AQA GCSE Mathematics Paper 2 (Foundation), June 2025: Question 18

4 marks · Easy difficulty · Short Answer

Solve three short ratio problems: convert a fraction to a ratio, simplify an algebraic ratio into the form n : 1, and find the value of x in an equivalent ratio equation.

Practise this question

Question

Question 18 with three parts. Part (a) states: Sam uses either the bus or the train to get to work. (number of days using bus) / (number of days using train) = 1/8. Write down the ratio number of days using bus : number of days using train. Part (b) asks to write the ratio 6a : 5a in the form n : 1 where n is a decimal, worth 2 marks. Part (c) gives the ratio equation 1 : x = x : 4 and asks to work out the value of x, worth 1 mark.

Mark scheme

Show the mark scheme Mark scheme for Question 18. Part (a) gives 1 : 8 or 0.125 : 1 (or equivalent like 0.25 : 2 or 2 : 16) for B1; 8 : 1 scores B0. Part (b) gives 1.2 (: 1) for B2, with B1 for 1.2a or 6/5 or 1 1/5 or 6 divided by 5. Part (c) gives 2 for B1 (accept -2), with additional guidance accepting clear indication such as 1 : 2 = 2 : 4 in working space.

How to answer it

Ratio Forms and Solving Equivalent Ratios

WHAT THIS QUESTION TESTS

This three-part question assesses your core fluency with ratios across three distinct skills:

  • Part (a): Converting a fractional relationship into a standard ratio expression A : B .
  • Part (b): Simplifying an algebraic ratio and expressing it in unit form n : 1 , where n is a decimal.
  • Part (c): Solving an algebraic proportion (equivalent ratio equation) involving an unknown term.

Question 18 (a)

Writing a fraction as a ratio [1 mark]

✅ Correct Answer

Answer: 1 : 8

Also acceptable: 0.125 : 1 , 2 : 16 , or any other equivalent ratio in the correct order.

💡 Key Knowledge

A fraction represents a comparison of two quantities:

Numerator / Denominator = Numerator : Denominator

The top of the fraction is the first part of the ratio; the bottom is the second part.

❌ Common Errors

  • Reversing the order: Writing 8 : 1 scores 0 marks. The question explicitly asks for bus : train, matching the fraction bus / train = 1 / 8.

🧠 Exam Technique

Always match the labels directly:
• Numerator = "number of days using bus" = 1
• Denominator = "number of days using train" = 8
Therefore, bus : train = 1 : 8.

Mark Scheme: [B1] for 1 : 8 or equivalent ratio (e.g. 0.125 : 1 , 2 : 16 ).

Question 18 (b)

Expressing an algebraic ratio in the form n : 1 [2 marks]

✅ Correct Answer

Answer: 1.2 (given on the line as 1.2 : 1 )

📐 Step-by-Step Calculation

Step 1: Cancel the common factor a from both sides.
6a : 5a = 6 : 5
Step 2: To make the right-hand side equal to 1 , divide both sides by 5 .
(6 ÷ 5) : (5 ÷ 5) = 1.2 : 1
Step 3: Check the condition: n must be a decimal.
n = 1.2 (terminating decimal).

🧠 Exam Technique & Partial Marks

Read the exact form requested: "where n is a decimal".

A partial credit mark [B1] is awarded for finding an unsimplified or fractional equivalent:
• 1.2a
• 6/5 or 1 1/5
• Showing the division 6 ÷ 5

❌ Common Errors

  • Leaving the fraction: Writing 6/5 : 1 loses the final accuracy mark because the question specifically asked for a decimal.
  • Leaving the variable: Writing 1.2a : 1 forgets that a divides out completely.
Mark Scheme: [B2] for fully correct 1.2 on the answer line. [B1] for 1.2a , 6/5 , 1 1/5 , or 6 ÷ 5 .

Question 18 (c)

Solving equivalent ratios [1 mark]

✅ Correct Answer

Answer: x = 2 (or -2 )

📐 Step-by-Step Calculation

Given the statement 1 : x = x : 4 :

Method 1: Fractions
Write as equal fractions: 1 / x = x / 4
Cross-multiply: x × x = 1 × 4
x² = 4 → x = 2
Method 2: Scale Factor
To go from 1 to x , multiply by x .
Applying this to the second part: x × x = 4 , so x = 2 .

🧠 Examiner Insight

Examiners will accept a clear indication of the answer in your working (for instance, writing 1 : 2 = 2 : 4 ) even if you forget to write 2 on the final answer line!

❌ Common Errors

  • Treating it as addition: Thinking the difference between terms is constant (e.g. guessing x = 2.5 because it is halfway between 1 and 4). Ratios are based on multiplication, not addition.
  • Arithmetic slips: Forgetting to take the square root and leaving the answer as 4 .
Mark Scheme: [B1] for 2 (accept -2 ).

Topics

Ratio, proportion and rates of change · Algebra · 3.3 Ratio, proportion and rates of change · 3.2.3 Solving equations and inequalities

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