GCSE Problem Solving
GCSE Problem Bank
Let Maths take you Further…
PRC v4
25/02/15
Always Fives
Marek is investigating even numbers. He finds out that if you pick an even number
and multiply it by three and then subtract half of the original number, you always get
a multiple of five.
Explain why this happens.
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Can you draw it? – Description 1
Triangle ABC has a right angle at C.
AC is the shortest side of triangle ABC.
M is the mid-point of the side AB.
N is the midpoint of the side BC
D is a point outside of the triangle ABC
DMN is an equilateral triangle.
Draw a diagram showing what the two triangles ABC and DMN could look like.
Can you draw it? – Description 2
Triangle ABC has a right angle at B.
Triangle LMN has a right angle at M
Triangle PQR has a right angle at Q
M is the mid-point of AC
L is the mid-point of BC
N lies outside the triangle ABC
P is the mid-point of MN
Q is the mid-point of LN
R Lies outside of triangle LMN but inside of triangle ABC
Draw a diagram showing what the three triangles ABC, LMN and PQR could look
like.
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Circly
What proportion of the large circle is shaded?
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Cube Slice
A cube is sliced vertically along the line shown in the diagram and the smaller part is
thrown away.
A
B
D
C
30 cm
15 cm
The prism that is formed is going to be sliced vertically downwards by a line going
through corner D.
Where would the slice have to be to split the new prism into two equal volumes?
A
D
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C
5 and 7 difference
Four positive whole numbers add up to 100.
Two of the numbers have a difference of 7.
The other two have a difference of 5.
Two of the numbers are identical.
What are the four numbers?
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Dodecagon and Hexagon
This diagram shows a regular dodecagon and a regular hexagon with a common
side PQ.
A
B
P
Q
Show that the points A, B and P all lie on the same straight line.
PRC v4
25/02/15
Exchange Rates (PISA)
Mei-Ling from Singapore was preparing to go to South Africa for 3 months as an
exchange student. She needed to change some Singapore dollars (SGD) into South
African rand (ZAR).
Question 1: Mei-Ling found out that the exchange rate between Singapore dollars
and South African rand was:
1 SGD = 4.2 ZAR
Mei-Ling changed 3000 Singapore dollars into South African rand at this exchange
rate.
How much money in South African rand did Mei-Ling get?
Question 2: On returning to Singapore after 3 months, Mei-Ling had 3 900 ZAR left.
She changed this back to Singapore dollars, noting that the exchange rate had
changed to:
1 SGD = 4.0 ZAR
How much money in Singapore dollars did Mei-Ling get?
Question 3: During these 3 months the exchange rate had changed from 4.2 to 4.0
ZAR per SGD.
Was it in Mei-Ling’s favour that the exchange rate now was 4.0 ZAR instead of 4.2
ZAR, when she changed her South African rand back to Singapore dollars?
Give an explanation to support your answer.
PRC v4
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Fill it up 1
0.2 m
This fuel tank is filled at a constant rate.
0.2 m
0.05 m
0.1 m
0.05 m
0.2 m
0.1 m
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Draw a graph showing how the height of the fuel
surface changes over time.
Fill it up 2
120 cm
45 cm
30 cm
25 cm
40 cm
20 cm
30 cm
45 cm
This fuel tank is going to be filled at a constant rate.
The fuel is poured in down through the hole marked with an arrow.
Sketch a graph showing how the maximum height of the fuel surface changes over
time.
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Growing Up (PISA)
The average height of both young males and young females in 1998 in the
Netherlands is represented in this graph:
Since 1980 the average height of 20-year-old females has increased by 2.3 cm, to
170.6 cm. What was the average height of a 20-year-old female in 1980?
Explain how the graph shows that on average the growth rate for girls slows down
after 12 years of age.
According to this graph, on average, during which period in their life are females
taller than males of the same age?
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Hexabubble
27 cm
This diagram shows 19 identical circles arranged in a hexagon.
All of the vertices of the trapezium are in the centre of a circle.
Find the area of the trapezium.
PRC v4
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Line Sketch 1
Sachin sketches the lines 𝑦 = 2𝑥 + 1 and 6𝑥 − 3𝑦 = 4 like this:
Is Sachin’s sketch a good representation of the lines?
Explain your answer.
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Line Sketch 2
Ganika sketches these two lines:
Where would the lines meet if they were extended?
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Mystery Number
Seven times a number gives the same answer as adding 36 to the number.
What is the number?
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Mystery Quadratic
𝑦
𝑥
This is part of the graph of a quadratic function.
What is the equation of the graph?
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Overlaps
6 cm
84 cm
These identical tiles have the same overlap for the whole design.
What is the length of a tile?
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How long is a piece of string?
This piece of string is 79 cm long.
It is going to be cut into three pieces.
The longest piece will be twice as long as the shortest piece.
The middle piece will be 7 cm longer than the shortest piece.
What three lengths will the string be cut into?
PRC v4
25/02/15
Postal Charges (PISA)
The postal charges in Zedland are based on the weight of the items (to the nearest
gram), as shown in the table below:
Which one of these graphs is the best representation of the postal charges in
Zedland?
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President (PISA)
In Zedland, opinion polls were conducted to find out the level of support for the
President in the forthcoming election. Four newspaper publishers did separate
nationwide polls. The results for the four newspaper polls are shown below:
Newspaper 1: 36.5% (poll conducted on January 6, with a sample of 500 randomly
selected citizens with voting rights)
Newspaper 2: 41.0% (poll conducted on January 20, with a sample of 500 randomly
selected citizens with voting rights)
Newspaper 3: 39.0% (poll conducted on January 20, with a sample of 1000
randomly selected citizens with voting rights)
Newspaper 4: 44.5% (poll conducted on January 20, with 1000 readers phoning in to
vote).
Which newspaper’s result is likely to be the best for predicting the level of support for
the President if the election is held on January 25?
Give two reasons to support your answer.
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Reaction Time (PISA)
In a Sprinting event, the ‘reaction time’ is the time interval between the starter’s gun
firing and the athlete leaving the starting block. The ‘final time’ includes both this
reaction time, and the running time.
The table gives the reaction time and the final time of 8 runners in a 100 metre sprint
race.
To date, no humans have been able to react to a starter’s gun in less than 0.110
seconds.
If the recorded reaction time for a runner is less than 0.110 seconds, then a false
start is considered to have occurred because the runner must have left before
hearing the gun.
If the Bronze medallist had a faster reaction time, would he have had a chance to
win the Silver medal? Give an explanation to support your answer.
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Scaled
64
A
B
This scale is divided into equal sections.
Find the numbers represented by A, B and C
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144
C
Shaded squares 1
This design is made up of squares.
What percentage of the design is shaded?
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Shaded squares 2
This design is made up of squares.
What percentage of the design is shaded?
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Tornado alley
The severity of a tornado is measured on the Fujita scale.
The most destructive tornados are given a rating of F5.
A reporter who researches the occurrence of tornados in
the USA’s tornado alley states, “In the next 20 years the
chance that an F5 tornado will occur in Kansas is two out of
seven.”
Which of the statements below best matches the meaning of the reporter’s
statement?
2
A) 20 × 7 = 5.7 so between 5 and 6 years from now there will be an F5 tornado
in Kansas
B) The likelihood that there will be an F5 tornado at some time during the next 20
years is lower than the likelihood of no tornado.
C)
2
1
is less than 2 so you can be sure there will not be an F5 tornado in Kansas
7
at any time during the next 20 years.
D) You cannot tell what will happen, because nobody can be sure when an F5
tornado will occur.
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Triangle trip 1
A
80 cm
B
This pattern is made up of equilateral triangles.
Find the length of the route marked from A to B
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Triangle trip 2
A
80 cm
B
This pattern is made up of equilateral triangles.
One possible route from A to B is marked on the pattern.
What is the length longest route you can find from A to B?
You can only travel along the side of each of the small triangles once.
Your route is not allowed to come into contact with itself at any stage (e.g. by
crossing or meeting itself at a point)
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Unfair Shares
Jon and Sandy share out £640 so that Sandy gets
How much do they each get?
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3
5
of what Jon gets.
Wire Bending
Aishe has a piece of wire that is 75 cm long.
She bends it into this shape made up of equilateral triangles.
What is the length of the side of a triangle?
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Zag Area 1
The area of this shape is 1.44 m2.
What is the perimeter of the shape?
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Zag Area 2
The area of this shape is 450 cm2.
What is the perimeter of the shape?
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Circle area 1
This diagram shows an equilateral triangle of side length 6 cm drawn inside a circle
so that each corner touches the circumference of the circle.
6 cm
What area of the circle is shaded?
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Shaded circles 1
𝑎
2𝑎
What proportion of the larger circle is shaded?
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Is Greenland bigger than Africa?
Here are the outlines of Greenland and Africa taken from a Mercator projection world map.
Use the map scale to estimate the area of both Greenland and Africa. Which appears to have the largest surface area?
Greenland
Africa
scale
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1000km
Is Greenland bigger than Africa?
Here are the outlines of Greenland and Africa taken from a Mercator projection world map.
Use the map scale to estimate the area of both Greenland and Africa. Which appears to have the largest surface area?
Greenland
Africa
scale
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0
1000km
Is Greenland bigger than Africa?
Greenland
Here are the outlines of Greenland and Africa taken
from a Peter’s projection world map.
This type of projection keeps the areas in correct
proportion.
Africa
Use the scale to estimate the area of both Greenland
and Africa.
scale
0
PRC v4
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2000km
Is Greenland bigger than Africa?
Greenland
Here are the outlines of Greenland and Africa taken
from a Peter’s projection world map.
This type of projection keeps the areas in correct
proportion.
Africa
Use the scale to estimate the area of both Greenland
and Africa.
scale
0
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2000km