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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.
therefore ac + b(c + 1) = 184
therefore ac + bc + b = 184
therefore ac + bc = 184 minus b
therefore c(a + b) = 184 minus b

But a + b = 15,
therefore 15c = 184 minus b

Hence 184 minus b must be divisible by 15; as 15 times 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

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