Close but Wrong!
Just because an answer seems obvious doesn't mean it's right
08 May 2025
Recently, a senior professional person (it was a lawyer) asked me a question about VAT. "Sorry to sound dim, but if a product costs £120 including VAT and I want to work out the price without VAT, why can't I just take off 20%?" (to give £96). I explained that this isn't how it works, and that you have to divide £120 by 1.2 instead, meaning the price before VAT was £100. Her response: "Yes, but WHY?"
I could see her point. When you add VAT to a price it seems like you just add 20%, yes? So when subtracting VAT, why not just subtract 20%? The answer: because sometimes our mathematical intuition when solving a problem is wrong. The "adding" we're doing here isn't the same as when you add, say, £25 to the bill.
Some of the most counter-intuitive maths problems are ones where the wrong answer is very close to the correct one. I think it’s often this closeness that makes the problems confusing. If the VAT rate is set at 100% instead of 20%, then the flaw in the reasoning becomes more obvious. Add 100% to £100 and you get £200. Take off 100% and the price is zero. (And as for Trump's tariffs of 145%....)
Here's a table of examples where the closeness of the wrong answer to the correct one increases the confusion:
| Problem | Wrong | Correct |
| If the price including VAT is £120 what is | £96 | £100 |
| If your investment of £100 grows by 4% each year, | £120 | £121.67 |
| If you drive to Brighton at 20mph and drive back at 30mph, | 25mph | 24mph |
| If a bat and ball cost £110, and the bat costs £100 | £10 | £5 |
As with the VAT example earlier, sometimes the way to understand these mistakes is to use extreme examples to reveal the flaw in reasoning. If your investment grows by 100% each year instead of 4%, it's easier to see the dramatic effect of compound interest (£100, £200, £400, £800...). If you drive to Brighton at 20mph and return at 0mph your average speed isn’t 10mph, it’s zero – you would never get home! And if the bat costs £0 more than the ball, you wouldn't say that the ball therefore costs £110.*
BONUS PUZZLES
Here are two classic puzzles where incorrect reasoning gets really close to what seems to be the 'right' answer.
- Three friends share a meal for £25. They give the waiter £30, he gives them back £5, they leave a £2 tip. The friends have now paid £9 each, the waiter has £2 as a tip. £27+£2=£29. What happened to the missing pound? Wrong answer: *Somebody stole it!* Right answer: They paid 3x£9=27, of which £25 went to the restaurant and £2 to the waiter. There is no missing pound.
- A sheikh leaves 17 camels to his three children, 1/2 to the eldest, 1/3 to the second and 1/9 to the third. When he dies the children realise they can’t divide up the camels. A passing traveller gives them his camel, so there are now 18. The eldest takes 9, the next takes 6 and the third takes 2, which makes 17. “So you didn’t need my camel after all,” says the traveller and he takes it back. Wrong answer: *What magic is this?* Right answer: the problem was that the sheikh's fractions didn't add up to 1, the flaw would have been much more obvious if the fractions had been, say, 1/3, 1/6 and 1/9.
* thanks to Colin Beveridge for suggesting this extreme example
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