AQA GCSE Mathematics Paper 3 (Foundation), June 2025: Question 19

5 marks · Medium difficulty · Multi-step Problem

Calculate the actual distance between two towns from a centimetre grid scale drawing and state the bearing between two locations.

Practise this question

Question

A centimetre grid with towns A, B, and C marked with crosses, accompanied by a North arrow pointing directly upwards and a given scale of 1 : 200 000. Town A is positioned near the centre, town C is lower down to the left, and town B is horizontally aligned with C to the right. Question 19(a) asks for the actual distance from B to C in kilometres for 4 marks. Question 19(b) states that B is South East of A and asks for the bearing of B from A for 1 mark.

Mark scheme

Show the mark scheme Mark scheme table for question 19. Part (a) awards B1 for a measured distance of [5.3, 5.7] cm, M2 for converting using scale and units (their [5.3, 5.7] × 200 000 ÷ (100 × 1000)), and A1 for an actual distance between 10.6 and 11.4 km. Part (b) awards B1 for the bearing 135°.

How to answer it

Scale Diagrams, Unit Conversions & Bearings

📌 WHAT THIS QUESTION TESTS

This question assesses your ability to combine practical measurement with proportional reasoning and geometric navigation:

  • Measuring on a grid: Accurately measuring straight-line lengths using a ruler in centimetres.
  • Ratio & Scale: Multiplying map measurements by a map scale ratio ( 1 : 200 000 ).
  • Metric conversions: Converting centimetres to metres and then to kilometres ( ÷ 100 000 ).
  • Three-figure bearings: Recalling cardinal and intercardinal compass directions and converting them into standard 3-digit clockwise angles from North.

Question 19 (a)

Work out the actual distance from B to C in kilometres [4 marks]

📐 Step-by-Step Calculation

Step 1: Measure map distance C to B
Using a ruler, measure the distance between the center of cross C and cross B.
Map distance = 5.5 cm (Acceptable range: 5.3 cm to 5.7 cm )
Step 2: Apply the map ratio scale
Multiply by 200 000 to find the real-life distance in cm:
5.5 cm × 200 000 = 1 100 000 cm
Step 3: Convert cm to metres
Divide by 100 (since 100 cm = 1 m):
1 100 000 ÷ 100 = 11 000 m
Step 4: Convert metres to kilometres
Divide by 1000 (since 1000 m = 1 km):
11 000 ÷ 1000 = 11 km
Combined shortcut: Divide by 100 000 directly ( 100 × 1000 ).

✅ Correct Answer

Any value in the range:

10.6 km to 11.4 km

(Using 5.5 cm gives exactly 11 km)

Mark Breakdown:
• B1: Measuring map length in range [5.3, 5.7] cm
• M1: One correct operation (e.g. their length × 200 000 or ÷ 100 or 100 × 1000 )
• M1: Fully correct method: (their length × 200 000) ÷ 100 000
• A1: Correct answer in range [10.6, 11.4]

❌ Common Errors

  • Incorrect unit conversion: Dividing by 1000 only (forgetting to convert cm to m first) giving an answer like 1100 km .
  • Counting squares instead of measuring: Points B and C do not lie exactly on grid line intersections, so counting grid intervals leads to inaccuracies.
  • Misreading the ruler: Forgetting that tolerance is strict (±2 mm), so measure carefully between the center of both crosses.

🧠 Exam Technique

  • Write down your measurement first: Writing 5.5 cm secures the B1 mark immediately, even if your later arithmetic has a slip.
  • Remember the master conversion: Memorise that 1 km = 100 000 cm . You can simply cross off five zeros when converting cm to km!

Question 19 (b)

B is South East of A. Write down the bearing of B from A. [1 mark]

✅ Correct Answer

135°

• B1: 135 (or 135° )

💡 Key Knowledge: The 3 Rules of Bearings

  1. Always measured from the North line (000°).
  2. Always measured in a clockwise direction.
  3. Always written using three digits (e.g. 045°, 090°, 135°).

📐 Angle Deduction

  • North = 000°
  • East = 090°
  • South = 180°
  • South East (SE) lies exactly midway between East and South:
    90° + 45° = 135° (or 180° - 45° = 135° )

❌ Common Traps

  • Writing just 45°: 45° is the angle from the East line or South line, not measured clockwise from North!
  • Measuring the wrong direction: The question asks for the bearing of B from A (stand at A, look towards North, rotate clockwise to B).

Topics

Geometry and measures · Ratio, proportion and rates of change · 3.3 Ratio, proportion and rates of change · 3.4.2 Mensuration and calculation

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 3 (Foundation), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.