For each window, the grey line around the areas represents the frame and the greyish area inside the grey lines represents the glass.
I used X to represent the price of the frame per square and Y to represent the the price of the glass per square. For each window I wrote an equation using X and Y:
A. 28X + 32Y = £880 (The frame borders 28 squares and the area of the glass is 32 squares)
B. 16X + 15Y = £470 (The frame borders 16 squares and the area of the glass is 15 squares)
C. 12X + 8Y = £320 (etc.)
D. 20X + 16Y = £560
E. 18X + 12Y = £550
F. 12X + 9Y = £330
G. 26X + 24Y = £760
H. 8X + 3Y= £190
I. 14X + 12Y = £400
J. 8X + 4Y = £200
K. 17X + 12Y = £460
L. 23X + 20Y = £660
M. 24X + 36Y = £840
N. 20X + 24Y = £640
O. 16X + 12Y = £440
I then looked for equations which had equal X or Y figures. I used these equations to explore some simultaneous equations:
F. 12X + 9Y = 330
C. 12X + 8Y = 320
F-C:
Y = 10
J. 8X + 4Y = 200
H. 8X + 3Y = 190
J-H:
Y = 10
N. 20X + 24Y = 640
D. 20X + 16Y = 560
N-D:
8Y = 80
Y = 10
B. 16X + 15Y = 470
O. 16X + 12Y = 440
B-O:
3Y = 30
Y = 10
As all the answers to the simultaneous equations which I investigated are £10, and there is only one incorrect equation, then Y must equal £10. It also indicates that equations F, C, J, H, N, D, B and O must be correct.
G. 26X + 24Y = 760
N. 20X + 24Y = 640
G-N:
6X = 120
X = 20
K. 17X + 12Y = 460
I. 14X + 12Y = 400
K-I:
3X = 60
X = 20
E. 18X + 12Y = 550
M. 24X + 36Y = 840
M. 8X + 12Y = 280 ...(M/3)
E- M/3: 10X = 270
X = 27
C. 12X + 8Y = 320
D. 20X + 16Y= 560
D. 10X + 8Y = 280 ...(D/2)
E- D/2: 2X = 40
X = 20
As 3 out of 4 of the answers to the simultaneous equations investigated are £20, I assume that X must £20. It also indicates that either equation E or M is the incorrect one because when they are solved in a simultaneous equation, they produce a different answer to all the others.
X= £20 and Y= £10; I entered these values into all the equations to see if they fitted in with the figures. All of them except E proved to be correct:
E. 360+ 120< 550; so the price for window E is incorrect. This makes sense because E was the only equation which didn't produce the right answer when put in a simultaneous equation.
Ans. Price of the frame which borders the square= £20
Price of the glass per square= £10
The window which has been given an incorrect price= window E