Secretive Shopkeeper IV

Algebra Level 3

This is part of the Secretive Shopkeeper set

You enter a shop, and see from the display that kayaks cost $£k$ , light bulbs cost $£l$ , mirrors cost $£m$ , necklaces cost $£n$ and ovens cost $£o$ . You tell the shopkeeper that you want to buy 1 kayak, 1 light bulb, 1 mirror, 1 necklace and 1 oven, but you don't know what the cost is. He tells you the following:

If you buy $2m$ kayaks, $o$ light bulbs, $2l$ mirrors, $3k$ necklaces and $k$ ovens, it will cost you £15,674.
If you buy $2o$ kayaks, $m$ light bulbs, $k$ mirrors, $3l$ necklaces and $2l$ ovens, it will cost you £14,929.

What is the cost of 1 kayak, 1 light bulb, 1 mirror, 1 necklace and 1 oven?

Details:

$k,l,m,n,o$ are positive integers
If you underpay him, he will call the police
If you overpay him, he will keep the extra money

The answer is 202.

1 solution

Stephen Mellor
Jan 5, 2018

From the information in the question, $2mk + ol + 2lm + 3kn + ko = 15674$ $2ok + ml + km + 3ln + 2lo = 14929$

Adding these equations together, $3km + 3kn + 3ko + 3lm + 3ln + 3lo = 30603$ $km + kn + ko + lm + ln + lo = 10201$ $(k+l)(m+n+o) = 10201$

As the variables are positive integers, we only have to consider the positive factorisations of $10201$ . As $10201 = 101^2$ , and $101$ is prime, we only have 2 possible pairs of factors: $(1,10201)$ or $101,101$ . However, as the variables are positive integers, $k+l \geq 2$ and $m+n+o \geq 3$ . Therefore, the pair of brackets must both equal $101$ .

Therefore, $k+l+m+n+o = 101 + 101 = \boxed{202}$

Secretive Shopkeeper IV

The answer is 202.

1 solution

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