Profit and Loss: Formulas, Markup, Margin and Break-Even Point

Every commercial activity can be summarised in three fundamental quantities. Revenue (R) is the total value obtained from sales. Cost (C) is the amount spent to produce or acquire what was sold. Profit (P) is the positive difference between revenue and cost: P = R − C. If the difference is negative, the result is a loss.

Cost can be divided into fixed cost — which does not vary with the quantity produced (rent, fixed salaries, insurance) — and variable cost — which increases proportionally with production (raw materials, commissions, packaging). This distinction is essential for more sophisticated analyses such as the break-even point.

Percentage profit (or profit rate) is the ratio between profit and a reference quantity, expressed as a percentage. When the reference is the cost, we calculate the markup. When the reference is the revenue, we calculate the profit margin. Both forms exist and are useful in different contexts.

Markup is the percentage added to the cost to obtain the selling price. Formula: Markup = (Profit ÷ Cost) × 100 = [(Price − Cost) ÷ Cost] × 100. Example: if a product costs $80 and sells for $100, the markup is (20 ÷ 80) × 100 = 25%.

Profit margin is the percentage of the selling price that represents profit. Formula: Margin = (Profit ÷ Revenue) × 100 = [(Price − Cost) ÷ Price] × 100. In the same example: (20 ÷ 100) × 100 = 20%. The markup is 25%, the margin is 20% — on the same product.

Confusing markup and margin is one of the most common errors in business. A retailer who wants a 30% margin should not add 30% to the cost — doing so produces a 30% markup, which corresponds to a lower margin. To achieve a 30% margin, calculate: Price = Cost ÷ (1 − 0.30) = Cost ÷ 0.70 ≈ Cost × 1.4286. That is, approximately 42.86% must be added to the cost.

In math problems, profit and loss questions frequently ask for the percentage relative to the purchase price (cost price). The formulas are: % profit = [(SP − CP) ÷ CP] × 100 and % loss = [(CP − SP) ÷ CP] × 100, where SP is the selling price and CP is the cost price.

Typical example: a trader buys a product for $240 and sells it for $300. Profit = $60. Profit rate = (60 ÷ 240) × 100 = 25% on cost. Another example: bought for $500 and sold for $425. Loss = $75. Loss rate = (75 ÷ 500) × 100 = 15%.

When a discount is applied to the original selling price, you must identify the cost price and the effective selling price (after the discount) before calculating profit or loss. A product with a list price of $400, a cost of $280, and a 10% discount is sold for $360. Profit = 360 − 280 = $80, that is, 80/280 ≈ 28.6% on cost.

The break-even point is the quantity of sales needed for total revenue to cover all costs, resulting in zero profit. Below this point there is a loss; above it there is a profit. To calculate it, we separate fixed costs (FC) from variable costs (VC per unit) and know the unit selling price (SP).

Break-even formula in units: Q_eq = FC ÷ (SP − VC). The denominator (SP − VC) is called the unit contribution margin — the amount each unit sold contributes toward covering fixed costs. Example: a shop has monthly fixed costs of $6,000, sells T-shirts for $80 each, and the variable cost per T-shirt is $35. Contribution margin = 80 − 35 = $45. Break-even point = 6,000 ÷ 45 ≈ 134 T-shirts per month.

Knowing the break-even point is essential for pricing decisions and planning. If the monthly sales target is below the break-even point, the business is at risk. If it is well above, there is room to absorb demand fluctuations. It is also possible to calculate the break-even point in dollars: R_eq = FC ÷ (1 − VC/SP) = FC ÷ contribution margin ratio.

In product pricing, a classic mistake is setting prices by 'adding a percentage to cost' without considering all fixed and variable costs. A product with a raw-material cost of $50 does not have a total cost of $50 — labour, packaging, freight, an allocation of fixed costs, and the desired profit margin must all be included.

Another frequent error: when offering a discount, failing to check whether the profit still justifies it. A product sold at a 20% discount may result in a loss if the original margin was below 20%. Always calculate the profit or loss after the discount before approving a promotion.

In buy-and-sell exercises, pay attention to which base the profit or loss percentage is calculated on. Most problems use the cost price (purchase price) as the base. If the problem does not specify, assume the percentage is on cost.