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Close but Wrong!

Just because an answer seems obvious doesn't mean it's right


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 was 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
Answer

Correct
Answer

If the price with VAT is £120 what is
the price without VAT?

£96
take off 20%

£100
divide by 1.2

If your investment of £100 grows by 4% each year,
what’s it worth after five years?

£120
£100 x 1.2

£121.67
£100 x 1.04^5

If you drive to Brighton at 20mph and drive back at 30mph,
what's your average speed over the whole journey?

25mph
( 20+30)/2

24mph
2/ ( 1/20 + 1/30)

If a bat and ball cost £110, and the bat costs £100
more than the ball, what’s the price of the ball?

£10
£110 - £100

£5
£105 - £100

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, it's easier to see the dramatic effects 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.