Dear Uncle Colin

I have a percentages problem: I’m told that in an election, 95.74% of the electorate voted for the winning side. What is the minimum possible size of the electorate?

- Percentages Often Lack Logic

Hi, POLL, and thank you for your message!

There are two possible answers to this, depending on whether the reported percentage is exact, or if it has been rounded.

It’s exact!

If it’s exact, there’s no real difficulty: 95.74% = $\frac{9574}{10000}$, which simplifies to $\frac{4787}{5000}$ - and a minimum electorate of 5000 people.

That’s fine, but it’s a bit dull, don’t you think? How about if we just know that between 95.735% and 95.745% of the population voted for the winners?

It’s rounded!

My approach here was to think about the losing side, who got 4.26% of the vote (to two decimal places). What fractions round to 0.0426?

One way to approach this is to look at the reciprocal, which is $\frac{1}{0.0426}\approx 23.474$ - which looks fairly close to 23.5, or $\frac{47}{2}$. In fact, $\frac{2}{47}\approx 0.0426$, and $\frac{45}{47}\approx 95.74$%, correct to two decimal places.

I’m on the fence about whether this is a good question - it should certainly be stated more clearly. Unless you’re studying continued fractions, it does depend on spotting the fraction, which can look a bit like magic.

Hope that helps!

- Uncle Colin