📚 Profit and Loss: Mastering Money Calculations for IGCSE Maths | 利润与亏损：掌握IGCSE数学中的金钱计算
Profit and loss is one of the most frequently tested topics in IGCSE Mathematics, appearing in both Paper 2 (Core) and Paper 4 (Extended). This topic bridges arithmetic, percentages, and real-world financial literacy, making it essential not only for examinations but also for everyday life decision-making.
利润与亏损是IGCSE数学中最常考查的主题之一，出现在Paper 2（核心）和Paper 4（扩展）两套试卷中。这一主题将算术、百分比与现实金融素养紧密相连，不仅对考试至关重要，也与日常生活中的决策息息相关。
1. Core Definitions | 核心定义
Before we dive into calculations, we must understand the fundamental terms. The cost price (CP) is the amount paid to acquire an item, while the selling price (SP) is the amount received when the item is sold. The difference between these two values determines whether we make a profit or incur a loss.
在深入计算之前，我们必须理解基本术语。成本价（CP）是购买一件物品所支付的金额，而售价（SP）是出售该物品时收到的金额。这两个数值之间的差异决定了我们是盈利还是亏损。
- Profit = SP − CP (when SP > CP) | 利润 = 售价 − 成本价（当售价大于成本价时）
- Loss = CP − SP (when CP > SP) | 亏损 = 成本价 − 售价（当成本价大于售价时）
- Profit percentage is always calculated based on the cost price | 利润率始终以成本价为基准计算
- Loss percentage is also calculated based on the cost price | 亏损率同样以成本价为基准计算
Profit % = (Profit ÷ CP) × 100% | 利润率 = （利润 ÷ 成本价） × 100%
Loss % = (Loss ÷ CP) × 100% | 亏损率 = （亏损 ÷ 成本价） × 100%
2. Finding Selling Price Given Profit Percentage | 已知利润率求售价
When a shopkeeper wants to earn a specific percentage of profit, we can calculate the selling price directly. For instance, if an item costs \$80 and the shopkeeper wants a 25% profit, we need to find the selling price that gives a 25% return on the cost.
当商家希望获得特定百分比的利润时，我们可以直接计算售价。例如，如果一件商品成本为80美元，商家希望获得25%的利润，我们需要找到能带来25%成本回报率的售价。
Method 1: Calculate the profit amount first. Profit = 25% of \$80 = 0.25 × 80 = \$20. Then SP = CP + Profit = 80 + 20 = \$100.
方法一：先计算利润金额。利润 = 80美元的25% = 0.25 × 80 = 20美元。然后售价 = 成本价 + 利润 = 80 + 20 = 100美元。
Method 2: Use the multiplier method. If CP represents 100%, then SP represents 125% (since we add the 25% profit). SP = \$80 × 1.25 = \$100.
方法二：使用倍数法。如果成本价代表100%，那么售价代表125%（因为我们加上25%的利润）。售价 = 80美元 × 1.25 = 100美元。
The multiplier method is faster and less prone to error, especially when dealing with decimal percentages or compound scenarios.
倍数法更快且更不容易出错，尤其是在处理小数百分比或复合情境时。
- Profit of r% ⇒ SP = CP × (1 + r/100) | r%的利润 ⇒ 售价 = 成本价 ×（1 + r/100）
- Loss of r% ⇒ SP = CP × (1 − r/100) | r%的亏损 ⇒ 售价 = 成本价 ×（1 − r/100）
3. Finding Cost Price Given Selling Price and Profit/Loss Percentage | 已知售价和利润率/亏损率求成本价
This reverse calculation is slightly more challenging. Suppose an item is sold for \$150 at a profit of 20%. What was the cost price? Many students make the mistake of subtracting 20% from the selling price. This is incorrect because the 20% is based on the cost price, not the selling price.
这种反向计算稍有挑战性。假设一件商品以150美元的价格出售，利润率为20%。成本价是多少？许多学生犯的错误是从售价中减去20%。这是不正确的，因为20%是基于成本价而非售价计算的。
Using the multiplier approach: SP = CP × 1.20, so CP = SP ÷ 1.20 = 150 ÷ 1.20 = \$125.
使用倍数法：售价 = 成本价 × 1.20，因此成本价 = 售价 ÷ 1.20 = 150 ÷ 1.20 = 125美元。
Key rule: To find the cost price, always divide the selling price by the multiplier (1 + r/100 for profit, or 1 − r/100 for loss).
关键规则：要求成本价，始终用售价除以倍数（盈利除以1 + r/100，亏损除以1 − r/100）。
| Scenario | 情境 | Multiplier | 倍数 | Calculation | 计算 |
| Sold at 10% profit | 盈利10%售出 | 1.10 | CP = SP ÷ 1.10 |
| Sold at 15% loss | 亏损15%售出 | 0.85 | CP = SP ÷ 0.85 |
| Sold at 200% profit | 盈利200%售出 | 3.00 | CP = SP ÷ 3.00 |
4. Discount and Marked Price | 折扣与标价
In retail, items often have a marked price (or list price), and the shopkeeper offers a discount on this price. The selling price is the marked price minus the discount.
在零售业中，商品通常有标价（或挂牌价），商家在此价格上提供折扣。售价等于标价减去折扣金额。
For example, a jacket is marked at \$80 with a 15% discount. The discount amount = 15% × 80 = \$12, so the selling price = 80 − 12 = \$68. Using the multiplier: SP = 80 × 0.85 = \$68.
例如，一件夹克标价80美元，享有15%的折扣。折扣金额 = 80的15% = 12美元，因此售价 = 80 − 12 = 68美元。使用倍数法：售价 = 80 × 0.85 = 68美元。
SP = Marked Price × (1 − Discount %) | 售价 = 标价 ×（1 − 折扣率）
Sometimes the question combines both discount and profit. For instance, a shopkeeper marks up an item by 30% above cost, then offers a 10% discount. The final selling price should be compared to the cost price to determine the overall profit percentage.
有时题目同时涉及折扣和利润。例如，商家将商品成本上浮30%作为标价，然后提供10%的折扣。最终售价应与成本价比较，以确定整体利润率。
- Marked price = CP × 1.30 | 标价 = 成本价 × 1.30
- Final SP = Marked price × 0.90 = CP × 1.30 × 0.90 = CP × 1.17 | 最终售价 = 标价 × 0.90 = 成本价 × 1.30 × 0.90 = 成本价 × 1.17
- Overall profit = 17% | 整体利润率 = 17%
5. Value Added Tax (VAT) | 增值税
VAT is a consumption tax added to the price of goods and services. In IGCSE questions, VAT is typically quoted as a percentage (such as 15%, 17%, or 20%). The price excluding VAT is the original price, and the price including VAT is what the customer actually pays.
增值税是对商品和服务征收的消费税。在IGCSE题目中，增值税通常以百分比形式给出（如15%、17%或20%）。不含税价格是原始价格，含税价格是顾客实际支付的金额。
If the price excluding VAT is \$250 and VAT is 15%, then the VAT amount = 0.15 × 250 = \$37.50, and the price including VAT = 250 + 37.50 = \$287.50.
如果不含税价格为250美元，增值税为15%，则增值税金额 = 0.15 × 250 = 37.50美元，含税价格 = 250 + 37.50 = 287.50美元。
To find the original price from the VAT-inclusive price: If the final price is \$287.50 and VAT is 15%, then 287.50 = Original price × 1.15, so Original price = 287.50 ÷ 1.15 = \$250.
从含税价格反推原始价格：如果最终价格为287.50美元，增值税为15%，那么287.50 = 原始价格 × 1.15，因此原始价格 = 287.50 ÷ 1.15 = 250美元。
Price incl. VAT = Price excl. VAT × (1 + VAT rate) | 含税价格 = 不含税价格 ×（1 + 增值税率）
6. Successive Discounts and Compound Percentage Change | 连续折扣与复合百分比变化
When two discounts are applied successively (one after another), we cannot simply add the percentages. For example, a "20% off" followed by a further "10% off" does not mean 30% total discount. Instead, the first discount reduces the price, and the second discount is applied to the already reduced price.
当两个折扣连续应用时（一个接一个），我们不能简单地将百分比相加。例如，"8折优惠"之后再"9折优惠"并不等于总折扣为30%。第一个折扣降低了价格，第二个折扣在已降低的价格上再应用。
| Step | 步骤 | Calculation | 计算 | Result | 结果 |
| Original price | 原价 | — | \$100 |
| After 20% discount | 应用8折后 | 100 × 0.80 | \$80 |
| After further 10% discount | 再应用9折后 | 80 × 0.90 | \$72 |
| Total equivalent discount | 等效总折扣 | (100 − 72) ÷ 100 | 28% |
The combined multiplier is 0.80 × 0.90 = 0.72, which corresponds to a 28% ightly less than the sum of the individual discount percentages when applied successively.
组合乘数为0.80 × 0.90 = 0.72，对应28%的总折扣。这表明连续应用时，组合折扣总是略小于各个折扣率之和。
Important note: If we apply a 10% increase followed by a 10% decrease, the result is NOT the original value. It is 0.99 times the original (a 1% net loss). This is a common trap question in IGCSE exams.
重要提示：如果先增加10%再减少10%，结果不是原始值。它是原值的0.99倍（净亏损1%）。这是IGCSE考试中常见的陷阱题。
7. Word Problems Involving Multiple Items | 涉及多件物品的文字题
IGCSE exams frequently present scenarios where a vendor sells different quantities of items or mixes items purchased at different prices. These problems require careful organisation of information.
IGCSE考试经常出现商家以不同价格购买不同数量物品的场景。这些问题需要仔细组织信息。
Example: A fruit seller buys 50 apples at \$0.30 each and 30 oranges at \$0.50 each. She sells all the fruit at \$0.60 per item. Find her total profit and profit percentage.
例题：一个水果商以每个0.30美元的价格购买50个苹果，以每个0.50美元的价格购买30个橙子。她以每个0.60美元的价格出售所有水果。求她的总利润和利润率。
- Total cost = (50 × 0.30) + (30 × 0.50) = 15 + 15 = \$30 | 总成本 =（50 × 0.30）+（30 × 0.50）= 15 + 15 = 30美元
- Total revenue = 80 × 0.60 = \$48 | 总收入 = 80 × 0.60 = 48美元
- Profit = 48 − 30 = \$18 | 利润 = 48 − 30 = 18美元
- Profit % = (18 ÷ 30) × 100% = 60% | 利润率 =（18 ÷ 30）× 100% = 60%
Notice that when calculating the profit percentage for the entire transaction, we use the total cost price as the base, not the sum of individual bases.
注意，在计算整体交易的利润率时，我们使用总成本价作为基准，而不是各个基准之和。
8. Exchange Rates and Profit in Different Currencies | 汇率与不同货币中的利润
Sometimes, profit and loss calculations involve currency conversions. For example, a trader buys goods in USD and sells them in GBP or EUR. The exchange rate must be applied consistently.
有时，利润和亏损计算涉及货币兑换。例如，商人以美元购买商品，以英镑或欧元出售。汇率必须一致地应用。
Example: A company buys US goods worth \$500. The exchange rate is £1 = \$1.25. The company sells the goods in the UK for £450. Determine the profit or loss in pounds.
例题：一家公司购买价值500美元的美国商品。汇率为1英镑 = 1.25美元。该公司在英国以450英镑的价格出售商品。以英镑计算利润或亏损。
Cost in pounds = 500 ÷ 1.25 = £400. Selling price = £450. Profit = 450 − 400 = £50. Profit % = (50 ÷ 400) × 100% = 12.5%.
英镑成本 = 500 ÷ 1.25 = 400英镑。售价 = 450英镑。利润 = 450 − 400 = 50英镑。利润率 =（50 ÷ 400）× 100% = 12.5%。
Converting currency: Amount in GBP = Amount in USD ÷ Exchange Rate | 货币换算：英镑金额 = 美元金额 ÷ 汇率
9. Simple Interest and Profit Comparison | 单利与利润的对比
While not strictly profit and loss on goods, IGCSE questions sometimes link the concept of simple interest to investment decisions. Simple interest is calculated as a percentage of the principal amount over a period of time.
虽然严格来说不属于商品利润与亏损，但IGCSE题目有时将单利概念与投资决策联系起来。单利是基于本金金额在一定时期内按百分比计算的利息。
Simple Interest = P × r × t | 单利 = 本金 × 利率 × 时间
interest rate per year (as a decimal), and t is the time in years. For example, investing \$2,000 at 4% per annum for 2 years gives interest = 2000 × 0.04 × 2 = \$160.
其中P为本金，r为年利率（小数形式），t为年数。例如，以4%的年利率投资2,000美元两年，利息 = 2000 × 0.04 × 2 = 160美元。
Exam tip: When a question asks "which is the better deal — a 15% discount or a \$12 discount on a \$70 item?", always compare the actual amounts or percentages carefully. In this case, 15% of \$70 = \$10.50, so the \$12 flat discount is better.
考试提示：当题目问"哪个更划算——70美元商品享受15%折扣还是12美元固定折扣？"时，务必仔细比较实际金额或百分比。在此例中，70美元的15% = 10.50美元，因此12美元的固定折扣更划算。
10. Common Pitfalls and Exam Strategies | 常见误区与考试策略
Students often lose marks in profit and loss questions due to avoidable errors. The most common mistake is calculating percentages based on the selling price instead of the cost price. Always ask yourself: "What is the base of this percentage?"
学生因可避免的错误在利润与亏损题目中失分。最常见的错误是基于售价而非成本价计算百分比。始终问自己："这个百分比的基准是什么？"
- Pitfall 1: Adding discount percentages directly when they are successive. Always multiply the multipliers. | 误区一：当折扣连续出现时直接相加。始终乘上倍数。
- Pitfall 2: Assuming that a 10% increase followed by a 10% decrease returns to the original value. It does not. | 误区二：假设先增加10%再减少10%会回到原始值。事实上不会。
- Pitfall 3: Forgetting to include all items when calculating cost price. | 误区三：计算成本价时忘记包括所有物品。
- Pitfall 4: Rounding intermediate answers too early, leading to inaccurate final results. | 误区四：过早舍入中间答案，导致最终结果不准确。
Strategy: For multi-step problems, write down each step clearly. Let the examiner see your working. Use the multiplier method whenever possible — it is more efficient and directly leads to the correct mathematical expression.
策略：对于多步问题，清晰地写下每一步。让考官看到你的计算过程。尽可能使用倍数法——它更高效，且直接通向正确的数学表达式。
Finally, always check your answer for reasonableness. If you buy something for \$80 and sell it for \$100, the profit percentage must be 25%, not 20%. The phrase "profit on cost" is the standard convention in IGCSE.
最后，始终检查答案的合理性。如果你以80美元购买某物并以100美元出售，利润率必须是25%，而不是20%。"基于成本的利润"是IGCSE中的标准惯例。
11. Practice Questions | 练习题
Test your understanding with these typical IGCSE-style questions. Solve each one carefully before checking the method described.
通过这些典型的IGCSE风格题目测试你的理解。先仔细解答每道题，再查看方法说明。
Question 1: A book is sold for \$36 at a profit of 20%. Find the cost price.
题目1：一本书以36美元的价格出售，利润率为20%。求成本价。
Question 2: A retailer buys a TV for \$500 and marks it up by 40%. He then offers a 15% discount on the marked price. Find the final selling price and the overall profit percentage.
题目2：零售商以500美元购买电视机，在标价上上浮40%。然后他对标价提供15%的折扣。求最终售价和整体利润率。
Question 3: A store sells a chair at a loss of 10%. If the selling price is \$54, find the cost price and the loss amount.
题目3：一家商店以10%的亏损率出售椅子。若售价为54美元，求成本价和亏损金额。
- Q1 Answer: CP = 36 ÷ 1.20 = \$30 | 题1答案：成本价 = 36 ÷ 1.20 = 30美元
- Q2 Answer: Marked price = 500 × 1.40 = \$700; SP = 700 × 0.85 = \$595; Profit % = (595 − 500) ÷ 500 × 100% = 19% | 题2答案：标价 = 500 × 1.40 = 700美元；售价 = 700 × 0.85 = 595美元；利润率 =（595 − 500）÷ 500 × 100% = 19%
- Q3 Answer: CP = 54 ÷ 0.90 = \$60; Loss = 60 − 54 = \$6 | 题3答案：成本价 = 54 ÷ 0.90 = 60美元；亏损 = 60 − 54 = 6美元
12. Summary of Key Formulas | 关键公式总结
To succeed in profit and loss questions, memorise the following formulas and understand when to apply each one. The framework below consolidates everything covered in this article.
要在利润与亏损题目中取得好成绩，请牢记以下公式并理解何时应用每个公式。以下框架总结了本文涵盖的所有内容。
| Concept | 概念 | Formula | 公式 |
| Profit | 利润 | SP − CP |
| Loss | 亏损 | CP − SP |
| Profit percentage | 利润率 | (Profit ÷ CP) × 100% |
| Loss percentage | 亏损率 | (Loss ÷ CP) × 100% |
| SP with profit r% | 盈利r%的售价 | CP × (1 + r/100) |
| SP with loss r% | 亏损r%的售价 | CP × (1 − r/100) |
| Discount | 折扣 | Marked Price × Discount % |
| VAT-inclusive price | 含税价格 | Price excl. VAT × (1 + VAT rate) |
| Combined multiplier | 组合倍数 | m₁ × m₂ (multiply all multipliers) |
The concept of profit and loss is fundamentally about understanding what base a percentage refers to. Once you master this, you can tackle any question in this topic, no matter how the problem is worded.
利润与亏损的概念本质上在于理解百分比所基于的基准。一旦你掌握了这一点，无论题目如何表述，你都能解决这一主题中的任何问题。
Mastering these calculations will not only help you score well in IGCSE Mathematics but also equip you with practical financial skills for your future. Keep practicing, and remember — every percentage question deserves careful attention to detail.
掌握这些计算不仅有助于你在IGCSE数学中取得优异成绩，还将使你具备未来实用的财务技能。持续练习，并记住——每个百分比问题都值得对细节的仔细关注。
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图