AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 5

3 marks · Medium difficulty · Multi-step Problem

Calculate the number of matches lost using a pie chart showing that 30 matches were won (180°), with 132° representing matches lost.

Practise this question

Question

A pie chart showing results of matches: Won, Lost, and Drew. A vertical diameter divides the circle into two halves; the left half is labelled 'Won'. The right half is divided into two sectors: 'Lost' with an angle of 132 degrees and 'Drew' with an angle of 48 degrees. The text states that 30 matches were won and asks to find how many matches were lost.

Mark scheme

Show the mark scheme Mark scheme with 3 alternative methods for finding 22 lost matches. Method 1 calculates 132/180 times 30 (M1, M1dep, A1). Method 2 finds degrees per match as 180 divided by 30 equals 6, then calculates 132 divided by 6 (M1, M1dep, A1). Method 3 calculates matches per degree as 30/180 times 132 (M1, M1dep, A1). Special case SC1 is awarded for an answer of 11 (assuming 30 total matches).

How to answer it

Pie Chart Proportions: Finding Quantities from Angles

📋 What this question tests
  • Interpreting pie charts: Understanding that the total angle in a circle is 360° and a straight line through the centre forms a semi-circle of 180°.
  • Direct proportion: Finding the constant relationship between degrees and the frequency (matches).
  • Unitary method: Calculating either "degrees per match" or "matches per degree" to find unknown quantities.
Question 5 (3 Marks)

Calculating the Number of Lost Matches

Given: Sector 'Lost' = 132°, Sector 'Drew' = 48°, Sector 'Won' = 30 matches

💡 Key Knowledge

  • A full circle contains 360°.
  • The vertical line passes straight through the centre, dividing the pie chart in half.
  • Therefore, the angle for Won = 180° (or 360° − 132° − 48° = 180° ).
  • In any pie chart, frequency is directly proportional to sector angle:
    Angle ÷ Frequency = Degrees per item

📐 Step-by-Step Calculation

  1. Find the angle for 'Won':
    Angle for Won = 180°
  2. Find degrees per match:
    180° ÷ 30 matches = 6° per match
  3. Calculate matches lost:
    The angle for lost is 132°.
    132° ÷ 6° = 22 matches
Alternative Method (Fractions):
Fraction of circle won = 180/360 = 1/2.
Total matches = 30 × 2 = 60.
Matches lost = (132/360) × 60 = 22

✅ Correct Answer & Marks

  • Final Answer: 22
  • [M1]: Correct method to establish relationship, e.g. 180 ÷ 30 (= 6) or 132/180 or 30 × 2 (= 60 total matches).
  • [M1dep]: Fully correct method to find the lost matches, e.g. 132 ÷ 6 or (132/180) × 30 or (132/360) × 60 .
  • [A1]: Accurate final integer value of 22.

🧠 Exam Technique & Examiner Insight

  • Read carefully: The question states "30 matches were won", NOT that there were 30 matches in total!
  • Sanity check: 132° is a bit less than 180° (which is 30 matches), and much more than half of 180° (15 matches). An answer of 22 is completely sensible!
  • Show intermediate values: Writing down 180 ÷ 30 = 6 secures method marks even if you make an arithmetic slip later.

❌ Common Misconceptions & Traps

  • Assuming 30 was the total number of matches:
    Many students calculated (132 ÷ 360) × 30 = 11 . The mark scheme only awards a Special Case SC1 for an answer of 11.
  • Dividing 360 by 30 directly:
    Calculating 360 ÷ 30 = 12 scores 0 marks (M0) because 30 only represents the 180° half of the pie chart, not the full 360°.
  • Premature rounding:
    Finding matches per degree as 30 ÷ 180 = 0.1666... and rounding to 0.17 before multiplying gives 0.17 × 132 = 22.44 . Rounding this to 22 loses the accuracy mark (M1 M1 A0) because of incorrect working. Always keep exact values or work with degrees per match ( 180 ÷ 30 = 6 ) which gives a clean whole number!

Topics

Statistics · Ratio, proportion and rates of change · 3.6 Statistics · 3.3 Ratio, proportion and rates of change

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.