The months are drawing in

February was two days shorter than January. "I'm worried", I tweeted, "If this carries on, how long will December 2012 be?"

Another way of looking at this is that February is about 93.5% the length of January, so I asked which would produce a shorter December:
A. losing a fixed 2 days each month; or,
B. each month being 93.5% of the previous.

It's possible, of course, to simply calculate the answer. However, it is possible to come to an answer as to which is shorter without recourse to such a messy technique.

Under B, we know 93.5% of January is two days, the amount by which February is shorter. If March is 93.5% of February we know this decrease must be less than two days because February was shorter than January. And so on. The decrease in A is always 2 days, but the decrease in B is 2 days in the first month and less for later months. Since the overall decrease has been greater, A gives a shorter December.

I suppose there's a niggle that we don't usually allow fractions of days on the calendar, so if you're going to be all 'real world' about it then each month should be rounded and this rounding will occur before the 93.5% is calculated to form the next month. So I suppose we will have to do a messy calculation after all.

Under A, losing 2 days per month for 11 months is 22 days, so December will be 9 days long.

Under B, taking each month to be 93.5% of the previous, and then rounding to the nearest integer in the normal way, I get a sequence for the number of days for each month: 31, 29, 27, 25, 23, 22, 21, 20, 19, 18, 17, 16.

So my December is a full seven days shorter by the 'fixed two days' method.

Did you get 14 or 15 days for December? If you simply take each month to be 93.5% of the previous without rounding, you calculate 0.935^11*31 and get December as 14.8 days. You can round this to 15, or take the whole days to get 14, but this requires nonsensical things like 18.5 days in November to have happened on the way.

I feel as though this could be a nice, silly way in at various levels to either some basic arithmetic, exponential decrease, through to boundedness in geometric sequences. Even, into some discussion about translating mathematics to real world answers, as the quick 0.935^11*31 calculation masks a whole mess of unreality along the way.