AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 12

4 marks · Medium difficulty · Multi-step Problem

Estimate the percentage increase in Rob's mean driving time compared to last year using grouped frequency data.

Practise this question

Question

A grouped frequency table with two columns: 'Time, t (minutes)' and 'Frequency'. The classes are 20 ≤ t < 25 with frequency 16, 25 ≤ t < 30 with frequency 32, 30 ≤ t < 40 with frequency 24, and 40 ≤ t < 60 with frequency 8, giving a total frequency of 80. The text below states that last year, the mean time Rob took to drive to work was 25 minutes. The question asks to estimate the percentage increase in the mean driving time for these 80 days for 4 marks.

Mark scheme

Show the mark scheme Mark scheme for question 12 showing a 4-mark allocation: M1 for finding the sum of midpoints multiplied by frequencies (22.5 × 16 + 27.5 × 32 + 35 × 24 + 50 × 8 = 2480); M1 for dividing by 80 to find the estimated mean of 31; M1 for the percentage increase calculation ((31 - 25) ÷ 25 × 100); A1 for the final answer of 24%. Includes an additional guidance section showing marking for common errors.

How to answer it

Estimating the Mean from Grouped Data & Percentage Increase

📋 What this question tests

This multi-step question assesses your ability to:

  • Find the midpoint of unequal grouped class intervals.
  • Calculate an estimate of the mean using Σ(midpoint × frequency) ÷ total frequency .
  • Calculate the percentage increase from an original baseline value.

Question 12 Breakdown (4 Marks)

A multi-step problem connecting statistics and percentages

💡 Key Knowledge

  • Class Midpoint: Add the lower and upper bounds and divide by 2.
    e.g., for 30 ≤ t < 40: (30 + 40) ÷ 2 = 35 .
  • Unequal Class Widths: Notice that the class widths are 5, 5, 10, and 20. Do not assume all intervals are the same size!
  • Percentage Increase:
    (Difference ÷ Original) × 100

🧠 Exam Technique

  • Annotate the table: Add two extra columns directly on the exam paper — one for Midpoint (m) and one for m × f .
  • Show full substitution: Even if you make an arithmetic error, writing total ÷ 80 guarantees method marks.
  • Identify the original: The question states last year's mean was 25 minutes. 25 is your base (denominator) for percentage change.

📐 Step-by-Step Solution

Step 1: Find midpoints and calculate products (Midpoint × Frequency)

Time interval, t Frequency (f) Midpoint (m) m × f
20 ≤ t < 25 16 22.5 22.5 × 16 = 360
25 ≤ t < 30 32 27.5 27.5 × 32 = 880
30 ≤ t < 40 24 35.0 35 × 24 = 840
40 ≤ t < 60 8 50.0 50 × 8 = 400
Total 80 — Σ(m × f) = 2480
awarded M1 for products using midpoints (allows at most 1 incorrect midpoint) or total of 2480

Step 2: Calculate the estimated mean for these 80 days

Estimated Mean = 2480 ÷ 80 = 31 minutes

awarded M1 for dividing their total sum by 80

Step 3: Calculate the percentage increase from last year (25 minutes)

Increase = 31 − 25 = 6 minutes

Percentage Increase = (6 ÷ 25) × 100 = 24%

Alternative method: (31 ÷ 25) × 100 = 124%, so increase is 124% − 100% = 24%

awarded M1 for setting up (their 31 − 25) ÷ 25 × 100 (where their mean > 25)

✅ Final Answer

24%

awarded A1 (accurate final value)

❌ Common Errors to Avoid

  • Unequal intervals trap: Thinking midpoints go up in 5s (e.g., guessing 32.5 or 37.5 for the last two rows). Always calculate (lower + upper) ÷ 2 !
  • Dividing by 4: Dividing 2480 by 4 (the number of rows) instead of 80 (the total frequency).
  • Wrong base in percentage: Dividing the change (6) by 31 instead of the original value (25). Remember: Change ÷ Original .
  • Using class boundaries: Multiplying lower or upper bounds instead of midpoints loses the first method mark.

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.