Phone-Case Pop-Up: Finding the Break-Even Point
Cian is running a mini-company stall at a school fair. He plans to sell custom phone cases for €9.00 each. The stall and equipment cost €180, which he must pay even if he sells nothing. Each case costs €3.20 in materials and €0.50 in packaging. He can make at most 60 cases and expects to sell 45. Before committing, he wants to know how many cases he must sell to cover his costs, what profit he would make at his expected sales, and what happens if some cases are faulty. The school requires the mini-company to make a profit, not just break even. Cian writes his profit as a formula in terms of n, the number of cases sold. He will not receive any other income and has no other costs.
1. Write an expression for Cian's profit in euro when he sells n cases, and explain what each part represents.
Read the worked answer
Revenue from n cases = 9n euro.
Variable cost per case = 3.20 + 0.50 = €3.70.
Total variable cost = 3.70n euro.
Fixed cost = €180.
Profit = revenue − total costs:
Profit = 9n − (180 + 3.70n) = 5.30n − 180
Meaning of the parts:
- 5.30n is the contribution from the cases: each case brings in €9.00 and uses €3.70 of variable cost, leaving €5.30 towards the fixed cost and then profit.
- 180 is the fixed stall and equipment cost, which does not depend on n.
- The expression is in euro because both 5.30 and 180 are euro amounts.
2. Solve for the break-even number of cases. Explain why the answer must be rounded up.
Read the worked answer
Break-even means profit = 0:
5.30n − 180 = 0
5.30n = 180
n = 180 ÷ 5.30 = 33.962...
Cian cannot sell a fraction of a case, so check whole numbers:
- At n = 33: profit = 5.30 × 33 − 180 = 174.90 − 180 = −€5.10 (a loss).
- At n = 34: profit = 5.30 × 34 − 180 = 180.20 − 180 = €0.20 (a small profit).
So the break-even number is 34 cases. The exact answer 33.96 must be rounded up, not to the nearest whole number, because 33 cases still lose money. Rounding down would leave Cian below break-even.
3. Cian sells 45 cases, but 5 are faulty and are refunded in full. He cannot reuse the materials. Calculate his actual profit.
Read the worked answer
Cian made 45 cases, so he paid variable costs on all 45, including the faulty ones.
Revenue: he sells 45 but refunds 5 in full, so he keeps payment for 45 − 5 = 40 cases.
Revenue = 40 × €9.00 = €360.00.
Variable costs = 45 × €3.70 = €166.50.
Fixed costs = €180.00.
Total costs = 180.00 + 166.50 = €346.50.
Actual profit = 360.00 − 346.50 = €13.50.
This is much lower than the expected profit at 45 sales, because the faulty cases still cost money to make but earn no revenue.