AQA GCSE Mathematics Paper 1 (Higher), June 2025: Question 13

2 marks · Easy difficulty · Short Answer

Find the difference between two consecutive cube numbers that lie on either side of 4.5 cubed.

Practise this question

Question

Question 13 states: 'c and d are consecutive cube numbers, where c < 4.5³ < d. Work out the value of d − c' with an answer line at the bottom, worth 2 marks.
Question text

13 c and d are consecutive cube numbers, where c < 4.53 < d

Work out the value of d – c

[2 marks]

Answer

Mark scheme

Show the mark scheme Mark scheme for Question 13: Final answer 61 awarded B2. B1 is awarded for identifying c = 4³ (or 64) and d = 5³ (or 125), or a correctly calculated difference of two other consecutive cubes x³ − (x − 1)³. Special credit SC1 is given for −61.

Q Answer Mark Comments

61 B1 (c =) 43 oe or (c =) 64

and (d =) 53 oe or (d =) 125

or

B2 3 3

their answer comes from x – (x – 1)

correctly calculated, where x is an integer

13 ≠ 5 eg answer 91 from 216 – 125

SC1 –61

Additional Guidance

SC1 is for c – d

Do not allow 64 or 125 seen in a list if not identified or used

How to answer it

Difference Between Consecutive Cube Numbers

📌 What this question tests

This question assesses your ability to:

  • Understand the concept of consecutive integers and cube numbers.
  • Interpret inequality notation involving powers: c < 4.5³ < d .
  • Identify which integer cubes surround a non-integer base cubed.
  • Perform accurate subtraction without a calculator.
Question 13 • 2 Marks

Full Question Walkthrough & Strategy

Finding the value of d − c where c < 4.5³ < d

💡 Key Knowledge

  • Cube number: The result of multiplying an integer by itself three times (e.g. n³ = n × n × n).
  • First few cubes:
    1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 6³ = 216.
  • Consecutive: Numbers that follow each other in order, without gaps (e.g. 4 and 5).
  • Since 4.5 lies strictly between the consecutive integers 4 and 5, 4.5³ must lie strictly between 4³ and 5³.

📐 Step-by-Step Calculation

  1. Identify the surrounding integers:
    4 < 4.5 < 5
  2. Apply the cubes:
    4³ < 4.5³ < 5³
  3. Calculate values of c and d:
    c = 4³ = 4 × 4 × 4 = 64
    d = 5³ = 5 × 5 × 5 = 125
  4. Subtract c from d:
    d − c = 125 − 64 = 61

✅ Final Answer & Mark Allocation

Answer: 61

Mark Scheme Breakdown:
  • B1: Identifies c = 4³ (or 64) AND d = 5³ (or 125), OR evaluates x³ − (x − 1)³ correctly for any other integer x.
  • B2: Correct final answer of 61.
  • Special Case (SC1): Awarded for −61 (calculating c − d instead of d − c).

🧠 Exam Technique

  • Do NOT calculate 4.5³: You do not need to work out 4.5 × 4.5 × 4.5 (which is 91.125). That wastes valuable exam time!
  • Look at the base numbers: The base 4.5 directly tells you that the consecutive whole numbers on either side are 4 and 5.
  • Order matters: Check the question carefully—it asks for d − c (larger minus smaller), not c − d .

❌ Common Errors & Pitfalls

  • Listing without identifying: Writing out a list of cube numbers (e.g. 1, 8, 27, 64, 125...) does not gain marks unless 64 and 125 are explicitly chosen or used in the subtraction.
  • Subtracting the wrong way round: Calculating c − d = 64 − 125 = −61 loses a mark (only gets SC1).
  • Confusing square and cube numbers: Writing 4² = 16 and 5² = 25 instead of 4³ and 5³.
  • Arithmetic slips: Making a column subtraction error when working out 125 − 64 (e.g. writing 71 or 51). Double-check your column subtraction!

Topics

Number · 3.1.1 Structure and calculation

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 1 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.