AQA GCSE Mathematics Paper 1 (Foundation), June 2025: Question 24

1 mark · Medium difficulty · Short Answer

Identify which value of x gives the greatest value of y for the equation y = 1/x.

Practise this question

Question

Question 24 shows the equation y = 1 over x. It asks: 'Which of these values of x gives the greatest value of y? Circle your answer.' The four options given are 9 over 20, 2 over 5, negative 80, and 95.

Mark scheme

Show the mark scheme Mark scheme table for question 24 shows the correct answer as 2 over 5 with 1 mark awarded as B1.

How to answer it

Reciprocal Functions: Finding the Maximum Value

📌 What this question tests

This question assesses your understanding of the reciprocal function y = 1 / x. Specifically, it tests your ability to evaluate the reciprocal of fractions, whole numbers, and negative numbers, and understand how the size of a positive denominator affects the overall value of a fraction.

Question 24

y = 1 / x — Which value of x gives the greatest value of y?

✅ Correct Answer

The correct option to circle is:

2/5

Mark Scheme: B1 for circling or clearly indicating 2/5.

💡 Key Knowledge

  • Reciprocal definition: The reciprocal of any number n is 1 / n.
  • Reciprocal of a fraction: For any fraction a/b, its reciprocal is simply flipped: 1 ÷ (a/b) = b/a.
  • Denominator rule: For positive values of x, the smaller the value of x, the larger the value of 1/x.
  • Negatives: A negative input gives a negative output, which can never be the greatest value when positive options exist.

📐 Step-by-Step Evaluation of Each Option

Substitute each given value of x into the formula y = 1 / x:

  1. Option 1: x = 9/20
    y = 1 ÷ (9/20) = 20/9 = 2.22... (or 2 2/9)
  2. Option 2: x = 2/5
    y = 1 ÷ (2/5) = 5/2 = 2.5 (or 2 1/2)
  3. Option 3: x = -80
    y = 1 ÷ (-80) = -1/80 = -0.0125 (negative value)
  4. Option 4: x = 95
    y = 1 ÷ 95 = 0.0105... (very small positive fraction)

Comparing the outputs: 2.5 > 2.22... > 0.0105 > -0.0125.
Therefore, x = 2/5 produces the largest value of y.

🧠 Exam Technique: The Quick Shortcut

  • Rule out negative numbers: Eliminate -80 immediately since dividing 1 by a negative gives a negative.
  • Rule out large integers: Eliminate 95 because dividing 1 by a large positive number yields a tiny decimal close to 0.
  • Compare the remaining fractions: We want the smallest positive x to get the largest reciprocal:
    • 2/5 = 8/20 = 0.4
    • 9/20 = 0.45
    Since 2/5 < 9/20, the reciprocal of 2/5 will be strictly larger than the reciprocal of 9/20!

❌ Common Traps & Misconceptions

  • Confusing x with y: Students often mistakenly choose 95 because 95 is the largest number in the list. Remember, the question asks which x gives the greatest y, not which x is the largest.
  • Thinking fractions make smaller results: Believing that dividing by a fraction makes a number smaller. In fact, dividing 1 by any proper fraction between 0 and 1 results in a number greater than 1.
  • Comparing fractions incorrectly: Thinking 9/20 is smaller than 2/5 because 2 and 5 are smaller numbers than 9 and 20. Always find a common denominator: 2/5 = 8/20.

Topics

Number · Algebra · 3.1.1 Structure and calculation · 3.1.2 Fractions, decimals and percentages · 3.2.1 Notation, vocabulary and manipulation

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