sh60 + Sh0 = Sh510. So the question is: where did the Sh10 come from?

This riddle takes advantage of the fact that many people don’t stop to think about what they are calculating – they just blindly follow instructions to add numbers without worrying about what those figures represent. You are told to add the expenses and then the balances and compare the two sums. Of course the answers are not the same!

My question to Haron when he sent me the puzzle was: “Why are you expecting the two sums to be equal?” Think about it this way; if each expenditure was done on a different day, would you add up the daily balances? If yes, what would be your reason for doing that? What information would you expect to get?

A second example: when you get you bank statement, do you add up the balances remaining after each transaction? Of course not; that would be ridiculous, wouldn’t it?

The same logic applies in the puzzle: the balances remaining after each expense should NEVER be added. If you are still not convinced; suppose you spent the whole amount in one instance, your balance would be zero immediately. In that case, your total expenses would be Sh500 and your balance Sh0. Would you then ask why the two are not the same? Obviously not!

I invited Haron to find out what happens when if he spent one shilling at a time. So the balances will be Sh499, Sh498, Sh497…..Sh0. When you them all up, you get Sh124,750; and now the question is: where did Sh124,250 come from?

Incidentally, there is an elegant mathematical trick for adding up a consecutive series of numbers. Suppose you wish to add 5 + 4 + 3 + 2 + 1: You start by doing the sum twice, that is, 5 + 4 + 3 + 2 + 1 + 5 + 4 + 3 + 2 + 1. But don’t start punching the numbers into the calculator yet!

Rearrange them a little: (5 +1) + (4 + 2) + (3 + 3) + (2 + 4) + (1 + 5). You immediately notice that each sum in brackets comes to 6. Secondly, there are 5 pairs of sums in brackets. So the total is 6 x 5 = 30

However, we are only interested in half of this sum (remember, we summed twice), so the result of 5 + 4 + 3 + 2 + 1 is 30 divided by 2; that is, 15. Now check with your calculator.

Of course it seems like a roundabout way to add five small numbers; but when you have 499 + 498 + 497 + 496 + … + 1, this trick comes in very handy. You simply do (499 + 1) x 499 and the divide the result by two; that is 500x499/2 = 124,750. You can check it if you have the time!