AQA GCSE Mathematics Paper 2 (Higher), June 2025: Question 19

4 marks · Medium difficulty · Multi-step Problem

Use the histogram showing the ages of 500 gym members to estimate the total annual fees paid, given that members under 65 pay £275 and members 65 or over pay £129.

Practise this question

Question

A histogram titled 'Ages of Members' with 'Age (years)' on the horizontal axis from 0 to 80 and 'Frequency density' on the vertical axis from 0 to 15. Five bars are shown: 16 to 20 with height 11.5, 20 to 30 with height 13.2, 30 to 40 with height 14.6, 40 to 60 with height 6, and 60 to 80 with height 2.8. A table gives annual fees: £275 for 'Under 65' and £129 for '65 or over'. The question asks to estimate the total annual fees paid by these members.

Mark scheme

Show the mark scheme Mark scheme for Question 19: M1 for (80 - 65) x 2.8 or (65 - 60) x 2.8; A1 for 42 (frequency >= 65) or 14 (frequency 60 to 65); M1dep for their 42 x 129 (5418) or their 458 x 275 (125950); A1 for 131 368. Includes special cases and additional guidance.

How to answer it

Estimating Total Membership Fees from a Histogram

📋 What this question tests

This 4-mark problem assesses your ability to interpret histograms with unequal class widths, calculate frequencies using the relation Frequency = Class Width × Frequency Density , split an interval to estimate a subgroup frequency, and apply two-tier pricing to find a total sum of money.

Question 19 Walkthrough & Strategy

Total Marks: 4

💡 Key Knowledge

  • Core formula: Frequency = Class Width × Frequency Density (Area of bar represents frequency).
  • Reading the scale: On the vertical axis, each small square = 0.2 units. On the horizontal axis, each small square = 1 year.
  • Splitting a bar: When a boundary (age 65) falls inside a bar (60 to 80), calculate the proportion by splitting the width:
    • Age 60 to 65: width = 5
    • Age 65 to 80: width = 15
  • Total frequency shortcut: The question states there are 500 members in total. Once you know members aged 65+, members under 65 = 500 − (members aged 65+) .

🧠 Exam Technique

  • Work smart to save time: Instead of calculating the frequency of every single bar to get the members under 65, calculate just the 65+ group and subtract from 500!
  • Show all sub-totals clearly: Writing down 42 or 14 locks in the first 2 marks even if you make an arithmetic error later.
  • Check total sense: If total members = 500, make sure your split groups add up to 500.

📐 Step-by-Step Calculation

  1. Read the height of the 60 to 80 bar:
    The bar extends from 60 to 80 with a height of 2.8 (14 small squares up).
  2. Find the frequency of members aged 65 or over:
    Interval = 65 to 80 → Width = 80 − 65 = 15
    Frequency (65+) = 15 × 2.8 = 42 members
    [Awards M1 A1]
  3. Find the frequency of members aged under 65:
    Method A (Fast): Total members = 500 → Under 65 = 500 − 42 = 458 members
    Method B (Verification via all bars):
    • 16 to 20: 4 × 11.5 = 46
    • 20 to 30: 10 × 13.2 = 132
    • 30 to 40: 10 × 14.6 = 146
    • 40 to 60: 20 × 6.0 = 120
    • 60 to 65: 5 × 2.8 = 14
    Sum = 46 + 132 + 146 + 120 + 14 = 458
  4. Calculate the total fees:
    • Fee for 65 and over: 42 × £129 = £5,418
    • Fee for under 65: 458 × £275 = £125,950
    [Awards M1dep]
    • Total Fee = £5,418 + £125,950 = £131,368
    [Awards A1]

❌ Common Errors & Pitfalls

  • Misreading the scale: Reading 2.8 as 2.4 or 3.4 by miscounting small grid squares (remember 5 squares = 1 whole unit on the vertical axis, so 1 square = 0.2).
  • Using height as frequency: Simply multiplying the height 2.8 by £129 instead of finding area/frequency first.
  • Incorrect bar width: Using width = 5 for the 65+ portion instead of 80 − 65 = 15 .
  • Forgetting to sum: Finding £5,418 and £125,950 but forgetting to add them together for the grand total.

✅ Final Answer & Mark Allocation

£131,368

Mark Scheme Breakdown:
  • M1: For (80 − 65) × 2.8 or (65 − 60) × 2.8
  • A1: For calculating 42 (members ≥ 65) or 14 (members 60–64)
  • M1 (dep): For calculating fee correctly: their 42 × 129 (£5,418) or (500 − their 42) × 275 (£125,950)
  • A1: Correct total of 131 368

Topics

Statistics · 3.6 Statistics

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 2 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.