Secretive Shopkeeper I

Algebra Level 2

This is part of the Secretive Shopkeeper set

You enter a shop, and see from the display that armchairs cost $£a$ , bikes cost $£b$ and cameras cost $£c$ . You tell the shopkeeper that you want to buy 1 armchair, 1 bike and 1 camera, but you don't know what the cost is. He tells you the following:

If you buy $a$ armchairs, $b$ bikes and $c$ cameras, it will cost you $£33,530$
If you buy $b$ armchairs, $c$ bikes and $a$ cameras, it will cost you $£30,053$

How much money (in pounds) should you give to the shopkeeper for 1 armchair, 1 bike and 1 camera?

Details:

$a,b,c$ 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 306.

1 solution

Stephen Mellor
Dec 30, 2017

From the information from the question: $a^2 + b^2 + c^2 = 33530$ $ab + ac + bc = 30053$ If you add the first equation to 2 times the second equation you get: $(a^2 + b^2 + c^2) + (ab + ac + bc) + (ab + ac + bc) = 33530 + 30053 + 30053$ $a^2 + b^2 + c^2 + 2ab + 2ac + 2bc = 93636$ $(a + b + c)^2 = 93636$ $a + b + c = 306$

Hence 1 armchair, 1 bike and 1 camera costs $£306$

For reference, numbers satisfying the problem are 72,95,139 in any order

Stephen Mellor - 3 years, 5 months ago

Secretive Shopkeeper I

The answer is 306.

1 solution

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