Profit and Loss

1Summary 重點摘要

Learn how to calculate whether boss makes or loses money, know relationship between cost, selling price and profit.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The basic relationshipThe basic relation of profit and loss: selling price − cost = profit. When the selling price exceeds the cost the shop gains (positive profit); when it falls short, the shop loses. If a shopkeeper buys an item at $80 and sells it at $100, the profit is 100 − 80 = $20; if it sells at only $70, the loss is 80 − 70 = $10. This relation starts every profit calculation, with cost and selling price as the standard terms. In any question, first separate the two prices: the cost is what the shopkeeper pays, the selling price is what the customer pays — confusing them ruins everything that follows.
2. Step 2: Profit per centProfit per cent = profit ÷ cost × 100%, and the denominator is always the cost, never the selling price. Continuing the example: cost $80 and profit $20 give 20 ÷ 80 × 100% = 25%. Another case: an item costing $200 sells for $250; the profit is 250 − 200 = $50 and the profit per cent is 50 ÷ 200 × 100% = 25% — the same percentage can carry different profit amounts. Why the cost as denominator? Because the cost is the owner's investment, and profit per cent measures the return on it: how much is earned per $100 put in. The fraction 20 ÷ 80 says 'earn 20 per 80 invested', which scales to 25 per 100, exactly 25%.
3. Step 3: Loss per centLoss per cent follows the same pattern: loss per cent = loss ÷ cost × 100%. An item costing $100 that sells at $80 records a loss of 100 − 80 = $20 and a loss per cent of 20 ÷ 100 × 100% = 20%. Note the denominator is still the cost: here $20 against $100 is 20%; mistakenly using the selling price gives 20 ÷ 80 = 25% and is wrong. A daily-life case: a bakery clears unsold bread at half price; if a loaf costs $12 and clears at $6, each loaf loses $6, a loss per cent of 50% — only by computing it can the owner judge whether the clearance price is wise.
4. Step 4: Finding the selling price from a rateThe reverse application: given the cost and the profit or loss per cent, find the selling price. Earning x% means selling price = cost × (1 + x%); losing x% means selling price = cost × (1 − x%). An item costing $200 sold to earn 30% prices at 200 × 1.3 = $260; cleared at a 10% loss it prices at 200 × 0.9 = $180. The logic treats the cost as 100%, so earning 30% makes the price 130% of cost. The common error is dropping the minus sign: writing the loss formula as (1 + x%) produces the absurdity of a loss selling higher than the cost; sanity-check the price against the cost at the end.
5. Step 5: Marked price and discountShops usually set a marked price above the intended selling price, then offer a discount; '20% off' means paying 80% of the marked price. An item marked $300 sold at 20% off sells for 300 × 0.8 = $240; if the cost is $180, the profit is 240 − 180 = $60 and the profit per cent is 60 ÷ 180 × 100% ≈ 33.3%. Watch chained discounts: 20% off followed by 10% off gives marked × 0.8 × 0.9 = marked × 0.72 — a 72%, not 70%. Such questions have three layers: marked price, selling price and cost. Marked × discount = selling price; selling price − cost = profit; write the layers separately and the working stays clean.
6. Step 6: Common pitfallsThe biggest trap in profit questions: the denominator of profit per cent is the cost, not the selling price. Selling at $100 with a $20 profit gives 20 ÷ 80 = 25% (cost = 100 − 20 = 80), not 20 ÷ 100 = 20% — the examiners' favourite spot. Second trap: when marked price, selling price and cost all appear, remember the discount acts on the marked price and the profit per cent on the cost; never divide the profit by the marked price. Third trap: in the loss formula (1 − x%), a forgotten minus sign produces nonsense. Sanity-check at the end: a profit must give a positive percentage with the selling price above the cost; a loss, the reverse.
7. Step 7: Applications in daily lifeProfit calculations fill real business life. A tea restaurant's iced milk tea costs $6 (tea, milk, sugar, cup) and sells at $18: profit $12, profit per cent 200%. A lunch promotion at $14 lowers the profit per cent to 8 ÷ 6 × 100% ≈ 133.3%, and the owner judges the promotion from this figure. At a season-end half-price clearance, a $400 jacket costing $150 sells at $200 and still earns $50; but one costing $220 would lose $20 — which is why some goods never reach the deepest discount. Knowing profit per cent also sharpens your shopping sense: whether '30% off' is a bargain depends on how high the shop set the marked price to begin with.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Profit = Selling price - Cost
• Profit per cent = Profit ÷ Cost × 100%
• Always based on cost, not selling price
• Earn x%: selling = cost × (1 + x%)
• Lose x%: selling = cost × (1 - x%)