9 min readConcept sheet10-mark problem, often with a break-even chart

Cost, revenue, profit and break-even

Profit is revenue minus cost. At break-even, revenue equals cost and profit is zero: divide the fixed cost by the contribution per unit.

CostC(x) = FC + vx

RevenueR(x) = px

ProfitP(x) = R(x) − C(x)

Break-even unitsFC ÷ (p − v)

Linear cost-volume-profit formulas

Total costC(x) = FC + vx (FC fixed cost, v variable cost per unit)

RevenueR(x) = px (p selling price)

ProfitP(x) = (p − v)x − FC

Contribution per unitp − v

Break-even quantityx = FC ÷ (p − v)

Break-even revenueBreak-even quantity × p

Units for a target profit T(FC + T) ÷ (p − v)

Margin of safetyActual sales − break-even sales

Break-even analysis in five steps

Break-even for a bakery

Problem

A Kandy bakery sells cakes at Rs. 1,500 each. Variable cost is Rs. 900 a cake and fixed cost Rs. 360,000 a month. Find C(x), R(x), P(x), the break-even point, and the profit on 800 cakes.

Working

C(x) = 360,000 + 900x; R(x) = 1,500x.
P(x) = 1,500x − (360,000 + 900x) = 600x − 360,000.
Break-even: 600x = 360,000, so x = 600 cakes.
Break-even revenue = 600 × 1,500 = 900,000.
Profit on 800 cakes: P(800) = 600 × 800 − 360,000 = 480,000 − 360,000 = 120,000.

Answer

Break-even at 600 cakes (Rs. 900,000 of sales); profit on 800 cakes is Rs. 120,000.

Bakery: revenue, cost and profit at different outputs

Cakes (x)	R(x) (Rs.)	C(x) (Rs.)	P(x) (Rs.)
0	0	360,000	−360,000
300	450,000	630,000	−180,000
600	900,000	900,000	0
900	1,350,000	1,170,000	180,000

Target profit and margin of safety

Problem

For the same bakery: (a) how many cakes give a profit of Rs. 150,000? (b) If it sells 800 cakes, what is the margin of safety?

Working

(a) Units = (FC + target) ÷ contribution = (360,000 + 150,000) ÷ 600 = 510,000 ÷ 600 = 850.
(b) Margin of safety = 800 − 600 = 200 cakes.
As a percentage of sales: 200 ÷ 800 = 0.25 = 25%. Sales can fall 25% before a loss.

Answer

(a) 850 cakes (b) 200 cakes, i.e. 25% of sales.

Choosing between two cost options

Problem

A firm can rent a machine for Rs. 40,000 a month plus Rs. 10 per unit, or outsource at Rs. 30 per unit. At what output do they cost the same, and which is cheaper above it?

Working

Rent: C₁ = 40,000 + 10x. Outsource: C₂ = 30x.
Equal when 40,000 + 10x = 30x, so 20x = 40,000 and x = 2,000.
Check at 2,000: 40,000 + 20,000 = 60,000 and 30 × 2,000 = 60,000. ✓
At 3,000 units: rent 70,000 vs outsource 90,000, so renting is cheaper above 2,000.

Answer

Same cost at 2,000 units; rent above 2,000, outsource below.

Reading a break-even chart

Plot units on the x-axis and rupees on the y-axis. The revenue line starts at the origin. The cost line starts at the fixed cost on the y-axis and is less steep. They cross at the break-even point. Left of it is the loss area; right of it is the profit area.

In Sri Lankan business

A Sri Lankan small business (a bakery, a garment unit, a tuk-tuk owner) uses break-even to answer 'how many must I sell to cover my costs?' before taking a bank loan or setting a price.

Exam angle

Short answer: “Define the break-even point.” The output where total revenue equals total cost, so profit is zero.

Structured question: “FC = Rs. 200,000, price Rs. 500, variable cost Rs. 300. Find the break-even units and revenue.” 200,000 ÷ 200 = 1,000 units; revenue Rs. 500,000.

Essay: “A bakery: price Rs. 1,500, variable cost Rs. 900, fixed cost Rs. 360,000. Find C, R, P, break-even, profit at 800, units for Rs. 150,000 profit, and margin of safety.” P = 600x − 360,000; BE 600 cakes (Rs. 900,000); Rs. 120,000; 850 cakes; 200 cakes (25%).

Remember it

Break-even = Fixed cost ÷ Contribution. Contribution (p − v) is what each unit adds towards fixed costs, then profit.