99p Shop

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. 100P + p pence and 0 \le p < 100). That we have:

100P + p = \sum_{i=1}^N (100a_i + 99)

For some integers a_i. Reducing mod 100 we have that:

p = \sum_{i=1}^N 99 = 99N \pmod{100}

Of course, the inverse of 99 is again 99 (as it equals -1) and so we have that 99p = N \pmod{100}. It hence the follows that:

N = 99p + 100k

(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 N = 8316 + 100k. By choice of k, this is equal to N = 16 + 100k'. It follows that k' = 0 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 N = 88+ 100k 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.

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