The break-even formula, derived from first principles.

1. The starting equation

At break-even, total revenue equals total cost. Total revenue is unit price P multiplied by quantity sold Q. Total cost is fixed cost F (a flat sum independent of Q) plus variable cost per unit V multiplied by Q:

P · Q = F + V · Q

Solve for Q: subtract V·Q from both sides, factor Q, divide:

Q = F / (P − V)

The denominator P − V is the contribution margin per unit: the amount each unit sold contributes toward covering fixed cost and, beyond break-even, toward profit. The break-even unit count is therefore fixed cost divided by contribution margin per unit.

2. Break-even in revenue (the dollar version)

For service businesses with no clear “unit,” or for multi-product businesses, break-even in revenue is more useful. Multiply both sides by P:

Revenue_BE = F / [(P − V) / P] = F / CM%

Where CM% is the contribution-margin ratio — contribution margin expressed as a percentage of price. A 40 % contribution-margin business with $40,000 of monthly fixed cost breaks even at $40,000 / 0.40 = $100,000 of monthly revenue.

3. Worked example: Columbus coffee cart

Monthly fixed costs (cart lease, permit, insurance, owner’s draw): $4,800. Variable cost per cup (beans, milk, cup, lid, sleeve, sugar, payment-processor fee): $1.10. Price per cup: $5.00. Contribution margin per cup: $3.90. Break-even units per month: $4,800 / $3.90 = 1,231 cups. At 25 operating days that is 49 cups per day. The owner sets a target of 60 cups per day to clear the break-even threshold with a margin of safety.

4. Worked example: SaaS startup

Monthly fixed cost (engineering payroll, office, founder draw): $72,000. Variable cost per subscription (hosting allocation, payment processor, support cost amortised across customer base): $3.40. Price per subscription: $29.00. Contribution margin per subscription: $25.60. Break-even subscriptions: 72,000 / 25.60 = 2,813 active subscriptions. The founders compare this to their current 920 subscribers and determine they need 3.06× growth to break even — a useful concrete number that recasts the runway question.

5. Worked example: graphic-design freelancer

Monthly fixed costs (rent on shared studio, software subscriptions, owner’s draw): $6,500. Variable cost per project (stock-photo licences, courier, payment processing): $120. Price per project: $1,400. Contribution margin per project: $1,280. Break-even projects per month: 6,500 / 1,280 = 5.08 projects. Round up to six. Pipeline planning becomes: deliver six paid projects per month, or restructure fixed cost.

6. The assumptions that quietly do the work

Four assumptions hold the formula together:

1. Linear cost behaviour. Fixed cost is constant, variable cost per unit is constant. In reality fixed costs step (you hire a second person at some volume threshold) and variable costs curve (bulk discounts, learning effects).
2. Single product or stable mix. Multi-product break-even uses a weighted-average contribution margin; if your sales mix shifts, the break-even number shifts even if no individual product’s economics changes.
3. Production equals sales. The model treats every unit produced as sold. If you build inventory ahead of demand, the cash break-even is later than the accounting break-even.
4. Costs are correctly classified. The single most common error in practice. The fixed vs variable page exists for exactly this reason.

7. When the formula stops being useful

The break-even formula is most useful in the planning horizon over which the linear-cost assumption is approximately true — typically one to three months for a growing business, longer for a mature one. Beyond that horizon, fixed costs step, variable costs curve, and the sales mix shifts. Re-run the calculation quarterly with refreshed inputs rather than treating one number as a permanent target.