Secretive Shopkeeper III

Algebra Level 4

This is part of the Secretive Shopkeeper set

You enter a shop, and see from the display that garden sets cost £ g £g , headphones cost £ h £h , ironing boards cost £ i £i and jackets cost £ j £j . You tell the shopkeeper that you want to buy 1 garden set, 1 pair of headphones, 1 ironing board and 1 jacket, but you don't know what the cost is. He tells you the following:

  1. If you buy g + h g+h garden sets, h + i h+i pairs of headphones, i + j i+j ironing boards and j + g j+g jackets, it will cost you £ 65 , 313 £65,313
  2. If you buy g g garden sets, h h pairs of headphones, i i ironing boards and j j jackets, it will cost you £ 41 , 578 £41,578
  3. If you buy i i garden sets, j j pairs of headphones, g g ironing boards and h h jackets, it will cost you £ 23 , 848 £23,848

How much money (in pounds) should you give to the shopkeeper for 1 garden set, 1 pair of headphones, 1 ironing board and 1 jacket?

Details:

  • g , h , i , j g,h,i,j 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 336.

Stephen Mellor by Stephen Mellor , 20, United Kingdom

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1 solution

Stephen Mellor
Jan 3, 2018

From the information in the question, g 2 + g h + h 2 + h i + i 2 + i j + j 2 + j g = 65313 (1) g^2 + gh + h^2 + hi + i^2 + ij + j^2 + jg = 65313 \tag{1} g 2 + h 2 + i 2 + j 2 = 41578 (2) g^2 + h^2 + i^2 + j^2 = 41578 \tag{2} 2 g i + 2 h j = 23848 (3) 2gi + 2hj = 23848 \tag{3}

( 1 ) − ( 2 ) (1) - (2) : g h + h i + i j + j g = 23735 gh + hi + ij + jg = 23735 and doubling, 2 g h + 2 h i + 2 i j + 2 j g = 47470 (4) 2gh + 2hi + 2ij + 2jg = 47470 \tag{4}

( 3 ) + ( 4 ) (3) + (4) : 2 g h + 2 g i + 2 g j + 2 h i + 2 h j + 2 i j = 71318 (5) 2gh + 2gi + 2gj + 2hi + 2hj + 2ij = 71318 \tag{5}

Now, ( g + h + i + j ) 2 = g 2 + h 2 + i 2 + j 2 + 2 g h + 2 g i + 2 g j + 2 h i + 2 h j + 2 i j (g+h+i+j)^2 = g^2 + h^2 + i^2 + j^2 + 2gh + 2gi + 2gj + 2hi + 2hj + 2ij Substituting in from ( 2 ) (2) and ( 5 ) (5) : ( g + h + i + j ) 2 = 41578 + 71318 (g+h+i+j)^2 = 41578 + 71318 ( g + h + i + j ) 2 = 112896 (g+h+i+j)^2 = 112896 g + h + i + j = 336 g+h+i+j = 336

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