Recurring 10%-off problem: solution

Last week I posed this problem:

If a shop reduces the price of something by 10% every day – so every day after the first 10% reduction there is an 'extra 10% off' – assuming no-one buys it, how many days would it take before it costs nothing at all?

The problem provides a good example of the difference between theoretical mathematics and mathematics in the real world.

I assume that the 'extra 10% off' is applied each time to the reduced price, since this is what shops actually do. So each time that 10% is deducted the price becomes 90% of what it was, which is the previous price multiplied by 0.9.

So, theoretically, if the starting price is £P the prices day after day (in pounds) go like this:

P, 0.9P, (0.9)²P, (0.9)³P, (0.9)⁴P, (0.9)⁵P, and so on.

If the starting price is £100, the price goes down like this:

£100, £90, £81, £72.90, £65.61, £59.049, £53.1441 ...

This sequence will never get to zero! It will eventually get smaller than any positive number you care to write down, but it will never reach zero. For example, after 94 reductions the price would be less than half of a penny. After 110 reductions it would be less than a tenth of a penny.

However, in the real world, prices have to be rounded to a whole number of pence. This gives us three possibilities, depending on how the rounding is applied.

(1) If we assume that the price is always rounded up to the next whole number of pence, then once the price gets down to 9p, it stays at that price for ever. A price of 9p reduced by 10% is 8.1p, but this is rounded up back to 9p.

(2) If we assume that the price is always rounded to the nearest penny (with 0.5 rounded up), then once the price gets down to 5p then it stays at that price for ever. A price of 5p reduced by 10% is 4.5p, which is rounded back up to 5p.

(3) However ... if the shop always rounds the price down to the next whole number of pence, then it will eventually cost nothing! For example, if the starting price is £100, then after 62 reductions it has got down to 10p. From here it goes down 1p a day, reaching zero on day 73.