GCSE Problem Solving GCSE Problem Bank

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1 GCSE Problem Solving GCSE Problem Bank Let Maths take you Further

2 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.

3 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.

4 Circly What proportion of the large circle is shaded?

5 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 C

6 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?

7 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.

8 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 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.

9 Fill it up m This fuel tank is filled at a constant rate. 0.2 m Draw a graph showing how the height of the fuel surface changes over time m 0.1 m 0.05 m 0.2 m 0.1 m

10 Fill it up cm 40 cm 45 cm 30 cm 25 cm 20 cm 45 cm 30 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.

11 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 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?

12 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.

13 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.

14 Line Sketch 2 Ganika sketches these two lines: Where would the lines meet if they were extended?

15 Mystery Number Seven times a number gives the same answer as adding 36 to the number. What is the number?

16 Mystery Quadratic y x This is part of the graph of a quadratic function. What is the equation of the graph?

17 Overlaps 6 cm 84 cm These identical tiles have the same overlap for the whole design. What is the length of a tile?

18 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?

19 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?

20 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.

21 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 seconds. If the recorded reaction time for a runner is less than 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.

22 Scaled A B C This scale is divided into equal sections. Find the numbers represented by A, B and C

23 Shaded squares 1 This design is made up of squares. What percentage of the design is shaded?

24 Shaded squares 2 This design is made up of squares. What percentage of the design is shaded?

25 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? A) 20 2 = 5.7 so between 5 and 6 years from now there will be an F5 tornado 7 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 is less than 1 so you can be sure there will not be an F5 tornado in Kansas 7 2 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.

26 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

27 Triangle trip 2 A 80 cm 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) B

28 Unfair Shares Jon and Sandy share out 640 so that Sandy gets 3 of what Jon gets. 5 How much do they each get?

29 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?

30 Zag Area 1 The area of this shape is 1.44 m 2. What is the perimeter of the shape?

31 Zag Area 2 The area of this shape is 450 cm 2. What is the perimeter of the shape?

32 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?

33 Shaded circles 1 a 2a What proportion of the larger circle is shaded?

34 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 km

35 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 km

36 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. Use the scale to estimate the area of both Greenland and Africa. Africa scale km

37 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. Use the scale to estimate the area of both Greenland and Africa. Africa scale km