AQA GCSE Mathematics Paper 3 (Foundation), June 2025: Question 9

4 marks · Easy difficulty · Short Answer

Find the smallest and largest whole numbers whose squares are two-digit numbers, and determine with a reason whether 0.8² or 0.8³ is larger.

Practise this question

Question

Question 9 consists of two parts. Part (a) states: 'The square of a whole number is a two-digit number. Work out the smallest number that could have been squared and the largest number that could have been squared.' Answer lines are provided for 'smallest' and 'largest', worth 3 marks. Part (b) asks: 'Which has the larger value 0.8² or 0.8³? Tick a box.' Two tick boxes are shown for 0.8² and 0.8³, followed by 'Give a reason for your answer', worth 1 mark.

Mark scheme

Show the mark scheme Mark scheme for Question 9. Part (a): B3 for 4 and 9 in correct positions. Partial credit B2 for reversed order, or only 4, or only 9, or 4² and 9², or 16 and 81; B1 for 4², 16, √16, 9², 81, or √81 seen. Part (b): B1 for ticking 0.8² with valid evaluation (0.64 and 0.512, or comparable fractions) or correct reasoning such as 0.8 < 1 or multiplying decimals below 1 makes numbers smaller.

How to answer it

Squares, Digits & Decimals Raised to Powers

📋 What this question tests

This question evaluates your foundational understanding of square numbers, place value constraints, and the behaviour of numbers between 0 and 1 when raised to powers.

  • Square numbers & place value: Identifying square numbers that produce a 2-digit result (values from 10 to 99).
  • Interpreting question wording: Finding the base number that was squared rather than writing the square itself.
  • Powers of decimals (< 1): Understanding why multiplying a number between 0 and 1 by itself results in a smaller value.
Part (a) • 3 Marks

Two-Digit Square Numbers

Find the smallest and largest whole numbers whose squares are two-digit numbers.

📐 Step-by-Step Calculation

  1. List the whole numbers and their squares:
    • 1² = 1 (1 digit)
    • 2² = 4 (1 digit)
    • 3² = 9 (1 digit)
    • 4² = 16 (2 digits) → Smallest two-digit square
    • 5² = 25, 6² = 36, 7² = 49, 8² = 64
    • 9² = 81 (2 digits) → Largest two-digit square
    • 10² = 100 (3 digits)
  2. Identify the required values:
    The question asks for the number that was squared, NOT the square itself.
    Smallest = 4
    Largest = 9

✅ Correct Answer

smallest: 4

largest: 9

Mark Scheme Breakdown (3 marks total):
• B3: 4 and 9 in correct positions.
• Partial credit (B2): 9 in smallest and 4 in largest; OR only 4 correct; OR only 9 correct; OR writing 4² and 9²; OR writing 16 and 81.
• Partial credit (B1): Mention of any one of 4², 16, √16, 9², 81, or √81.

🧠 Exam Technique

  • Read the prompt precisely: The question asks for the number that could have been squared. This means the base number (4 and 9), not the product (16 and 81).
  • If you write 16 and 81 on the answer line, you drop 1 mark (capping you at B2).

❌ Common Errors

  • Writing the squares instead: Putting 16 and 81 on the answer line instead of 4 and 9.
  • Swapping the positions: Writing 9 as smallest and 4 as largest (scores B2 instead of B3).
  • Confusing the base and power: Thinking 2² = 4 means 2 is the smallest, forgetting that 4 has only 1 digit.
Part (b) • 1 Mark

Comparing Powers of a Decimal

Which has the larger value: 0.8² or 0.8³? Tick a box and give a reason.

✅ Correct Answer & Accepted Reasons

Tick the box for: 0.8²

Any one of the following reasons:

  • By evaluation: 0.64 and 0.512 (showing 0.64 > 0.512).
  • By fractions: 64/100 and 64/125 (or matching denominators: 80/125 and 64/125 ).
  • By inequality/property: Stating that 0.8 < 1 .
  • In words: "Multiplying by a number less than 1 makes it smaller."
B1: 0.8² ticked with a correct reason or working in comparable form.

💡 Key Knowledge

  • For any positive number x > 1:
    Higher powers produce larger numbers (e.g., 2² = 4, 2³ = 8; 8 > 4).
  • For any positive number 0 < x < 1:
    Higher powers produce smaller numbers (e.g., 0.8² = 0.64, but 0.8³ = 0.512; 0.64 > 0.512).
  • Multiplying by a decimal between 0 and 1 finds a fraction/part of the number, decreasing its size.

📐 Calculation Check

0.8² = 0.8 × 0.8 = 0.64

0.8³ = 0.64 × 0.8 = 0.512

Comparing: 0.640 > 0.512, so 0.8² is definitely larger.

❌ Common Errors

  • False generalisation: Writing "squaring always makes numbers bigger" (scores 0 marks because it is mathematically false for decimals < 1).
  • Incomparable fractions: Writing 16/25 and 64/125 without converting them to a common denominator or common numerator (scores 0 marks).
  • Ticking 0.8³: Assuming higher powers always give larger results.

Topics

Number · 3.1.1 Structure and calculation

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 3 (Foundation), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.