AQA GCSE Mathematics Paper 3 (Foundation), June 2025: Question 21

3 marks · Medium difficulty · Multi-step Problem

Work out how much money Ary has given Cat's amount, the fractional relationship between Cat and Bea, and the ratio of Ary's amount to Bea's amount.

Practise this question

Question

Question 21: Ary, Bea and Cat each have an amount of money. Cat has £280. Bullet point 1: Cat's amount is 2/3 of Bea's amount. Bullet point 2: Ary's amount to Bea's amount is 5 to 12. Text asks: 'Work out how much money Ary has.' An answer line is provided with a pound sign, worth 3 marks.

Mark scheme

Show the mark scheme Mark scheme for Question 21: Alternative method 1 shows 280 divided by 2 times 3 to get 420 for M1, then their 420 divided by 12 times 5 for M1 dependent, giving an answer of 175 for A1. Alternative method 2 finds the combined ratio parts: 2/3 times 12 = 8 for M1, then 280 divided by their 8 times 5 for M1 dependent, resulting in 175 for A1. Special case SC2 is given for finding 77.77 or 77.78.

How to answer it

Linking Fractions & Ratios: Finding an Unknown Share

📋 What this question tests
  • Reverse fraction calculations: Finding an original whole quantity when a fractional part is known.
  • Ratio division: Using an individual's known value to determine the value of one ratio part.
  • Multi-step problem solving: Connecting two related conditions (Cat-to-Bea fraction and Ary-to-Bea ratio) to find an unknown value.
Question 21 • [3 marks]

Problem Breakdown & Step-by-Step Solutions

Calculate Ary's amount of money given Cat's amount and the relationships between Cat, Bea, and Ary.

📐 Method 1: Work with Money First (Recommended)

Step 1: Find Bea's total money

Cat has £280, which is 2/3 of Bea's amount.

Bea = 280 ÷ 2 × 3 = £420

M1: For 280 ÷ 2 × 3 or obtaining 420

Step 2: Find the value of 1 ratio part

Ary : Bea = 5 : 12. Bea's share corresponds to 12 parts.

1 part = 420 ÷ 12 = £35


Step 3: Calculate Ary's share (5 parts)

Ary = 35 × 5 = £175

M1 (dep): For their 420 ÷ 12 × 5
A1: Final answer of 175

📐 Method 2: Combine the Ratios First

Step 1: Express Cat's share as ratio parts

Bea has 12 parts. Cat has 2/3 of Bea's parts:

Cat's parts = 2/3 × 12 = 8 parts

The three-way ratio is Ary : Bea : Cat = 5 : 12 : 8.

M1: For 2/3 × 12 , obtaining 8, or stating 5 : 12 : 8

Step 2: Find the value of 1 part

Cat's 8 parts equal £280:

1 part = 280 ÷ 8 = £35


Step 3: Calculate Ary's share

Ary = 35 × 5 = £175

M1 (dep): For 280 ÷ 8 × 5
A1: Final answer of 175

✅ Final Answer

Answer: £175 (or simply 175 on the answer line as £ is already printed).

Examiner Insights & Study Tips

💡 Key Knowledge

  • Fraction of an unknown: If A is 2/3 of B, then B = A ÷ 2/3 = A × 3/2.
  • Unitary Method in Ratios: In a ratio A : B, divide the known amount by its corresponding number of parts to find 1 part, then multiply by the other person's parts.
  • Combined Ratio: You can scale quantities into a single unified ratio (Ary : Bea : Cat = 5 : 12 : 8).

🧠 Exam Technique

  • Check reasonableness: Cat has £280 and has less than Bea (since 2/3 < 1). Therefore, Bea must have more than £280 (£420 makes sense; £186.67 does not).
  • Show full working: Even if your final calculation goes wrong, writing 280 ÷ 2 × 3 secures the first method mark (M1).
  • Trace the names: Always label your lines of work clearly (e.g. "Bea =", "1 part =", "Ary =") to avoid confusing who has which share.

❌ Common Errors & Traps

  • Multiplying by 2/3 instead of dividing: Calculating 280 × 2/3 = 186.67 . Cat's amount is 2/3 of Bea's, not the other way around! (Yields an answer of 77.78, which gets a special case SC2, losing a method mark).
  • Dividing by total parts (17): Dividing £420 by (5 + 12 = 17). £420 is Bea's money alone, not the sum of Ary and Bea. Divide only by 12.
  • Inverting the ratio: Multiplying £420 by 12 and dividing by 5. Check the question order: Ary : Bea = 5 : 12, so Ary must have less than Bea.

Topics

Number · Ratio, proportion and rates of change · 3.1.2 Fractions, decimals and percentages · 3.3 Ratio, proportion and rates of change

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.