Making money or losing it
Every business compares money coming in with money going out.
- Revenue = price × quantity sold
- Total cost = fixed costs + variable costs
- Profit = revenue − total cost
If profit is negative, it’s a loss.
Costs
| Type | Changes with output? | Examples |
|---|---|---|
| Fixed | no | rent, insurance, equipment, salaries |
| Variable | yes | materials, packaging, shipping per item |
Example: A T-shirt business:
- Fixed costs: $1,200/month (rent, equipment)
- Variable cost: $8 per shirt
- Price: $20 per shirt
Cost: C(x) = 1,200 + 8x
Revenue: R(x) = 20x
Profit: P(x) = 20x − (1,200 + 8x) = 12x − 1,200
The break-even point
The break-even point is where revenue = cost (profit = 0).
20x = 1,200 + 8x → 12x = 1,200 → x = 100 shirts
- Sell fewer than 100 → loss
- Sell more than 100 → profit
Shortcut: break-even quantity = fixed costs ÷ contribution margin, where contribution margin = price − variable cost per unit = $12.
Graphing
Plot C(x) and R(x). They cross at the break-even point (100, 2,000). To the right, revenue is above cost: the gap is profit.
Using the model
| Shirts sold | Revenue | Cost | Profit |
|---|---|---|---|
| 50 | $1,000 | $1,600 | −$600 |
| 100 | $2,000 | $2,000 | $0 |
| 150 | $3,000 | $2,400 | $600 |
| 200 | $4,000 | $2,800 | $1,200 |
Target profit: to earn $1,800, solve 12x − 1,200 = 1,800 → x = 250 shirts.
Profit margin
Profit margin = profit ÷ revenue × 100%. At 200 shirts: 1,200/4,000 = 30%.
Changing the numbers
- Raise the price → break-even comes sooner (but sales may drop).
- Lower variable costs → higher contribution margin.
- Cut fixed costs → lower break-even.
Worked examples
Example 1: Fixed costs $3,000; price $50; variable cost $30. Break-even? 3,000/20 = 150 units.
Example 2: Same business sells 200 units. Profit? 200 × 20 − 3,000 = $1,000.
Example 3: Revenue $8,000; total cost $9,500. Profit or loss? Loss of $1,500.
Common mistakes
- Forgetting fixed costs.
- Using price instead of contribution margin in the break-even shortcut.
- Confusing revenue with profit.
Try it yourself
- Fixed $600, variable $4/unit, price $10. Break-even?
- Profit at 150 units for question 1?
- A bakery’s costs C = 400 + 1.5x and revenue R = 3.5x. Break-even?
- Name one fixed and one variable cost for a pizza shop.
Answers: 1) 100 units 2) $300 3) 200 items 4) e.g. rent (fixed), cheese (variable)
Quick recap
- Profit = revenue − (fixed costs + variable costs).
- The break-even point is where revenue = cost.
- Break-even quantity = fixed costs ÷ contribution margin.