Back to basics

Forgive my arbitrary topic of algebra in today’s post, but I simply couldn’t resist. Back in my student days when I’d do anything for money, tutoring Maths / Science was my staple in the income-earning department, and many times students asked “but why do you solve it like that?” and the answer often had to be “I’ll think about the explanation and come back to you next week.” Because sometimes it’s bloody hard to explain why someone else’s logic is wrong.

And then, last week, in the course of an ordinary work day, a bit of a mathematical row broke out (as it so often does).

Our office manager V has started hosting big dj parties as an additional income earner, together with 4 friends. The 5 of them organize the venue, arrange the dj lineup, the free cocktails and snacks, and market the event to the general public. The venue owner calculates the profit of the party, takes 20% of it, while the remaining 80% gets shared among the 5 organisers.

One of V’s friends (N) called her last week to say that he’d been through the accounts, and had discovered the venue owner had been cheating them. “No he hasn’t! I checked the accounts myself!” V exclaimed. “Oh yes he has! He’s taking more than 20% of the profit!” Now, N is a lawyer – for a big, renowned firm – working in the corporate realm, and it seems he often has to work out VAT amounts for things. His take on the split of profits was as follows:

R18,800 was the total profit of the party (correct)
This should be split into two sums: 20% for the venue and 80% for the 5 organisers (correct)
80% of R18,800 = R15,667 (what the f%#&%*?????)

To which V responded that 20% of R18,800 (calculated by typing ’18,800 – 20%’ on her calculator, or equivalently ‘0.8*18,800) was equal to R15,040.

I overheard their argument on the phone and could not resist the urge to join the battle.

N’s reasoning was that when you calculate a VAT amount for a number, you use the following calculation – for argument’s sake, the number is 100: (100*114)/100 (us normal folk would’ve just said 100*1.14, but hey). The answer to this sum is 114. Now, if you have a number 114, and want to remove the VAT from it, you use the equation (114*100)/114 (again, I’d simply have said 114/1.14, but us simple folk don’t know the fancy equations lawyers do, it seems). So N proposed to do the same thing with the amount of R18,800:

80% of R18,800 is the same as removing 20% of R18,800, i.e. (18,800*100)/120 = R15,667.

The argument went back and forth about how you calculate percentages (at this point we’d moved to Google Talk), and he kept pulling out the argument that if you remove x% from a number y, then try to add x% back to the new number, the answer never gives you y again. [Yes, N, but what the hell does that have to do with this? We’re not working with VAT!]

Our Google chat went something like this:

N: I can’t believe your firm of supposedly smart people does not know how to work out a basic percentage!!!
Ant: I can’t believe you work with financial aspects of companies and can’t do basic arithmetic!
N: Any moment now, you’re going to realize you’re wrong and humiliate yourself!
Ant: Do me a favour – please please PLEASE go and ask your manager to help you with this calculation!
Ant [again, typing furiously]: But be sure you have another job lined up first, he’s gonna kick your ass out of there for your stupidity!
N: Fine, I will! You should ask yours too!
…

Of course, he came back a few minutes later, humbly apologising because he’d “got confused and was working from an interest-based perspective” (what’s that? Do you even know, N?) to which one of our colleagues told us we should ask whether he’d considered nominal vs real terms in his calculation!

That night, I spent some time thinking how to explain why he was wrong, and this is the reasoning: the formula he used to remove 20% is wrong, because that is a formula used to remove a percentage when that same percentage has before been added to a number. In this instance, the base number, R18,800, has not had any percentage added to it initially, so you can’t ‘remove’ anything from it using his VAT equation. In algebraic terms, let’s assume that R18,800 = x.

From his formula to remove 20%: (x x 100)/120 = x/1.2 = 0.83x (which is why he’d thought they’d been cheated by the owner, who had given them only 0.8x instead of what N had calculated).

Anyway, it allowed me to sleep better that night – I hope this has the same effect on you.