OCR GCSE Maths
Topic questions Ratio, proportion and rates of change Scale factors, scale diagrams and maps
37 questions · 201 marks · about 3 hours
The scale drawing shows the positions of ship A and ship B.
Scale: 1 cm represents 6 km
Find the actual distance between ship A and ship B.
Ship C is 33 km from ship B on a bearing of 285°.
On the scale drawing, mark and label the position of ship C with a cross.
Also involves Converting between units
Write the ratio 14: 35 in its simplest form.
The scale of a map is 1: 25 000.
Two castles are 6 kilometres apart on the ground.
Work out how far apart the two castles are on the map. Give your answer in centimetres.
1 cm on a map represents 150 km in real life.
Below is part of the map showing City A and City B.
Heidi is planning to walk from City A to City B for charity.
Heidi measures the straight-line distance on the map from City A to City B accurately as 5 cm. Heidi says she will walk from City A to City B at an average speed of 2.5 km / h for 10 hours per day.
Use this information to work out how many days Heidi needs to complete the walk from City A to City B.
Explain why the information used in part (i) is likely to give an underestimate for the number of days Heidi needs to complete the walk.
Yoshi writes the scale of the map, 1 cm represents 150 km, as the ratio 1: 150.
Explain Yoshi’s error and write the scale correctly in the form 1: n.
The correct answer is 1:........
1 cm on a map represents 150 km in real life.
Below is part of the map showing City A and City B.
Heidi is planning to walk from City A to City B for charity.
Heidi measures the straight-line distance on the map from City A to City B accurately as 5 cm. Heidi says she will walk from City A to City B at an average speed of 2.5 km / h for 10 hours per day.
Use this information to work out how many days Heidi needs to complete the walk from City A to City B.
Explain why the information used in part (i) is likely to give an underestimate for the number of days Heidi needs to complete the walk.
Yoshi writes the scale of the map, 1 cm represents 150 km, as the ratio 1: 150.
Explain Yoshi’s error and write the scale correctly in the form 1: n.
The correct answer is 1:........
Also involves Converting between units
Using ruler and compasses only, construct the road from town T to the road AB.
The new road costs £200 000 per kilometre to build. The road constructor says
The new road will cost over £600 000 to build.
Show that the road constructor is incorrect.
The diagram shows the positions of three towns A, B and C.
The bearing of town B from town A is 053°. The bearing of town C from town B is 108°. AB = 35 km and BC = 61 km.
Calculate AC. You must show your working.
Also involves Converting between units
Using ruler and compasses only, construct the road from town T to the road AB.
The new road costs £200 000 per kilometre to build. The road constructor says
The new road will cost over £600 000 to build.
Show that the road constructor is incorrect.
Also involves Measuring lengths and angles, bearings
The scale drawing shows the positions of two towns, A and B.
Scale: 1 cm represents 10 km
Find the actual distance between town A and town B.
Town C is 55 km from town B on a bearing of 330°.
On the scale drawing, mark and label the position of town C with a cross.
B is 12 km due east of A. C is south-east of A and on a bearing of 225° from B.
Complete the diagram to show the positions of A, B and C. Show clearly the values of all three angles in triangle ABC.
Scale: 1 cm represents 2 km
Also involves Converting between units
Write 28: 70 as a ratio in its simplest form.
A map has a scale of 8 centimetres represents 1 metre. The scale can be written as a ratio in the form 1: n.
Find the value of n.
B is 12 km due east of A. C is south-east of A and on a bearing of 225° from B.
Complete the diagram to show the positions of A, B and C. Show clearly the values of all three angles in triangle ABC.
Scale: 1 cm represents 2 km
A town square has a fountain (F) at the centre. There is also a bell tower (B) and a statue (S).
The bearing of the statue from the fountain is x °.
The bearing of the bell tower from the fountain is 140° more than the bearing of the statue from the fountain.
Write down, in terms of x, the bearing of the bell tower from the fountain.
The bearing of the bell tower from the fountain is also three times the bearing of the statue from the fountain.
Work out the bearing of the bell tower from the fountain.
The diagram shows the scale drawing of a sandpit, ABCD. It also shows the arc of all points in the sandpit that are 80 cm from corner A.
Scale: 1 cm represents 20 cm
A game is played by throwing a ball into the sandpit. Points may be scored when the ball lands in the sandpit.
• 1 point if the ball lands within 80 cm of corner A, and
• 1 point if the ball is closer to side AB than side AD, and
• 1 point if the ball is closer to corner A than corner B.
By completing the construction, find and shade the regions where 2 points can be scored.
Show all your construction lines.
The scale diagram below shows the position of two castles, J and K.
Scale: 1 cm represents 2 km
The bearing of a tower from castle J is 072°. The bearing of the tower from castle K is 116°.
Use construction to find the distance from castle J to the tower. Give your answer to the nearest 0.1 km.
The diagram shows the scale drawing of a sandpit, ABCD. It also shows the arc of all points in the sandpit that are 80 cm from corner A.
Scale: 1 cm represents 20 cm
A game is played by throwing a ball into the sandpit. Points may be scored when the ball lands in the sandpit.
• 1 point if the ball lands within 80 cm of corner A, and
• 1 point if the ball is closer to side AB than side AD, and
• 1 point if the ball is closer to corner A than corner B.
By completing the construction, find and shade the regions where 2 points can be scored.
Show all your construction lines.
The scale drawing represents a park, ABCD.
Scale: 1 cm represents 10 m
A straight path goes across the park from B. The path is always the same distance from side AB and side BC.
Construct the route followed by the path.
Show all your construction lines.
A bench is to be placed on the path. The bench must be no more than 50 m from C.
Construct the locus of the possible positions of the bench.
Indicate clearly on the diagram where the bench can be placed.
Also involves Converting between units
The scale on a map is 1: 50 000.
How many kilometres on the ground are represented by 8 cm on the map?
The scale 1 cm represents 25 m can be written in the form 1: k.
Find the value of k.
The scale drawing represents a harbour.
A boat leaves the harbour from point C and sails on a path that is equidistant from BC and CD. The harbour rules do not allow boats to sail within 75 m of point E.
Find by construction whether the path of the boat will follow the harbour rules. Show all your construction lines.
Also involves Converting between units
T is a radar tower. A and B are two aircraft.
At 3pm
• aircraft A is 3250 km from T on a bearing of 015°
• aircraft B is 4960 km from T on a bearing of 057°.
Aircraft A flies directly towards radar tower T at a speed of 890 km/h.
At what time will the aircraft pass over radar tower T? Give your answer to the nearest minute.
Calculate the distance that was between aircraft A and aircraft B at 3pm.
The scale drawing shows Katy’s garden ABCD.
Scale: 1 cm represents 5 m
A B
D C
Katy places a statue in the garden.
The statue is
• more than 30 m from D
• closer to CB than AB.
Construct and shade the region where the statue could be placed.
Show all your construction lines.
Also involves Converting between units
This map shows part of a village.
Neil knows that Packer Street is 180 m long in real life.
Neil measures the map.
He says
Packer Street is 3.5 cm long. High Street is 11.2 cm long.
Therefore, I calculate that High Street is 576 m long in real life.
Use Neil’s figures to show that the answer to his calculation is correct.
Jodie measures the same map.
She says
I think Packer Street is longer than Neil’s measurement of 3.5 cm. Therefore, High Street must be longer than 576 m in real life.
Is Jodie’s reasoning correct? Show how you decide.
On another map, Packer Street is 2.4 cm long.
Express the scale of this map in the form 1: n.
This map shows part of a village.
Neil knows that Packer Street is 180 m long in real life.
Neil measures the map.
He says
Packer Street is 3.5 cm long. High Street is 11.2 cm long.
Therefore, I calculate that High Street is 576 m long in real life.
Use Neil’s figures to show that the answer to his calculation is correct.
Jodie measures the same map.
She says
I think Packer Street is longer than Neil’s measurement of 3.5 cm. Therefore, High Street must be longer than 576 m in real life.
Is Jodie’s reasoning correct? Show how you decide.
On another map, Packer Street is 2.4 cm long.
Express the scale of this map in the form 1: n.
Also involves Measuring lengths and angles, bearings
The scale diagram below shows two cities, P and Q.
A plane departs from P at 09 47 and arrives at Q at 12 07.
Work out the average speed, in kilometres per hour, of the plane.
Give one reason why your answer may be inaccurate.
The scale diagram below shows towns, A, B and C. Line AB represents the road from A to B and line AC represents the road from A to C.
A shopping centre is to be built so that it is
• nearer to the road from A to B than the road from A to C,
• less than 14 km from town C.
Using construction, shade the region where the shopping centre could be built. Show all your construction lines.
Explain why the region found in part (a) may not be an appropriate site for the shopping centre.
Also involves Rounding · Measuring lengths and angles, bearings
The scale diagram below shows two cities, P and Q.
A plane departs from P at 09 47 and arrives at Q at 12 07.
Work out the average speed, in kilometres per hour, of the plane.
Give one reason why your answer may be inaccurate.
The scale diagram below shows towns, A, B and C. Line AB represents the road from A to B and line AC represents the road from A to C.
A shopping centre is to be built so that it is
• nearer to the road from A to B than the road from A to C,
• less than 14 km from town C.
Using construction, shade the region where the shopping centre could be built. Show all your construction lines.
Scale: 1 cm represents 2 km
Explain why the region found in part (a) may not be an appropriate site for the shopping centre.
Four points A, B, C and D are shown on the scale diagram below.
Scale: 1 cm represents 5 m
On the diagram, construct and mark the two points that are
• the same distance from A and B
and
• 15 m from C.
Show all your construction lines.
The points A, B, C and D represent the four corners of Monty’s garden. His garden is bounded by four straight fences A to B, B to C, C to D and D to A.
Monty wants to plant a tree in his garden at a place that satisfies the two conditions in part
Explain why there is only one position where Monty can plant his tree.
The diagram shows a scale drawing of a park, ABCD.
Scale: 1 cm represents 10 m
A straight water pipe runs across the park. The pipe runs equidistant from DA and DC.
Construct, using compasses and ruler only, the position of the water pipe.
You must show all your construction lines.
A straight path connects the entrance to the exit. This path is perpendicular to CB.
Construct, using compasses and ruler only, the position of the path.
Leave in all your construction lines.
Find the actual length of the path, in metres.
Also involves Forming and solving equations
The diagram shows the positions of two towns, Amton and Bisham.
The bearing of Bisham from Amton is b °. The bearing of Amton from Bisham is 6 b °.
Calculate the 3-figure bearing of Amton from Bisham.
Also involves Converting between units
A model railway is built using the scale 1: 87.
On the model railway, the distance between the rails is 16.5 mm.
Calculate, in metres, the distance between the rails for a full-size train.
The volume of a full-size train carriage is 220 m³. Trevor calculates the volume of a model train carriage to be 334 cm³ correct to 3 significant figures.
Is Trevor’s calculation correct? Show how you decide.
Also involves Converting between units
The ratio 20 minutes to 1 hour can be written in the form 1: n.
Find the value of n.
The scale on a map is 1: 25 000.
How many kilometres on the ground is represented by 6 cm on the map?
Kiri and Peter share some sweets in the ratio 6: 7.
What fraction of the sweets does Kiri receive?
The scale diagram shows the positions of town A and town B.
Scale: 1 cm represents 10 miles
Lucy’s house is nearer to town A than to town B. Her house is exactly 50 miles from town B.
On the scale diagram show all the possible positions of Lucy’s house.
You must show all your construction lines.
A, B, C and D are four towns.
Work out the bearing of B from C.
Calculate the bearing of D from B.
A and B are two farms on this map.
Use the map to complete these sentences.
The distance from A to B is........ km.
The bearing of B from A is........ °.
C is another farm. C is 2.5 km from B on a bearing of 230°.
Mark and label the position of C on the map with a cross.
Also involves Ratio
The scale of a map is 1 cm represents 25 m.
The length of a path is 240 m.
Work out the length, in centimetres, of the path on the map.
The scale 1 cm represents 25 m can be written in the form 1: k.
Find the value of k.
The scale drawing represents a park.
Scale: 1 cm represents 25 m
A new play area must be
• no more than 150 m from B
• closer to AD than to CD.
Construct and shade the region where the play area can be positioned.
Show all your construction lines.
Also involves Trigonometry · Angle facts
A, B, C and D are four towns.
Work out the bearing of B from C.
Calculate the bearing of D from B.
The scale of a map is 1 cm represents 25 m.
The length of a path is 240 m.
Work out the length, in centimetres, of the path on the map.
The scale 1 cm represents 25 m can be written in the form 1: k.
Find the value of k.
The scale drawing represents a park.
Scale: 1 cm represents 25 m
A new play area must be
• no more than 150 m from B
• closer to AD than to CD.
Construct and shade the region where the play area can be positioned.
Show all your construction lines.
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