Secretive Shopkeeper II

Algebra Level 3

This is part of the Secretive Shopkeeper set

You enter a shop, and see from the display that drones cost $£d$ , electronics kits cost $£e$ and flutes cost $£f$ . You tell the shopkeeper that you want to buy 1 drone, 1 electronics kit and 1 flute, but you don't know what the cost is. He tells you the following:

If you buy $d$ drones, $e$ electronics kits and $f$ flutes, it will cost you $£24,722$
If you buy $e+1$ drones, $f+1$ electronics kits and $d+1$ flutes, it will cost you $£18,639$

How much money (in pounds) should you give to the shopkeeper for 1 drone, 1 electronics kit and 1 flute?

Details:

$d,e,f$ 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 248.

1 solution

Stephen Mellor
Jan 2, 2018

From the information in the question, $d^2 + e^2 + f^2 = 24722$ $d(e+1) + e(f+1) + f(d+1) = 18369$ or $de + df + ef + d + e + f = 18639$

If we let $k = d + e + f$ , we can rearrange this equation to get, $de + df + ef = 18639 - k$

Now, $(d+e+f)^2 = d^2 + e^2 + f^2 + 2(de + df + ef)$ so $k^2 = (24722) + 2(18639 - k)$ or $k^2 + 2k - 62000 = 0$

Therefore $k = -250$ or $k = 248$ . As $d$ , $e$ and $f$ are positive integers, $k = 248$ , or $d + e + f = 248$ . Hence, the price of 1 drone, 1 electronics kit and 1 flute is $£248$

Secretive Shopkeeper II

The answer is 248.

1 solution

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