Secretive Shopkeeper V

Algebra Level 3

This is part of the Secretive Shopkeeper set

You enter a shop, and see from the display that pairs of pyjamas cost $£p$ , quilts cost $£q$ , rugs cost $£r$ , suitcases cost $£s$ and tables cost $£t$ . You tell the shopkeeper that you want to buy 1 pair of pyjamas, 1 quilt, 1 rug, 1 suitcase and 1 table, but you don't know what the cost is. He tells you the following:

If you buy $p$ pairs of pyjamas, $q$ quilts, $r$ rugs, $s$ suitcases and $t$ tables, it will cost you £25,591.
If you buy $q$ pairs of pyjamas, $r$ quilts, $s$ rugs, $t$ suitcases and $p$ tables, it will cost you £18,798.
If you buy $r$ pairs of pyjamas, $s$ quilts, $t$ rugs, $p$ suitcases and $q$ tables, it will cost you £14,919.

What is the cost of 1 pair of pyjamas, 1 quilt, 1 rug, 1 suitcase and 1 table?

Details:

$p,q,r,s,t$ 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 305.

1 solution

Zee Ell
Jan 14, 2018

Let:

$S = p+q+r+s+t$

$A = p^2 + q^2 + r^2 + s^2 + t^2 = 25591$

$B = pq + qr + rs + st+ pt = 18798$

$C = pr + qs + rt + ps+ qt = 14919$

Then:

$S^2 = A + 2B + 2C = 25591 + 2 × 18798 + 2 × 14919 = 93025$

$S = \sqrt { 93025 } = \boxed {305}$

Secretive Shopkeeper V

The answer is 305.

1 solution

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