AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 6

5 marks · Medium difficulty · Multi-step Problem

Use a scale grid to calculate the actual distance between two points in kilometres and determine a three-figure bearing.

Practise this question

Question

A scale diagram showing points A, B, and C plotted on a centimetre grid with a North arrow. The given scale is 1 : 200 000. Question 6(a) asks to work out the actual distance from B to C in kilometres, worth 4 marks. Question 6(b) states that B is South East of A and asks for the bearing of B from A, worth 1 mark.

Mark scheme

Show the mark scheme Mark scheme table for question 6. For part 6(a): B1 for measured distance [5.3, 5.7] cm; M2 for (their distance × 200 000) / (100 × 1000) with M1 for one correct operational step; A1 for an answer in the range [10.6, 11.4] km. For part 6(b): B1 for 135 degrees.

How to answer it

Scale Diagrams, Map Ratios & Bearings

📋 What This Question Tests
  • Reading scale diagrams: Accurately measuring map distances using a ruler on a centimetre grid.
  • Ratio scales: Interpreting representative scales (e.g. 1 : 200 000).
  • Metric conversions: Converting centimetres (cm) into kilometres (km) by dividing by 100 000.
  • Compass directions & bearings: Stating three-figure bearings measured clockwise from North.

Question 6 (a)

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

✅ Acceptable Final Answers

Any value between 10.6 km and 11.4 km inclusive (e.g. 11 km or 10.8 km).

Mark Breakdown:
• B1: Measuring map distance accurately: [5.3 cm to 5.7 cm]
• M1: Multiplying by 200 000 OR dividing by 100 / 1000
• M2: Complete method: (measurement × 200 000) ÷ 100 000
• A1: Correct answer in range [10.6, 11.4]

📐 Step-by-Step Calculation

  1. Measure on the grid: Using a ruler between the crosses at C and B gives 5.4 cm (or 5.5 cm).
  2. Apply the ratio (1 : 200 000):
    Actual cm = 5.4 × 200 000 = 1 080 000 cm
  3. Convert cm to metres:
    1 080 000 ÷ 100 = 10 800 m
  4. Convert metres to kilometres:
    10 800 ÷ 1000 = 10.8 km

Direct shortcut: Divide by 100 000 in one go, because 1 km = 100 000 cm.

💡 Key Metric Conversions

  • 1 m = 100 cm
  • 1 km = 1000 m = 100 000 cm
  • A scale of 1 : 200 000 means:
    1 cm on the map = 200 000 cm in real life = 2 km.
  • Knowing 1 cm = 2 km allows a quick calculation:
    5.4 cm × 2 km/cm = 10.8 km !

❌ Common Calculation Traps

  • Incorrect conversion factor: Dividing by 1 000 instead of 100 000, giving answers like 1 080 km.
  • Counting grid squares carelessly: Points B and C are along a horizontal line, but ensure your ruler aligns with the centres of the × symbols.
  • Stopping at cm: Leaving the answer as 1 080 000 without converting to km loses the final 2 marks.

Question 6 (b)

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

✅ Correct Answer

135°

Acceptable written as 135 or 135°.

Mark Breakdown:
• B1: Standalone mark for 135(°)

💡 Understanding Bearings & Compass Directions

  • Rule 1: Always measured from North (000°).
  • Rule 2: Always measured in a clockwise direction.
  • Rule 3: Always given as a 3-digit number (e.g. 045°, 135°).
  • Compass Points:
    • North = 000° / 360°
    • East = 090°
    • South = 180°
    • South East (SE) is halfway between East and South:
    90° + 45° = 135°

🧠 Exam Technique: "Bearing of B from A"

  • Look at the word "from": you start at Town A.
  • Imagine standing at A facing North (up the page along the arrow).
  • Turn clockwise towards point B. Since B is directly South East, you turn 90° to face East, plus another 45° towards South.
  • No protractor is required because "South East" specifies exactly half of a 90° right angle.

Topics

Ratio, proportion and rates of change · Geometry and measures · 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 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.