DAY 31

Comment

It is tempting to say the answer is 5 weeks, because 5 x 10 % is 50 %, but this would only work if the weekly reduction was always on the original price rather than on the previous week's price.
This means it will take more than 5 weeks for the price to halve.

Answer

After 7 weeks (when the price is set to about 48 % of the original price).

Readers' methods

CA of Amsterdam:
Say the original price is £100.
After one week it is reduced to £90.
It then becomes 0.9 x £90 which is £81.
It then becomes 0.9 x £81 which is £72.90.
It then becomes 0.9 x £72.90 which is £65.61.
It then becomes 0.9 x £65.61 which is about £59.05.
It then becomes 0.9 x £59.05 which is about £53.14.
It then becomes 0.9 x £53.14 which is about £47.83.
So after 7 weeks its price is less than half.

Brian S, Croydon CR0:
Take the price as p, the number of weeks as w then you have:
p*(0.9^w) = 0.5p (ie the price times 90 % to the poer of the number of weeks will equal half the price).
Simplifying gives 0.9^w = 0.5
Taking the log of both sides gives w(log0.9) = long0.5
so w = (log0.5) / (log0.9)
so w = -0.693 / -0.105
so w = 6.6
so the price will become less than half in the 7th week.