Scale drawings & maps – GCSE Maths exam questions with mark scheme

1Grade 1–3FoundationNon-calculator2 marks

The scale of a map is $1 : 25\,000$.

Two churches are 6 cm apart on the map.

Work out the real distance between the churches. Give your answer in kilometres. [2]

$6 \times 25\,000$ (= 150 000 cm)M1
1.5 kmA1
Answer: 1.5 km
Examiner tip: Convert carefully: 100 000 cm = 1 km.

2Grade 1–3FoundationCalculator3 marks

A plan of a room is drawn using a scale of $1 : 50$.

(a) The length of the room on the plan is 9 cm. Work out the real length of the room in metres. [2]

(b) The real width of the room is 3.2 m. Work out the width of the room on the plan in centimetres. [1]

(a) $9 \times 50$ (= 450 cm)M1
4.5 mA1
(b) 6.4 cmB1
Answer: (a) 4.5 m (b) 6.4 cm
Examiner tip: Real → plan means divide by 50; plan → real means multiply by 50.

3Grade 4–5Non-calculator2 marks

On a map, 2 cm represents 5 km.

Write the scale of the map in the form $1 : n$. [2]

5 km $= 500\,000$ cm, or $2 : 500\,000$M1
$1 : 250\,000$A1
Answer: $1 : 250\,000$
Examiner tip: A ratio scale has no units, so both parts must be in centimetres before you simplify.

4Grade 4–5Non-calculator4 marks

The diagram shows the positions of two harbours, A and B, on a map.

The scale of the map is $1 : 50\,000$. On the map, B is 8 cm from A on a bearing of 065°.

A boat sails directly from A to B at an average speed of 12 km/h.

How many minutes does the journey take? [4]

$8 \times 50\,000$ (= 400 000 cm)M1
4 km (allow 4000 m)A1
their $4 \div 12$ (= $\frac{1}{3}$ hour) or $60 \div 12 \times 4$M1
20 minutesA1
Answer: 20 minutes
Examiner tip: The bearing is not needed here – focus on converting the map distance to km before using speed.

5Grade 6–7HigherCalculator3 marks

The scale of a map is $1 : 20\,000$.

A lake has an area of 12 cm² on the map.

Work out the real area of the lake in m². [3]

1 cm represents 200 m (or 20 000 cm) (may be implied by later work)M1
$12 \times 200^2$ or $12 \times 20\,000^2$ cm² (= 4 800 000 000 cm²)M1
480 000 m²A1
Answer: 480 000 m²
Examiner tip: Areas scale by the square of the scale factor – $12 \times 200 = 2400$ m² is the classic wrong answer.

6Grade 8–9HigherCalculator5 marks

Two towns are 18 km apart. On map A the towns are 7.2 cm apart.

(a) Work out the scale of map A in the form $1 : n$. [2]

Map B has a scale of $1 : 400\,000$.

(b) A forest has an area of 6 cm² on map B. Work out the area of the same forest on map A. [3]

(a) $18 \times 100\,000 \div 7.2$ or $1\,800\,000 : 7.2$M1
$1 : 250\,000$A1
(b) linear scale factor from map B to map A: $400\,000 \div 250\,000 = 1.6$, or finds the real area of the forest, e.g. $6 \times 4^2 = 96$ km² (map B: 1 cm represents 4 km)M1
$6 \times 1.6^2$ (area factor 2.56), or their real area $\div 2.5^2$ (map A: 1 cm represents 2.5 km), e.g. $96 \div 6.25$M1
15.36 cm²A1
Answer: (a) $1 : 250\,000$ (b) 15.36 cm²
Examiner tip: Map A has the larger scale (fewer real cm per map cm), so the forest looks bigger – and areas use the square of 1.6.

7Grade 1–3FoundationNon-calculator3 marks

The scale of a map is $1 : 40\,000$.

On the map, a bus station and a library are 3.5 cm apart.

Zara works out the real distance between them. Here is her working.

$3.5 \times 40\,000 = 140\,000$ cm
$140\,000 \div 10\,000 = 14$
So the real distance is 14 km.

(a) Explain the mistake Zara has made. [1]

(b) Work out the correct real distance in kilometres. [2]

(a) she divided by 10 000 instead of 100 000 (accept: there are 100 000 cm in 1 km, not 10 000; ignore comments about the first line, which is correct)C1
(b) $140\,000 \div 100\,000$, or 1400 m seenM1
1.4 kmA1
Answer: (a) Should divide by 100 000 (100 000 cm = 1 km) (b) 1.4 km
Examiner tip: Learn the chain 100 cm = 1 m and 1000 m = 1 km, so 1 km = 100 000 cm – not 10 000.

8Grade 4–5Non-calculator3 marks

The diagram shows the scale bar on a walking map.

On the map, the whole scale bar is 6 cm long.

(a) Write the scale of the map in the form $1 : n$. [2]

(b) On the map, a footpath is 9.5 cm long. Work out the real length of the footpath in kilometres. [1]

(a) 2 cm represents 1 km, or $2 : 100\,000$, or $6 : 300\,000$M1
$1 : 50\,000$A1
(b) 4.75 km (accept 4750 m)B1
Answer: (a) $1 : 50\,000$ (b) 4.75 km
Examiner tip: Use the stated length of the bar, not a ruler on your screen: 6 cm for 3 km means 2 cm for each kilometre.

9Grade 4–5Non-calculator3 marks

Map P has a scale of $1 : 25\,000$.

On map Q, 4 cm represents 1 km.

A road is 6 cm long on map P.

Kai says, “The road will be longer on map Q than on map P.”

Is Kai correct? You must show your working. [3]

real length $6 \times 25\,000$ (= 150 000 cm or 1.5 km), or 1 km = 100 000 cm used with map QM1
$1.5 \times 4$ (= 6 cm on map Q), or map Q scale written as $4 : 100\,000$ or $1 : 25\,000$M1
No, with correct supporting figures, e.g. the road is 6 cm on both maps, or both maps have scale $1 : 25\,000$C1
Answer: No – both maps have scale $1 : 25\,000$, so the road is 6 cm long on each.
Examiner tip: Put both scales into the same form before comparing – they can look different but be identical.

10Grade 4–5Calculator6 marks

A plan of a rectangular field is drawn to a scale of $1 : 2500$.

On the plan, the field is 9.6 cm long and 6.4 cm wide.

(a) Fencing is sold in rolls. Each roll is 25 m long and costs £62.
Work out the cost of enough fencing to go all the way around the field. [3]

(b) A bag of grass seed covers 0.5 hectares. 1 hectare = 10 000 m².
How many bags are needed to cover the whole field? [3]

(a) $9.6 \times 2500$ or $6.4 \times 2500$ (= 24 000 cm or 16 000 cm), or 240 m or 160 m seenM1
perimeter 800 m and $800 \div 25$ (= 32 rolls)M1
£1984A1
(b) $240 \times 160$ (= 38 400 m²) (ft their real lengths)M1
$38\,400 \div 10\,000 \div 0.5$ (= 7.68), or $38\,400 \div 5000$M1
8 bags (must be a whole number rounded up)A1
Answer: (a) £1984 (b) 8 bags
Examiner tip: Find the real lengths before working out the area – multiplying 9.6 by 6.4 and then by 2500 gives a wrong area.

11Grade 6–7HigherCalculator4 marks

A model of a cargo ship is made to a scale of $1 : 200$.

(a) The real ship is 96 m long. Work out the length of the model in centimetres. [1]

(b) The cargo hold of the model has a volume of 45 cm³.
Work out the volume of the cargo hold of the real ship. Give your answer in m³. [3]

(a) 48 cmB1
(b) volume scale factor $200^3$ (= 8 000 000) used, e.g. $45 \times 200^3$ (= 360 000 000 cm³); or 1 cm represents 2 mM1
$\div 100^3$ (= 1 000 000) to convert cm³ to m³, or $45 \times 2^3$M1
360 m³A1
Answer: (a) 48 cm (b) 360 m³
Examiner tip: Volumes scale by the cube of the scale factor, and 1 m³ is 1 000 000 cm³, not 100 cm³.

12Grade 8–9HigherCalculator5 marks

The scale of a map is $1 : 25\,000$.

On the map, the straight path from a car park to a waterfall measures 8.4 cm, correct to the nearest millimetre.

Jas walks along the path at a speed of 4.5 km/h, correct to the nearest 0.5 km/h.

Can Jas be sure she will reach the waterfall in less than 30 minutes? You must show how you get your answer. [5]

8.45 (upper bound of map length) seenB1
4.25 (lower bound of speed) seenB1
$8.45 \times 25\,000$ (= 211 250 cm = 2.1125 km)M1
their upper bound distance $\div$ their lower bound speed, e.g. $2.1125 \div 4.25$ (= 0.497… h)M1
29.8 minutes (accept 29.82…, or 0.497 h) and YesA1
Answer: Yes – the longest possible time is 29.8 minutes (to 1 d.p.), which is under 30 minutes.
Examiner tip: Longest time needs the longest distance divided by the slowest speed – using the upper bound of speed loses both method marks.

13Grade 8–9HigherNon-calculator4 marks

A park has a real area of 0.72 km².

On a map, the area of the park is 18 cm².

Work out the scale of the map. Give your answer in the form $1 : n$. [4]

$0.72 \div 18$ (= 0.04 km² represented by 1 cm²), or 0.72 km² = 7 200 000 000 cm² seenM1
$\sqrt{0.04} = 0.2$ km (1 cm represents 0.2 km), or $7\,200\,000\,000 \div 18 = 400\,000\,000$M1
0.2 km = 20 000 cm, or $\sqrt{400\,000\,000}$M1
$1 : 20\,000$A1
Answer: $1 : 20\,000$
Examiner tip: The area ratio is the square of the length ratio – take the square root before writing the scale.

14Grade 1–3FoundationNon-calculator4 marks

The diagram shows a scale plan of Mia's garden, drawn on a centimetre grid.

The scale of the plan is 1 cm represents 2 m.

(a) Work out the real length of the longer side of the shed. [1]

(b) Mia wants to add a circular pond with a real diameter of 3 m. What will the diameter of the pond be on the plan? [1]

(a) 6 mB1
(b) 1.5 cm (accept 15 mm)B1
(c) correct method for the whole perimeter, e.g. plan perimeter $5+2+3+3+8+5$ (= 26 cm), or $26 \times 2$, or all six real side lengths (10, 4, 6, 6, 16, 10 m) foundM1
(c) 52 mA1
Answer: (a) 6 m (b) 1.5 cm (c) 52 m
Examiner tip: The L-shaped lawn has six sides – count every edge, including the two short ones next to the shed.

15Grade 4–5Calculator5 marks

The diagram shows part of a map drawn on a centimetre grid.

The scale of the map is $1 : 20\,000$.

A drone flies in a straight line from A to B.

(a) Work out the real distance the drone flies. Give your answer in kilometres. [3]

A van drives from A to B along the road shown.

(b) How much further, in metres, does the van travel than the drone? [2]

(a) $6^2 + 8^2$ (= 100) (horizontal 6 cm and vertical 8 cm used in Pythagoras), or real lengths first, e.g. $1.2^2 + 1.6^2$ (km) or $1200^2 + 1600^2$ (m)M1
$\sqrt{100} = 10$ cm and $10 \times 20\,000$ (= 200 000 cm), or $\sqrt{1.2^2 + 1.6^2}$ oeM1
2 kmA1
(b) $(6 + 8) \times 20\,000$ (= 280 000 cm or 2.8 km), or $(14 - 10) \times 20\,000$M1
800 m (ft their (a) if M1 earned)A1
Answer: (a) 2 km (b) 800 m
Examiner tip: Find the map distance first (10 cm) and convert once at the end – converting every step invites slips.

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