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?
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,
15c = 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?