Break-even analysis: practice problems with step-by-step solutions

Also called: break-even point · contribution margin · CVP analysis

Free Break-even analysis practice problems with diagrams, hints and full worked solutions. New values every time, at three levels: break-even units, target profit, margin of safety.

Free practice problemEasy
Q€revenuetotal costfixed costQ = ?

A company sells a product for €47 per unit. The variable cost is €45 per unit and fixed costs are €5 000 per month. How many units Q must it sell per month to break even?

F = €5 000p = €47v = €45
units
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What you practise

  1. Easybreak-even units
  2. Mediumtarget profit
  3. Hardmargin of safety

Key formulas

Contribution margin
CM = p − v
Break-even quantity
QBE = FC/(p − v)
Target profit
Q = (FC + profit)/(p − v)
Profit
π = (p − v)·Q − FC
Margin of safety
(Q − QBE)/Q

Learn Break-even analysis from scratch

1When does a business start making money?

A business has two kinds of cost. Fixed costs FC stay the same however much it sells: rent, insurance, salaries. Variable costs v are paid for every unit made: materials, packaging, delivery.

Each unit sells for a price p. After paying its own variable cost, every unit leaves something over. That amount is the contribution margin:

CM = p − v

The contributions of all units first pay off the fixed costs. Only after that does the business make a profit.

The break-even point is the number of units at which the contributions exactly cover the fixed costs. Profit is zero there: no loss, no gain.

Self-check A product sells for €20 and has a variable cost of €12 per unit. How much does each unit contribute towards the fixed costs?
  1. €12
  2. €20
  3. €32
  4. €8

Answer: €8. CM = p − v = 20 − 12 = €8. The full price of €20 is not available, because €12 of it pays for the unit’s own materials and other variable costs.

2Profit and the break-even point

With Q units sold, the total contribution is (p − v) · Q. Subtract the fixed costs to get the profit π:

π = (p − v) · Q − FC

At break-even π = 0, so (p − v) · Q = FC. Solve for Q:

QBE = FC / (p − v)

Example: p = €25, v = €15 and FC = €12 000 per month. Each unit contributes €10, so 12 000 / 10 = 1200 units are needed to break even.

Q€FCTCTRQBE​ = 1200

In the chart, total revenue TR = p · Q starts at zero. Total cost TC = FC + v · Q starts at the fixed costs. Left of the crossing the business makes a loss, right of it a profit.

Self-check Fixed costs are €6000 per month, the price is €10 and the variable cost is €7 per unit. What is the break-even quantity?
  1. 2000 units
  2. 353 units
  3. 600 units
  4. 857 units

Answer: 2000 units. Q = 6000 / (10 − 7) = 6000 / 3 = 2000 units. Dividing by the price (600) forgets that each unit’s variable cost must be paid first.

3The method step by step

  1. Write down p, v and FC from the question. Check they use the same period, for example per month.
  2. Find the contribution margin per unit, p − v.
  3. For break-even: Q = FC / (p − v).
  4. For a target profit: the contributions must cover the fixed costs and the profit, so Q = (FC + profit) / (p − v).
  5. Round up to whole units if needed, and check by putting Q back into π = (p − v) · Q − FC.

4A worked example

Follow every step. Try to predict the next one before you open it.

Worked example

A product sells for €25 per unit with a variable cost of €15 per unit. Fixed costs are €12 000 per month. How many units Q must be sold per month to make a profit of €6000?

1

List the data

p = 25 v = 15 FC = 12 000 π = 6000
2

Contribution margin

p − v = 25 − 15 = 10

Each unit sold adds €10 towards fixed costs and profit.

3

Contribution needed

The contributions must pay the fixed costs and leave the target profit.

FC + π = 12 000 + 6000 = 18 000
4

Units needed

Q = 18 000 / 10 = 1800
Q€FCTCTRQBE​ = 1200Q = 1800π
5

Check the answer

Profit at 1800 units: 10 · 1800 − 12 000 = 18 000 − 12 000 = €6000. That is the target.

Answer

Q = 1800 units per month.

5Margin of safety

The margin of safety MS tells you how far sales can fall below the budget before the business makes a loss.

MS = (Qbudget − QBE) / Qbudget · 100%

Example: the business above budgets 2000 units a month. Break-even is 1200 units, so MS = (2000 − 1200) / 2000 = 0.40 = 40%.

Sales could drop by 40% before the business makes a loss. A larger margin of safety means a less risky plan.

Self-check Budgeted sales are 1500 units and the break-even point is 1200 units. What is the margin of safety as a percentage of budgeted sales?
  1. 20%
  2. 25%
  3. 300%
  4. 80%

Answer: 20%. (1500 − 1200) / 1500 = 300 / 1500 = 20%. Dividing by the break-even quantity (300/1200 = 25%) is a common slip: the margin is a share of budgeted sales.

6Common mistakes

  • Dividing the fixed costs by the price instead of by the contribution margin p − v.
  • For a target profit, forgetting to add the profit to the fixed costs.
  • For a target profit, dividing only the profit by p − v and forgetting the fixed costs.
  • Mixing periods: fixed costs per year with sales per month.
  • Dividing the margin of safety by the break-even quantity instead of by budgeted sales.
Self-check p = €23, v = €15 and FC = €9600 per month. How many units give a profit of €5000?
  1. 1200 units
  2. 1825 units
  3. 625 units
  4. 635 units

Answer: 1825 units. (9600 + 5000) / (23 − 15) = 14 600 / 8 = 1825. 1200 is only the break-even point, 625 forgets the fixed costs, and 635 divides by the price instead of p − v.

7How exam questions look

  • Easy: break-even units. Price, variable cost and fixed costs are given: Q = FC / (p − v).
  • Medium: target profit. Find the units needed for a given profit: Q = (FC + profit) / (p − v).
  • Hard: margin of safety. Find the break-even point first, then compare it with budgeted sales as a percentage.

Give the answer in units (or percent for the margin of safety), and check it with the profit formula.

Learn it step by step in the appLearn it step by step in the app

Worked examples

Example 1Easy
Q€revenuetotal costfixed costQ = ?

A company sells a product for €36 per unit. The variable cost is €30 per unit and fixed costs are €15 000 per month. How many units Q must it sell per month to break even?

F = €15 000p = €36v = €30

Find Q (units)

Show hints
  1. At break-even, total revenue equals total cost.
  2. Each unit contributes p − v towards the fixed costs.
  3. p − v = €6.
Show the answer and the step-by-step solution

Cost-volume-profit: each unit sold earns a contribution margin, price minus variable cost per unit. Profit = Q·(p − v) − F, where F is the fixed cost. At the break-even point profit is zero, so Q = F/(p − v).

Contribution margin per unit

p − v = 36 − 30 = €6

Break-even quantity

Q = F/(p − v) = 15 000/6 = 2 500

Answer

Q = 2 500 units
Example 2Medium
Q€revenuetotal costfixed costQ = ?

A product sells for €31 per unit with a variable cost of €25 per unit. Fixed costs are €9 000 per month. How many units Q must be sold per month to make a profit of €6 000?

F = €9 000p = €31v = €25profit = €6 000

Find Q (units)

Show hints
  1. Profit = Q·(p − v) − F. Set it equal to the target and solve for Q.
  2. The contribution has to cover the fixed costs and the target profit.
  3. F + profit = €15 000, p − v = €6.
Show the answer and the step-by-step solution

Cost-volume-profit: each unit sold earns a contribution margin, price minus variable cost per unit. Profit = Q·(p − v) − F, where F is the fixed cost. At the break-even point profit is zero, so Q = F/(p − v).

Contribution margin per unit

p − v = 31 − 25 = €6

Units for the target profit

Q = (F + profit)/(p − v) = (9 000 + 6 000)/6 = 2 500

Answer

Q = 2 500 units
Example 3Hard
Q€revenuev = €12v = €13fixed cost2 500Q = ?

A company sells 2 500 units per month at €17 each, with a variable cost of €12 per unit and fixed costs of €5 000 per month. The variable cost rises by €1 per unit and the price stays the same. How many units Q must now be sold per month to keep the same profit?

Q = 2 500p = €17v = €12 → €13F = €5 000

Find Q (units)

Show hints
  1. First find the current monthly profit.
  2. Π = 2 500·5 − 5 000 = €7 500.
  3. New contribution margin per unit: €5 − €1 = €4.
Show the answer and the step-by-step solution

Cost-volume-profit: each unit sold earns a contribution margin, price minus variable cost per unit. Profit = Q·(p − v) − F, where F is the fixed cost. At the break-even point profit is zero, so Q = F/(p − v).

Current profit

Π = 2 500·(17 − 12) − 5 000 = €7 500

New contribution margin per unit

17 − 13 = €4

Units for the same profit

Q = (5 000 + 7 500)/4 = 3 125

Answer

Q = 3 125 units

Questions and answers

What is Break-even analysis?

A business has two kinds of cost. Fixed costs FC stay the same however much it sells: rent, insurance, salaries. Variable costs v are paid for every unit made: materials, packaging, delivery.

Each unit sells for a price p. After paying its own variable cost, every unit leaves something over. That amount is the contribution margin:

How do I solve Break-even analysis problems step by step?

Cost-volume-profit: each unit sold earns a contribution margin, price minus variable cost per unit. Profit = Q·(p − v) − F, where F is the fixed cost. At the break-even point profit is zero, so Q = F/(p − v).

What are common mistakes in Break-even analysis?
  • That is the new break-even point. Here the contribution must also cover the current profit.
  • Recalculate with the new contribution margin: Q = (F + profit)/(p − v new).
What do the Break-even analysis exercises cover?
  • Easy: break-even units
  • Medium: target profit
  • Hard: margin of safety

Every problem has new values, so you learn the method instead of an answer. Stuck? Open a small hint first, and the full explanation only when you need it.

How practising works

  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Your progress, mistakes and saved problems are kept, so you can review before the exam.

Calculators

Break-even point calculator

Courses with this topic

Corporate FinanceManagement AccountingBusiness Fundamentals

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Printable worksheet

Break-even analysis worksheet with answers
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