# 99p Shop

January 6, 2011 Leave a comment

Here is a problem from “*SYMmetry Plus+*” which is the magazine produced by the Mathematical Association for young mathematicians. This is from the Summer 2005 edition (#27).

*The supermarket Tesbury’s prices all its items at so many pounds and 99 pence. If the total bill comes to £35.84 is it possible to say how many items have been bought? What if the total were £135.84 or £47.12?*

To be clear, this means that every item is either 99p, £1.99, £2.99, etc. It’s a very simple to solve problem, but it is kind of still interesting I suppose. So let us consider the problem in general. Let us say the cost price overall pill is *P* pounds and *p* pence (i.e. pence and ). That we have:

For some integers . Reducing mod 100 we have that:

Of course, the inverse of 99 is again 99 (as it equals -1) and so we have that . It hence the follows that:

(where *p* is the number of pence spent (ignoring the pounds)*, **N* is the number of items bought and *k* is some integer.)

Take the original example, where *p = 84*. Then . By choice of *k*, this is equal to . It follows that as if there were 116 items the cost would be at least 99×116 = £114.84. Given that it is less than this at £35.84, we be sure that there are only 16 items. This also answers the second part – you cannot tell, there could be 16 or 116 items in your basket.

As for the final part, that means there is items. However, if there 88 items that must cost at least £87.12, and so there is no way that this total could be achieved.