# 5 + 5 + 5 – 5 + 5 + 5 – 5 + 5 x 0 = ?

I quite like puzzles and “brain teasers”. Although I would never describe myself as a mathematician, I do like playing with numbers and I am quite good at mental arithmetic [even though I am rarely far from a calculator, having bought my first one in 1973]. So I was interested in a recent puzzle that has been doing the rounds on Facebook.

The puzzle is expressed as **5 + 5 + 5 – 5 + 5 + 5 – 5 + 5 x 0 = ?** The options for answers were 0, 15, 20 and 40. What answer did you go for? …

When I last looked at this quiz, the number of people who had selected each answer was:

- 0: 2,353,899
- 15: 1,026,717
- 20: 542,206
- 40: 90,146

I will look at each possible answer in turn:

- 0 – This was most popular and is the answer that you arrive at if you simply do the calculation from left to right, one operator at a time – the way you would on a calculator. It is not the correct answer, as it ignores a fundamental rule of mathematics: the precedence of operators. The rule says that, in the absence of parentheses, you simply do multiplication and division first, working from left to right, the do the same for addition and subtraction.
- 15 – This is the correct answer, as it apply the precedence rule correctly.
- 20 – The only way that I can see to arrive at this answer is to ignore the “x 0” entirely. How might that be valid?
- 40 – This is worse. The only way I can get this answer is to ignore all the operators and add up the digits.

Here is the sequence of steps that lead to the correct answer:

- 5 + 5 + 5 – 5 + 5 + 5 – 5 + 5 x 0
- 5 + 5 + 5 – 5 + 5 + 5 – 5 + 0
- 10 + 5 – 5 + 5 + 5 – 5 + 0
- 15 – 5 + 5 + 5 – 5 + 0
- 10 + 5 + 5 – 5 + 0
- 15 + 5 – 5 + 0
- 20 – 5 + 0
- 15 + 0
- 15

If you want an easy way to check, enter the following formula into Excel: **=5+5+5-5+5+5-5+5*0**

Why did so many people, a large proportion of whom will have had a reasonable education, get it wrong?

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Colin – I’ve seen this “puzzle” for a couple of years now. Like you (and as a software engineer) I knew that the “correct” answer (as defined by the puzzle’s deviser) would depend on whether operator precedence was being applied, or simply left-to-right operator application. I suspect that the deviser was simply trying to be clever – anything multiplied by zero is zero, so the only bit you need to see is the final “x 0”. All this goes to show – as far as I am concerned – that the deviser isn’t as clever as he/she thinks after all! Mike

You are right Mike. It is essentially a bit of an ego trip for the deviser, who, to the best of my knowledge, has never announced the “correct” answer. I guess it is essentially the difference between mathematics and “sums” …

I would re-phrase your final question:

Why do less than 26% of Facebook users who took this quiz appear to have had a reasonable education?

Now there’s an interesting and controversial topic!

Perhaps a better wrong answer than 20 would have been -5, which is what you get if you take “+” as having higher precedence than “-” (rather than being equal precedence). If you take “-” as having higher precedence, you get the right answer — and you also get the right answer if you give everything equal precedence and apply the right-to-left.

Peter: fair point.

Brooks: interesting possibilities.