AQA GCSE Mathematics Paper 1 (Foundation), June 2025: Question 6

📋 What this question tests

This question assesses your ability to solve a real-life context problem involving money and business maths:

Calculating total cost (multiplication with money).
Finding the break-even quantity (division).
Interpreting the condition of making a profit rather than simply breaking even.
Rounding up to the nearest whole integer in a practical context.

Calculate Total Outlay (Cost):
Quin buys 200 toys at £4 each.
Total cost = 200 × £4 = £800
Find the Break-Even Number of Toys:
Each toy sells for £5. Find how many must be sold to recover £800:
£800 ÷ £5 = 160 toys
Note: At 160 toys, total income is £800. Profit is £0 (break-even).
Find the Least Number to Make a Profit:
To make a profit, total income must be strictly greater than £800.
Quin must sell at least one more toy:
160 + 1 = 161 toys
Income at 161 toys: 161 × £5 = £805 (Profit = £5).

Correct Answer: 161

Mark Scheme Breakdown:

M1: A correct first step using one operator, e.g. finding total cost 200 × 4 = 800 , or unit ratio 5 ÷ 4 = 1.25 .
M2: Complete method to reach break-even: 200 × 4 ÷ 5 (or finding 160 ), or showing both values 805 and 800 .
A1: Correct final answer of 161 .

Total Cost: Number of items × Cost per item .
Revenue: Number of items sold × Selling price per item .
Break-even: Revenue = Cost (profit is exactly zero).
Profit: Revenue > Cost. When items must be whole numbers, the next integer above the break-even point gives the first point of profit.

Watch the bold words: The question highlights the word least. It also specifically states "to make a profit", not "cover his costs".
Alternative Method:
Each toy costs £4 and sells for £5, so the profit per toy sold after covering costs is £1.
To cover the initial £800, he needs 800 ÷ 5 = 160 sales. The 161st toy provides the first £1 of profit!
Check by substitution:
If he sells 160: 160 × 5 = £800 (no profit).
If he sells 161: 161 × 5 = £805 (profit of £5).

Stopping at 160: The most frequent mistake is calculating 800 ÷ 5 = 160 and writing 160 on the answer line. Selling 160 toys leaves Quin with £0 profit (he has only broken even). This loses the final A1 mark.
Dividing 200 by 5 directly: Writing 200 ÷ 5 = 40 , ignoring the purchase price of £4 per toy.
Arithmetic slips: Miscalculating 200 × 4 or 800 ÷ 5 without showing working, resulting in 0 marks awarded.

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