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Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?
Take a Square
Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.
Semi-detached
A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.
Age 14 to 16
Challenge Level
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?
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NRICH is part of the family of activities in the Millennium Mathematics Project.
NRICH is part of the family of activities in the Millennium Mathematics Project.