AQA AS Level Mathematics Paper 2, June 2025: Question 15

2 marks · Easy difficulty · Short Answer

Find the unknown frequency density $k$ in a histogram given that the frequency of one class interval is double that of another.

Practise this question

Question

A histogram graph with frequency density on the vertical axis and variable x on the horizontal axis. It shows two bars: one extending from x = 10 to x = 30 with a height of 0.7, and an adjacent taller bar extending from x = 30 to x = 35 with an unknown height k. The text specifies that the frequency density for 10 < x ≤ 30 is 0.7, the frequency density for 30 < x ≤ 35 is k, and the frequency for the class 30 < x ≤ 35 is double the frequency for 10 < x ≤ 30. The question asks to find the value of k.
Question text

15 The diagram shows part of a histogram illustrating the distribution of a variable x

k

Frequency

density

0.7

O 10 30 35 x

The frequency density for the class 10 < x ≤ 30 is 0.7

The frequency density for the class 30 < x ≤ 35 is k

The frequency for the class 30 < x ≤ 35 is double the frequency for the class

10 < x ≤ 30

Find the value of k

[2 marks]

Mark scheme

Show the mark scheme Mark scheme for Question 15 worth 2 marks. Method mark M1 is awarded for using area to obtain 14 or 5k, possibly implied by 28 or 2.5k. Accuracy mark A1 is awarded for obtaining 28/5 or equivalent, with a note 'Do not ISW'. The typical solution shows: 5k = 2 * 20 * 0.7, leading to 5k = 28, so k = 28/5.

Q Marking instructions AO Marks Typical solution

15 Uses area to obtain 14 or 5k 1.1a M1

PI by 28 or 2.5k 5k = 2 × 20 × 0.7

5k = 28

28 1.1b A1 28

Obtains k =

ACF

Do not ISW

Question 15 Total 2

How to answer it

Histogram Frequency Density and Area Relationships

📋 What this question tests

This question assesses your understanding of the fundamental principle of histograms: Frequency is proportional to Area ( Frequency = Class Width × Frequency Density ). You must correctly identify class widths from continuous boundary data, set up an algebraic equation relating the frequencies of two classes, and solve for an unknown frequency density height.

Question 15 (Total: 2 Marks)

Finding the unknown frequency density k

💡 Key Knowledge

  • Histogram Core Formula:
    Area = Frequency (when standard scale is used)
    Frequency = Class Width × Frequency Density
  • Class Width calculation:
    For 10 < x ≤ 30 : Width = 30 − 10 = 20
    For 30 < x ≤ 35 : Width = 35 − 30 = 5
  • Ratio / Comparison: If frequency of class B is double class A, then:
    Area(B) = 2 × Area(A)

📐 Step-by-Step Calculation

  1. 1 Find Area / Frequency of first class:
    Width = 30 − 10 = 20
    Frequency Density = 0.7
    Frequency₁ = 20 × 0.7 = 14
  2. 2 Express Area / Frequency of second class:
    Width = 35 − 30 = 5
    Frequency Density = k
    Frequency₂ = 5 × k = 5k
  3. 3 Form and solve the equation:
    Since Frequency₂ is double Frequency₁:
    5k = 2 × 14
    5k = 28
    k = 28/5 = 5.6

✅ Correct Answer & Mark Breakdown

k = 28/5 (or 5.6, or 5 3/5)

M1 (Method, AO 1.1a):
Awarded for using area to obtain 14 or 5k .
Implied by seeing 28 (the doubled frequency) or 2.5k .
A1 (Accuracy, AO 1.1b):
Obtains 28/5 or equivalent fraction/decimal form (ACF = Any Correct Form).
Note: "Do not ISW" (Ignore Subsequent Working) applies — do not incorrectly round or alter your final exact value!

🧠 Exam Technique

  • Always write down class widths: Don't try to compute everything in your head; explicitly write 30 − 10 = 20 and 35 − 30 = 5 .
  • Leave answers as exact fractions: 28/5 is clean, exact, and fast to leave without risk of rounding errors.
  • Visual sense check: Look at the diagram. Class 2 is much narrower (width 5 vs 20), but has double the total area, so the bar must be significantly taller than 0.7. 5.6 makes complete visual sense.

❌ Common Errors to Avoid

  • Confusing Height with Frequency: Treating the vertical axis as frequency directly (i.e. writing k = 2 × 0.7 = 1.4 ). Remember: on a histogram, vertical axis is always Frequency Density!
  • Inverting the "Double" Relationship: Writing 14 = 2 × (5k) instead of 5k = 2 × 14 , resulting in k = 1.4 . Read carefully: "class 30 < x ≤ 35 is double class 10 < x ≤ 30".
  • Class Width subtraction slip: Using the coordinate value (e.g. 30 or 35) as the class width instead of finding the difference between upper and lower class boundaries.

Topics

Statistics · L: Data presentation and interpretation

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