As a start, I decided to use simultaneous equations to compare the perimeters, areas and prices of different windows. To make things easier to explain, I assumed that each square unit represents one square centimetre; even though windows are very unlikely to be this size, the units don't matter for this problem.
I used "x" for the area of glass used in square centimetres, and "y" as the length of frame in centimetres. I decided to compare windows C and F first because they would be easier to compare due to their similar prices.
Here is how I began working out the problem:
8x + 12y = 320
9x + 12y = 330
These were the two equations I compared. To make things easier, both the windows had the same length of frame, so I just had to compare the difference in price and area. There was a difference of 1 squared centimetre in area, and £10 in price, so I worked out that the cost is £10 per squared centimetre.
x = 10
Using this knowledge, I then substitued the respective values into the equation to find out y:
80 + 12y = 320
12y = 240
y = 20
Therefore the price of the window is 10 times the area of glass in square centimetres added to 20 times the length of the frame in centimetres. To make sure this was correct I tested it on Window H:
Area = 3 squared centimetres
Perimeter = 8 centimetres
Area x 10 = 3 x 10 = £30
Perimeter x 20 = 8 x 20 = £160
Finally, £30 + £160 = £190.
This worked, also proving that H has the correct price.
After that I went on to check each window to look for the incorrect one.
Here is the test for Window A:
Area = 32
Perimeter = 28
32 x 10 = 320
28 x 20 = 560
560 + 320 = £880
Here is the test for Window B:
Area = 15
Perimeter = 16
15 x 10 = 150
16 x 20 = 320
150 + 320 = £470
Here is the test for Window D:
Area = 16
Perimeter = 20
16 x 10 = 160
20 x 20 = 400
400 + 160 = £560
Here is the test for Window E:
Area = 12
Perimeter = 18
12 x 10 = 120
18 x 20 = 360
120 + 360 = £480
Window E is the window with the incorrect price. Another way to tell is by looking at whether the prices of different windows are multiples of 20 or not; if a window's price is not a multiple of 20 (ie ending in 10, 30, 50, 70 or 90) then its glass area must be an odd number. Window E's price is not a multiple of 20, but its area is an even number. Therefore Window E seemed a likely suspect. Its real price should be £480.
Something I may have overlooked is the fact that the windows are Double-glazed. This means they had two layers of glass and therefore double the area of glass. In that case, if you work out the total area of both layers of glass, you need to multiply it by 5, rather than 10, before adding on the perimeter, to get the correct price.