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Warmsnug Double Glazing

Stage: 3 Challenge Level: Challenge Level:1
We received many solutions in which people used pairs of similar windows to find the prices. One possibility is to find a pair with the same area, but different frame lengths. Harry & Roxana from Thorpe House Langley Preparatory School did this:

We looked at K (Area = 12, Frame = 17) and I (Area = 12, Frame = 14) and we used the £60 price difference to find out the cost of the frame (£20 per unit) and the cost of the glass (£10 for each 1 by 1 pane).

Millie and Kate's method for finding the costs of the frame and the glass is very neat:

K had 3 cm of extra frame and was £60 more so we divided it by 3 to find that each centimetre of frame was £20.

Similarly, Jake from Colyton Grammar School started by finding the price of each unit square of glass:

J (Area = 4, Frame = 8) and H (Area = 3, Frame = 8) each have the same length of frame, but J has one square of glass more. J costs £10 more than H so that means that a 1 by 1 pane of glass costs £10.

Once these prices have been found, the correct price of each shape can be found and compared to its price tag. But what would happen if one of K, I, J or H had the wrong price tag? Would this method work? Luckily, many students checked all the price tags and found that the only incorrect one is E.

E has 18 frame squares and 12 glass panes, so it should cost £360 + £120, which equals £480. The price marked is £550, so window E is wrong.

Rhea from Loughborough High School used a very systematic approach to make sure she found the window that was priced incorrectly. She used an algebraic method with simultaneuous equations:

I used X to represent the price of the frame per square and Y to represent 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 Y = 10, and there is only one incorrect equation, Y must equal £10.
It also indicates that equations (and the prices of) 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/3.  8X + 12Y = 280           
E - M/3: 
                10X = 270                         
                     X = 27 

C. 12X + 8Y = 320
D. 20X + 16Y= 560
D/2. 10X + 8Y = 280
C - D/2: 
                2X = 40
                   X = 20

As 3 out of 4 of the answers to the simultaneous equations investigated are X = 20, I assume that X must £20.
It also indicates that either equation (and the prices of) 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.

Let 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. 18X + 12Y = 360 + 120 = 480; so the price for window E is incorrect.
This makes sense because E didn't produce the right answer when put in a simultaneous equation.

Well done to everyone who found the solution.
Can you see the similarity between the algebraic method and the 'comparing pairs' method?