Copyright © University of Cambridge. All rights reserved.

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?

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?