## Pen Problem

#### Problem

A girl bought 15 pens costing £1.84. She paid one pence more for each red pen than each blue pen. How many of each kind did she buy and at what price?

#### Solution

Let `a` be the number of blue pens, `b` be the number of red pens and `c` the cost of a blue pen, hence a red pen costs `c` + 1.

`ac` + `b`(`c` + 1) = 184

`ac` + `bc` + `b` = 184

`ac` + `bc` = 184 `b`

`c`(`a` + `b`) = 184 `b`

But `a` + `b` = 15,

15`c` = 184 `b`

Hence 184 `b` must be divisible by 15; as 15 12 = 180, we get `b` = 4,

so `a` = 11 and `c` = 12.

That is, 11 blue pens at 12p each and 4 red pens at 13p each, costing £1.84 in total.

What must be special about the total cost (and number of pens bought) for this problem to have integral solutions?

Problem ID: 99 (Jan 2003) Difficulty: 2 Star