AQA GCSE Mathematics Paper 3 (Foundation), June 2025: Question 13

3 marks · Easy difficulty · Multi-step Problem

Work out the ratio of the area of a rectangle to the area of a triangle drawn on a square grid.

Practise this question

Question

A square grid containing two shapes: a rectangle with width 5 squares and height 3 squares, and a right-angled triangle with base 4 squares and height 4 squares. The question asks to work out the ratio of the area of the rectangle to the area of the triangle, with an answer line formatted as 'Answer ___ : ___'.

Mark scheme

Show the mark scheme Mark scheme for question 13: M1 for 3 multiplied by 5 or 15; M1 for 4 multiplied by 4 divided by 2 or 8; A1 for 15 : 8 or equivalent ratio. Additional guidance notes to ignore attempts to simplify after correct answer seen, award M1M1A0 for 8 : 15, and M0 for perimeter calculations such as 5 + 3 = 8 or 4 + 4 = 8.

How to answer it

Ratio of Areas on a Grid

What this question tests

This question evaluates your ability to combine geometric area formulas with ratio representation:

  • Finding the dimensions of 2D shapes by counting unit squares on a grid.
  • Calculating the area of a rectangle ( base × height ).
  • Calculating the area of a right-angled triangle ( ½ × base × height ).
  • Writing a comparison of two quantities as an unsimplified or simplified ratio in the correct order.

Question 13 (3 Marks)

Work out the ratio   area of rectangle : area of triangle

📐 Step-by-Step Calculation

  1. Find the area of the rectangle:
    Count the grid units: Base = 5 units, Height = 3 units.
    Area = 5 × 3 = 15
  2. Find the area of the triangle:
    Count the grid units: Base = 4 units, Vertical Height = 4 units.
    Area = ½ × 4 × 4 = 8
  3. Write down the ratio in the requested order:
    area of rectangle : area of triangle = 15 : 8

✅ Final Answer & Mark Breakdown

Answer: 15 : 8

  • M1: For finding the area of the rectangle ( 3 × 5 or 15 ).
  • M1: For finding the area of the triangle ( 4 × 4 ÷ 2 or 8 ).
  • A1: For the final ratio 15 : 8 (or equivalent ratio).

💡 Key Knowledge

  • Grid Counting: Treat each grid square as 1 unit in length and 1 unit² in area.
  • Triangle Area Formula: Always remember to halve the product: Area = (base × height) ÷ 2 .
  • Ratio Format: A ratio must be written with a colon ( : ) and match the exact order requested in the prompt.

🧠 Exam Technique & Examiner Commentary

  • Double-check order: The question asks for rectangle : triangle. Reversing the order to 8 : 15 loses the final accuracy mark ( M1 M1 A0 ).
  • Simplification: 15 : 8 cannot be simplified using whole numbers. Even if an unnecessary decimal simplification like 1.875 : 1 is attempted, examiners will ignore it once a correct answer is seen.
  • Always show method: Showing 5 × 3 = 15 and 4 × 4 ÷ 2 = 8 ensures method marks even if a copying error occurs later.

❌ Common Errors to Avoid

  • Forgetting to divide by 2 for the triangle: Calculating 4 × 4 = 16 gives an incorrect ratio of 15 : 16 .
  • Adding instead of multiplying: Writing 5 + 3 = 8 or 4 + 4 = 8 scores M0 . Perimeter is not area!
  • Miscounting squares: Always mark off the grid squares with tick marks or count the edges carefully to avoid being off by 1 square.

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.