When a business buys goods and sells them at a different price, the difference is either a profit or a loss. In maths, we express that difference as a percentage of the original cost price. The key rule: always divide by the cost price, not the selling price.
What is profit and loss in maths?
Profit occurs when you sell something for more than you paid for it.
Loss occurs when you sell something for less than you paid for it.
The starting amount — the price you originally paid — is called the cost price (CP).
The amount you receive when you sell is the selling price (SP).
- Profit = Selling Price − Cost Price (when SP > CP)
- Loss = Cost Price − Selling Price (when CP > SP)
These are always written as positive values. You then express either as a percentage of the cost price.
How do I calculate percentage profit?
Use this formula:
Percentage profit = (Profit ÷ Cost Price) × 100
Worked Example 1 — A trader buys a lamp for £40 and sells it for £50. Calculate the percentage profit.
- Find the profit: Profit = £50 − £40 = £10
- Divide by the cost price: 10 ÷ 40 = 0.25
- Multiply by 100: 0.25 × 100 = 25% profit
The answer tells you that the trader made a profit equal to 25% of what they originally paid.
How do I calculate percentage loss?
Use this formula:
Percentage loss = (Loss ÷ Cost Price) × 100
Worked Example 2 — A second-hand shop buys a laptop for £80 and sells it for £64. Calculate the percentage loss.
- Find the loss: Loss = £80 − £64 = £16
- Divide by the cost price: 16 ÷ 80 = 0.20
- Multiply by 100: 0.20 × 100 = 20% loss
Notice that you divide by the cost price (£80), not the selling price (£64). This is the single most common error on this topic.
What if I know the percentage and need to find a price?
When you know the percentage profit or loss and one of the prices, you work with multipliers.
A multiplier is the decimal you multiply the cost price by to get the selling price directly:
- A 30% profit means the selling price is 130% of the cost price → multiplier = 1.30
- A 20% loss means the selling price is 80% of the cost price → multiplier = 0.80
Worked Example 3 — A shirt costs a retailer £25. They add a 30% mark-up. What is the selling price?
- Multiplier for a 30% increase = 1.30
- Selling price = £25 × 1.30
- Selling price = £32.50
Worked Example 4 — A shop sells a bicycle for £180, making a 20% profit. What was the cost price?
- Multiplier for 20% profit = 1.20
- Cost price × 1.20 = £180
- Cost price = £180 ÷ 1.20
- Cost price = £150
Check: £150 × 1.20 = £180 ✓ and profit = £180 − £150 = £30; percentage profit = (30 ÷ 150) × 100 = 20% ✓
How does this link to multipliers?
Using multipliers is a faster, one-step method for both mark-up and reverse problems:
| Scenario | Multiplier | Calculation |
|---|---|---|
| 25% profit | × 1.25 | SP = CP × 1.25 |
| 20% loss | × 0.80 | SP = CP × 0.80 |
| Find CP from SP (25% profit) | ÷ 1.25 | CP = SP ÷ 1.25 |
| Find CP from SP (20% loss) | ÷ 0.80 | CP = SP ÷ 0.80 |
The reverse of multiplying by 1.25 is dividing by 1.25 — this gives you the reverse percentage technique, which you will meet again at GCSE.
Summary table of worked examples
| Scenario | Cost Price | Selling Price | Profit or Loss | Percentage |
|---|---|---|---|---|
| Lamp | £40 | £50 | Profit: £10 | 25% profit |
| Laptop | £80 | £64 | Loss: £16 | 20% loss |
| Shirt (30% mark-up) | £25 | £32.50 | Profit: £7.50 | 30% profit |
| Bicycle (20% profit given) | £150 | £180 | Profit: £30 | 20% profit |
What are the most common mistakes?
Mistake 1: Dividing by the selling price instead of the cost price.
This is by far the most frequent error. The percentage is always calculated relative to the original amount paid — the cost price. If you divide the profit by the selling price instead, you get a different (and incorrect) percentage.
Mistake 2: Subtracting the wrong way round for profit.
Profit = Selling Price − Cost Price. If you reverse this and get a negative number, it means there was a loss, not a profit. Always identify which is larger before subtracting.
Mistake 3: Confusing percentage profit with the multiplier.
A 25% profit does not mean the multiplier is 0.25. The multiplier is 1.25 (100% of the cost price plus the extra 25%). Mixing these up produces an answer that is nowhere near correct.
Mistake 4: Not checking the answer makes sense.
After calculating, ask yourself: if the seller made a profit, is the selling price higher than the cost price? If they made a loss, is it lower? This quick sense-check catches arithmetic errors immediately.
Frequently asked questions
Why do we always divide by the cost price and not the selling price?
The percentage tells us how much extra (or less) the seller received compared with what they originally spent. The cost price is the reference point — the baseline — so it goes on the bottom of the fraction. Dividing by the selling price would answer a different, less useful question and would give a smaller percentage for a profit situation.
What is the difference between profit and percentage profit?
Profit is an amount of money (for example, £10). Percentage profit is a proportion that tells you how significant that profit is relative to the cost (for example, 25%). A £10 profit on a £40 item is impressive; a £10 profit on a £10,000 item is negligible. The percentage makes comparisons meaningful.
Can the percentage profit ever be more than 100%?
Yes. If someone buys an item for £5 and sells it for £12, the profit is £7 and the percentage profit is (7 ÷ 5) × 100 = 140%. There is no upper limit on percentage profit. Percentage loss, however, cannot exceed 100% because you cannot lose more than you originally spent.
How do I know when to multiply and when to divide in reverse percentage questions?
If you are given the cost price and asked for the selling price, multiply by the multiplier. If you are given the selling price and asked for the cost price, divide by the same multiplier. A common mistake is to subtract a percentage from the selling price — for example, subtracting 20% of £180 to find the cost price when there is a 20% profit. This gives £144, not £150, because 20% of £180 is not the same as 20% of the cost price.
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