AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 10

2 marks · Medium difficulty · Short Answer

Work out the value of x given a Venn diagram with sets A and B, where n(A only) = 30, n(A ∩ B) = 20, n(outside) = 6, and P(A) = 1/2.

Practise this question

Question

Venn diagram showing universal set ξ containing two overlapping circles, A and B. The region in set A only contains the number 30, the intersection of A and B contains 20, the region in set B only contains the letter x, and the region outside both circles contains 6. The question states that P(A) = 1/2 and asks to work out the value of x.

Mark scheme

Show the mark scheme Mark scheme for question 10 showing: M1 for 2 × (30 + 20) or 100, or 20 + 30 - 6, or 50 - 6, or x + 6 = 50 (or equivalent equation); A1 for 44, which may be seen in the correct position on the diagram. Additional guidance notes that M1 may be awarded for correct work with no answer or incorrect answer.

How to answer it

AQA GCSE Mathematics • Higher & Foundation

Venn Diagrams: Finding an Unknown Frequency from Probability

What this question tests

  • Interpreting frequencies inside regions of a Venn diagram (subsets, intersection, and complement).
  • Applying the definition of probability: P(event) = Total in event ÷ Universal Total .
  • Setting up and solving a simple linear equation to find an unknown frequency ( x ).

Question 10 Walkthrough

Total marks: 2

📐 Step-by-Step Solution

Method 1: Using the Total Frequency

  1. Find total items in set A:
    Set A includes the elements only in A and the intersection.
    n(A) = 30 + 20 = 50
  2. Find the universal total (ξ):
    Since P(A) = 1/2 , set A represents half of everything.
    Total = 50 × 2 = 100 [M1]
  3. Set up the equation for all regions:
    30 + 20 + x + 6 = 100
    56 + x = 100
  4. Solve for x:
    x = 100 - 56 = 44 [A1]

Method 2: Comparing Halves Directly

If set A is half the diagram ( 50 items), then the region outside A must also equal 50 items:

x + 6 = 50 [M1]
x = 50 - 6 = 44 [A1]

💡 Key Knowledge

  • Total in a set: A circle labeled A consists of two parts: the region unique to A ( 30 ) and the shared intersection A ∩ B ( 20 ). Always add both!
  • Universal set (ξ): The rectangle contains every single item. Sum of all regions = total items.
  • Probability link:
    P(A) = n(A) ÷ n(ξ)
    If P(A) = 1/2 , then n(ξ) = 2 × n(A) .

✅ Final Answer

x = 44

Mark Scheme:
• M1: For 2 × (30 + 20) , 100 , 50 - 6 , or setting up x + 6 = 50 .
• A1: Correct answer of 44 (also credited if clearly written in the x region on the diagram).

❌ Common Errors

  • Ignoring the intersection: Thinking set A contains only 30 items instead of 30 + 20 = 50 .
  • Forgetting the outside items: Overlooking the 6 items sitting outside both circles when calculating the total.
  • Confusing values with elements: The numbers in the diagram represent counts (cardinalities), not individual data values.

🧠 Exam Technique

  • Show your working clearly: Writing down 100 or x + 6 = 50 secures the method mark ( M1 ) even if an arithmetic slip happens later.
  • Sanity check: Put your answer back in:
    Total = 30 + 20 + 44 + 6 = 100 .
    Set A = 50 .
    50 / 100 = 1/2 . It checks out perfectly!

Topics

Probability · Algebra · 3.5 Probability · 3.2.3 Solving equations and inequalities

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.