Practice Maths

Break-even Analysis

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Key Terms

Fixed costs (FC)
: costs that stay the same regardless of how many units are produced (e.g. rent, insurance, equipment).
Variable costs (VC)
: costs that increase with each unit produced (e.g. materials, ingredients per item).
Total Cost (TC)
= Fixed Costs + Variable Cost per unit × quantity:   TC = FC + VC×n
Revenue (R)
= Selling price per unit × quantity sold:   R = P×n
Break-even point
: the quantity where R = TC (profit = 0). Below break-even: loss. Above break-even: profit.
Profit (P)
= R − TC. Positive = profit; negative = loss.
Contribution margin
= Selling price − Variable cost per unit. Each unit sold contributes this amount toward covering fixed costs and eventually generating profit.
Key Formulas
• TC = FC + VC×n   (total cost, linear in n)
• R = price × n   (revenue, linear through origin)
• Break-even: R = TC → price×n = FC + VC×n → n = FC ÷ (price − VC)
• Profit = R − TC = (price − VC)×n − FC
• Contribution margin = price − VC per unit

Break-even analysis: TC = 3n + 600, R = 8n. Break-even at n = 120, R = TC = $960.

n (units) $ ($) 0 40 80 120 160 200 $300 $600 $960 $1200 $1500 TC R Break-even (120, $960) LOSS PROFIT FC=$600
Hot Tip At break-even, profit = 0 — you have not made any money yet, but you have covered all costs. The break-even quantity is the minimum you must sell to avoid a loss. Every unit sold beyond break-even earns a profit equal to the contribution margin (selling price − variable cost per unit).

Worked Example 1 — Finding the break-even quantity algebraically

Question: A business has fixed costs of $1 800, variable costs of $4.50 per unit, and sells units at $10 each. Write TC and R equations, find the break-even quantity, and find profit at 500 units.

Equations:
TC = 4.50n + 1800   (fixed cost $1800, variable cost $4.50/unit)
R = 10n   (revenue $10/unit)

Break-even: Set R = TC:
10n = 4.50n + 1800
5.50n = 1800
n = 1800 ÷ 5.50 ≈ 327.3
Break-even at 328 units (round up since you need to exceed costs).

Profit at 500 units:
Profit = R − TC = 10(500) − [4.50(500) + 1800] = 5000 − [2250 + 1800] = 5000 − 4050 = $950 profit

Contribution margin = $10 − $4.50 = $5.50/unit. Each unit beyond 328 contributes $5.50 to profit.

Worked Example 2 — Reading a break-even graph

Question: From the graph above (TC = 3n + 600, R = 8n): (a) state the break-even point; (b) find profit at n = 180; (c) what is the contribution margin?

(a) Break-even: read intersection at (120, $960). At 120 units, revenue = costs = $960.

(b) Profit at n = 180: R = 8(180) = $1440; TC = 3(180) + 600 = 540 + 600 = $1140.
Profit = 1440 − 1140 = $300

(c) Contribution margin = selling price − variable cost = $8 − $3 = $5 per unit. (Each unit above break-even adds $5 to profit.)

Break-even analysis answers the fundamental business question: “How many units do we need to sell before we start making money?” It connects two linear models — total cost and revenue — and finds their intersection.

Setting up the models

Total Cost (TC): Start with fixed costs (the y-intercept), then add variable costs per unit times the quantity. TC is a straight line starting at (0, FC).

Revenue (R): Selling price times quantity. R starts at the origin (0, 0) — if you sell nothing, you earn nothing. R is a straight line through the origin.

The gradient of TC is the variable cost per unit. The gradient of R is the selling price per unit. For a viable business, the selling price must exceed the variable cost (otherwise you never break even), so the R line must be steeper than the TC line.

Finding the break-even point

Set R = TC and solve for n:

price × n = FC + VC × n
(price − VC) × n = FC
n = FC ÷ (price − VC) = FC ÷ contribution margin

This formula is worth memorising: break-even quantity = Fixed Costs ÷ Contribution Margin.

Example: FC = $2 400, VC = $8/unit, price = $20/unit.
Contribution margin = 20 − 8 = $12/unit.
Break-even = 2 400 ÷ 12 = 200 units.
At 200 units: TC = 8(200) + 2400 = $4 000. R = 20(200) = $4 000 ✓
Tip: The break-even quantity is usually not a whole number. Always round UP (not to the nearest integer) because you need to sell at least that many whole units to cover all costs. Selling 327.3 units is not possible; you need 328 to be in profit.

Profit function

Once you have TC and R, profit is P = R − TC = (price − VC)×n − FC. This is also a linear function of n, with gradient = contribution margin and y-intercept = −FC. The break-even is where P = 0.

Tip: On a break-even graph, the y-intercept of TC (at n = 0) shows the fixed cost — this is the loss the business makes before selling a single unit. The gap between R and TC at any quantity n is the profit (positive) or loss (negative) at that output level.

Mastery Practice

See Answers ➔
  1. Fluency

    A business has Total Cost TC = 5x + 900 and Revenue R = 14x, where x is the number of units produced and sold.

    1. (a) Find the break-even quantity.
    2. (b) Find the profit or loss when x = 80 units.
    3. (c) Find the profit when x = 200 units.
  2. Fluency

    A small business has fixed costs of $2 400 per month, variable costs of $6 per unit, and sells units at $18 each.

    1. (a) Write the TC and R equations.
    2. (b) Find the break-even quantity.
    3. (c) State the contribution margin and explain what it represents.
  3. Fluency

    From a break-even graph, you can read: break-even point at 150 units, fixed costs of $1 800, and selling price of $20 per unit.

    1. (a) At break-even, TC = R. Use this to find the Revenue at 150 units.
    2. (b) Hence find the Total Cost equation TC = VC×n + FC, by finding the variable cost per unit.
    3. (c) Verify: at n = 150, does TC = R?
  4. Understanding

    A school canteen makes sandwiches with a variable cost of $2.80 each. The canteen’s daily fixed costs are $120. Sandwiches are sold for $6.50 each.

    1. (a) Write the TC and R equations for the number of sandwiches sold per day (n).
    2. (b) Find the break-even number of sandwiches per day.
    3. (c) Find the profit (or loss) if 60 sandwiches are sold in a day.
    4. (d) How many sandwiches must be sold to make a daily profit of $250?
  5. Understanding

    A local band hires a venue for $800 (fixed cost). There are no per-ticket costs. Tickets are sold at $25 each.

    1. (a) Write TC and R equations.
    2. (b) Find the break-even number of tickets sold.
    3. (c) Find the profit if 60 tickets are sold.
    4. (d) The venue has a maximum capacity of 120 people. Find the maximum possible profit.
    5. (e) If the ticket price is reduced to $18, find the new break-even quantity. Is a full house now profitable?
  6. Understanding

    A manufacturing business has fixed costs of $5 500, variable costs of $12.50 per unit, and a selling price of $35 per unit.

    1. (a) Find the break-even quantity.
    2. (b) Find the profit at 400 units produced and sold.
    3. (c) What selling price per unit would exactly halve the break-even quantity (i.e. result in break-even at half the original quantity)? (Assume fixed and variable costs remain the same.)
  7. Understanding

    Two business options are available for producing handmade items, both selling at $7 per item.

    Option A: Buy equipment for $8 000 (one-time fixed cost), ingredients cost $2 per item.
    Option B: Rent equipment (no fixed cost), but ingredients and rental combined cost $4.50 per item.

    1. (a) Write TC and R equations for each option.
    2. (b) Find the break-even quantity for each option.
    3. (c) Calculate profit at 2 000 units for each option. Which is more profitable?
    4. (d) At what quantity does Option A become more profitable than Option B? (Find the quantity where profit of A = profit of B.)
  8. Understanding

    Interpret the following break-even graph. The TC line has a y-intercept of $1 200 and passes through (400, $2 800). The R line passes through the origin and through (400, $3 200).

    1. (a) Find the gradient (variable cost per unit) of the TC line, and the gradient (selling price) of the R line.
    2. (b) Write the equations for TC and R.
    3. (c) Find the break-even quantity algebraically.
    4. (d) State the contribution margin.
    5. (e) Find the profit at 600 units.
  9. Problem Solving

    A food truck sells meals at $16 each. Daily fixed costs are $180 (vehicle depreciation, insurance, council permit). Each meal costs $5.80 in ingredients.

    1. (a) Write TC and R equations for daily operations (n = meals sold).
    2. (b) Find the daily break-even number of meals.
    3. (c) The food truck operates 6 days per week for 4 weeks and consistently sells 50 meals per day. Calculate the total profit over the 4-week period.
    4. (d) If daily fixed costs increase to $220, how many additional meals per day are needed to maintain the same daily profit as in part (c)? (Use 50 meals as the reference.)
  10. Problem Solving

    A business currently sells 300 units per month at $45 each. Fixed monthly costs are $4 200 and variable costs are $18 per unit.

    1. (a) Calculate the current monthly profit.
    2. (b) A marketing campaign costs an extra $1 500 per month and is projected to increase sales to 420 units per month (at the same $45 price). Calculate the profit with the campaign.
    3. (c) Is the marketing campaign worthwhile? Justify using your calculations.
    4. (d) As an alternative to the campaign, the business considers raising the selling price (keeping sales at 300 units, no campaign). What minimum selling price per unit would achieve the same monthly profit as option (b)?