How much is every other?

On a recent math test for Swedish 7th graders, one question asked: How many percent is every tenth? (Hur många procent är var tionde?) You’d think the answer is a straightforward 10%, but is it? Several students have had difficulties with that particular question. Why?

In order to arrive at 10% you have to assume infinity. The question requires you to imagine a situation where you can pick every 10th apple (for instance) out of an infinite amount of apples. Hence it requires a certain level of abstraction that isn’t necessarily intuitive, because in reality no one has an infinite amount of apples to chose from.

In fact, the expression every Nth can correspond to a whole number of varying percentages unless you have been taught to assume infinity. For instance, every 4th equals 25% only when the total amount is either four, eight, twelve, any other number divisible by four, or else if the total is (the hypothetical) infinity. If the total is, say, five, then picking every 4th apple may only give you 20%. That is, { 1, 2, 3, 4=pick, 1 }, leaving you with four unpicked apples, namely, the first three and the one remaining after the one you picked. If the total is seven, every 4th equals a mere 14.28%, i.e. { 1, 2, 3, 4=pick, 1, 2, 3 }, or one out of seven.

If the total is:	1	2	3	4	5	6	7	8	9	10	11	…
then ‘every 4th’ equals:	0%	0%	0%	25%	20%	16.7%	14.3%	25%	22%	20%	18.2%	…

An important point here is that every Nth is not the same as a Nth. While a 4th always equals 25%, irrespective of the assumed total, every 4th does not, simply because every Nth is unit-based, meaning it counts only whole units. That is, you don’t cut up any "remainders" as you would if you were to chose a 4th of all apples.

I don’t have any statistics showing how often students experience difficulties with this seemingly uncomplicated question, but I know that at least some do. This is undoubtedly a tricky question, if not a trick question. Answering it in the expected way is in any case a cultural feat (i.e. you have to be taught to assume the abstract concept of infinity) just as much as it is a logical or mathematical one.