Master Percentages: A Complete IGCSE Guide | 掌握百分数：IGCSE 完整指南

📚 Master Percentages: A Complete IGCSE Guide | 掌握百分数：IGCSE 完整指南

Percentages are one of the most frequently tested topics in IGCSE Mathematics. They appear in everyday contexts such as shopping discounts, bank interest, tax calculations, and statistical data. This guide gives you a structured, exam-focused revision of percentage skills, from the basic definition to advanced reverse and compound percentage problems.

百分数是 IGCSE 数学中最常考的主题之一，几乎出现在购物折扣、银行利息、税务计算和统计数据等所有日常场景中。本指南将带你系统复习百分数的核心技能，从基本定义到高难度的反向百分数与复利问题，紧扣考试重点。

1. What is a Percentage? | 什么是百分数？

A percentage is a number or ratio expressed as a fraction of 100. The word ‘percent’ means ‘per hundred’, so 25% means 25 out of 100, or the fraction 25/100.

百分数是一个以 100 为分母的数或比率。英文中的 “percent” 意为 “每百”，因此 25% 表示 100 份中的 25 份，即分数 25/100。

To convert a percentage to a decimal, divide by 100. To convert a decimal to a percentage, multiply by 100. For example:

将百分数转换为小数，需要除以 100；将小数转换为百分数，则需要乘以 100。例如：

75% = 75 ÷ 100 = 0.75 and 0.375 = 0.375 × 100 = 37.5%

Likewise, a fraction can be written as a percentage by first dividing its numerator by its denominator and then multiplying by 100.

同样地，将分数转换成百分数时，先让分子除以分母得到小数，再乘以 100 即可。

Example 1: Write 7/20 as a percentage. Since 7 ÷ 20 = 0.35, then 0.35 × 100 = 35%.
例 1：将 7/20 写成百分数。因为 7 ÷ 20 = 0.35，所以 0.35 × 100 = 35%。

2. Percentage of a Quantity | 求一个量的百分数

The most basic skill is finding a given percentage of a quantity. Multiply the quantity by the percentage written as a decimal or fraction.

最基础的技能是求某个量的给定百分数。用该量乘以百分数对应的小数或分数即可。

Value = Quantity × Percentage (as a decimal)

数值 = 量 × 百分数（化为小数）

For example, find 30% of $250. Write 30% as 0.30, then 250 × 0.30 = 75. So the answer is $75.

例如，求 250 美元的 30%。把 30% 写成 0.30，然后 250 × 0.30 = 75。因此答案是 75 美元。

Example 2: A school has 800 students. If 45% are girls, the number of girls is 800 × 0.45 = 360.
例 2：一所学校有 800 名学生。若女生占 45%，则女生人数为 800 × 0.45 = 360。

Always decide whether to use a calculator or mental arithmetic. A useful mental approach is to find 10% first, then build other percentages from it.

做这类题时要判断使用计算器还是心算。一个有用的心算技巧是先求 10%，再由此推出其他百分数。

3. Percentage Increase and Decrease | 百分数增减

Percentage change tells us how much a quantity has grown or shrunk relative to its original value. The general formula is:

百分数变化描述的是一个量相对于原值增长或减少了多少。其通用公式为：

Percentage change = (Change ÷ Original value) × 100%

百分数变化 = （变化量 ÷ 原值） × 100%

If a price rises from $50 to $62, the change is 12, so the percentage increase is (12 ÷ 50) × 100% = 24%.

如果价格从 50 美元涨到 62 美元，变化量为 12，则百分比增加为 (12 ÷ 50) × 100% = 24%。

To apply a percentage change directly, multiply by a decimal multiplier. A 15% increase means multiply by 1.15; a 15% decrease means multiply by 0.85. This method is faster and is essential for compound problems.

要直接进行百分数增减，可以乘以一个小数乘数。增加 15% 相当于乘以 1.15；减少 15% 相当于乘以 0.85。这种方法更快捷，也是解决复利问题的基础。

Example 3: Increase 200 by 8%: 200 × 1.08 = 216.
例 3：将 200 增加 8%：200 × 1.08 = 216。
Example 4: Decrease 350 by 20%: 350 × 0.80 = 280.
例 4：将 350 减少 20%：350 × 0.80 = 280。

4. Reverse Percentage | 反向百分数

A reverse percentage problem gives you the result after an increase or decrease, and asks you to find the original amount. For example, the price of a jacket is $56 after a 30% discount. What was the original price?

反向百分数问题给出增减后的结果，要求你求出原值。例如，一件夹克打七折后售价为 56 美元，原价是多少？

Here the multiplier after a 30% discount is 0.70. Since the final price equals original × 0.70, we divide the final price by 0.70: 56 ÷ 0.70 = 80. The original price was $80.

打七折后的乘数为 0.70。因为最终价格 = 原价 × 0.70，所以我们用最终价格除以 0.70：56 ÷ 0.70 = 80。原价为 80 美元。

Original = Final value ÷ decimal multiplier

原值 = 最终值 ÷ 小数乘数

Remember not to subtract the percentage directly. Many students mistakenly calculate 56 − 30% of 56, which is incorrect because the 30% discount applies to the original price, not to the final price.

切记不能直接用减法处理。很多学生错误地计算 56 − 56 的 30%，这是不对的，因为 30% 折扣是针对原价，而非最终售价。

5. Compound Percentage Change | 复合百分数变化

When a percentage change is applied repeatedly over several periods, the changes compound. This appears in bank interest, population growth, and inflation problems.

当百分数变化在多期内反复应用时，变化会进行复利累积。这在银行利息、人口增长和通货膨胀问题中十分常见。

For compound interest, the formula is:

复利的公式为：

Final value = Principal × (1 + rate)^n

最终值 = 本金 × （1 + 利率）^n

Here ‘n’ is the number of compounding periods and the rate is expressed as a decimal per period. For example, $1000 invested at 4% per year for 3 years grows to 1000 × 1.04³ = 1000 × 1.124864 ≈ $1124.86.

其中 n 是计息期数，rate 以每期小数表示。例如，1000 美元按年利率 4% 存 3 年，增长为 1000 × 1.04³ = 1000 × 1.124864 ≈ 1124.86 美元。

The same idea applies to repeated percentage decreases, such as the value of a car depreciating by 12% each year.

同理也适用于连续百分比下降，例如汽车价值每年贬值 12%。

Example 5: A car is worth $20,000. It depreciates by 15% per year. After 2 years, its value is 20000 × 0.85² = 20000 × 0.7225 = $14,450.
例 5：一辆汽车价值 20000 美元，每年贬值 15%。两年后，其价值为 20000 × 0.85² = 20000 × 0.7225 = 14450 美元。

6. Percentages in Ratios and Proportions | 百分数与比率、比例

Percentages are often linked to ratios. For instance, if a fruit mix contains apples and oranges in the ratio 3 : 2, then 3 out of 5 parts are apples, so apples represent 3/5 of the total, which is 60%. Oranges represent 2/5, or 40%.

百分数经常与比率联系。例如，若水果混合物中苹果和橙子的比为 3 : 2，则 5 份中有 3 份是苹果，所以苹果占总量的 3/5，即 60%。橙子占 2/5，即 40%。

When solving word problems, it is useful to translate ‘is’ to ‘=’, ‘of’ to ‘×’, and ‘what percent’ to ‘x/100’. For example: ‘What percent of 80 is 20?’ becomes x/100 × 80 = 20. Thus x = (20/80) × 100 = 25%.

在解应用题时，可以把 “is” 看作 “=”，”of” 看作 “×”，”what percent” 看作 x/100。例如：”80 的百分之几是 20？” 可写成 x/100 × 80 = 20，所以 x = (20/80) × 100 = 25%。

Careful translation prevents careless errors and helps you set up equations quickly during the exam.

仔细的转化可以避免粗心错误，并帮助你在考试中快速建立方程。

7. Applications: Discount, Tax, Profit and Loss | 实际应用：折扣、税费、利润与亏损

Everyday financial applications are common in IGCSE questions. The key is to identify which quantity represents the original 100%.

日常金融应用是 IGCSE 中的常见题型。关键是判断哪个量代表了 100% 的基准。

Application	Method	Example
Discount 折扣	Find the percentage of the original price, then subtract it.	20% off $45 → 45 × 0.20 = 9, so price = 45 − 9 = $36.
Sales tax 销售税	Find the percentage of the price and add it on.	5% tax on $80 → 80 × 0.05 = 4, total = 80 + 4 = $84.
Profit 利润	Profit percent = (profit / cost price) × 100%.	Cost $60, sold $75 → profit % = (15/60) × 100% = 25%.

A common confusion is whether to use cost price or selling price as the base for profit percentage. In IGCSE, profit percentage is almost always based on the cost price.

常见的混淆点是利润百分数该以成本价还是售价为基准。在 IGCSE 中，利润百分数几乎总是基于成本价。

8. Common Pitfalls and Exam Tips | 常见陷阱与考试技巧

Many percentage errors come from misusing the multiplier. Always write the decimal multiplier explicitly before calculating. For an increase of 25%, use 1.25; for a decrease of 25%, use 0.75.

许多百分数错误来自乘数的误用。计算前务必将小数乘数明确写出。增加 25% 用 1.25；减少 25% 用 0.75。

Pitfall 1: Adding or subtracting percentages instead of multiplying. ‘Increase 50 by 10%’ means 50 × 1.10, not 50 + 10.
陷阱 1：将百分数直接加减而不是乘法。”50 增加 10%” 意味着 50 × 1.10，而不是 50 + 10。
Pitfall 2: Reversing the base in percentage change. The original value is always the denominator, not the final value.
陷阱 2：在百分数变化中搞错基准。原值永远是分母，而不是最终值。
Pitfall 3: Forgetting to convert a percentage to a decimal before using it in a multiplier. 5% must be written as 0.05.
陷阱 3：忘记将百分数转化为小数再作为乘数。5% 必须写成 0.05。

In exam questions, read carefully whether the question asks for the percentage change or the new value. Circle key words such as ‘increase’, ‘decrease’, ‘original price’, ‘final value’.

考试时请仔细读题，确认题目要求的是百分数变化量还是新值。圈出关键词，例如 “increase”（增加）、”decrease”（减少）、”original price”（原价）、”final value”（最终值）。

9. Worked Practice Problem | 综合练习例解

Let us solve a mixed problem step by step.

让我们逐步解决一个综合题。

A laptop is priced at $850. In a sale, the price is reduced by 30%. A customer then pays an additional 8% sales tax on the sale price. What is the final price for the customer?

一台笔记本电脑标价 850 美元。促销时降价 30%。顾客需在促销价基础上再支付 8% 的销售税。顾客最终支付多少钱？

Step 1: The sale price multiplier is 1 − 0.30 = 0.70. Sale price = 850 × 0.70 = $595.

第 1 步：促销价乘数为 1 − 0.30 = 0.70。促销价 = 850 × 0.70 = 595 美元。

Step 2: The tax multiplier is 1 + 0.08 = 1.08. Final price = 595 × 1.08 = $642.60.

第 2 步：税费乘数为 1 + 0.08 = 1.08。最终价格 = 595 × 1.08 = 642.60 美元。

Notice that the final price is not simply 850 × 0.78. The discount and tax are applied to different bases, so the combined multiplier is 0.70 × 1.08, not 0.78.

注意，最终价格不是简单地 850 × 0.78。因为折扣和税费所基于的价格不同，所以综合乘数为 0.70 × 1.08，而不是 0.78。

10. Key Formula Summary | 核心公式汇总

Here is a rapid revision table of the most important percentage formulas.

以下是最重要百分数公式的快速复习表。

Situation	Formula
Percentage change	Change ÷ Original × 100%
Percentage increase	New value = Original × (1 + r)
Percentage decrease	New value = Original × (1 − r)
Reverse percentage	Original = Final ÷ (1 ± r)
Compound interest	Principal × (1 + r)^n

Make a copy of this table and test yourself, then try solving at least ten mixed past-paper questions to build confidence.

请抄写这份表格并自我检测，然后尝试解答至少十道混合历年真题来建立信心。

With these strategies, you can handle nearly every percentage problem on the IGCSE Mathematics exam.

掌握这些策略后，你几乎可以应对 IGCSE 数学考试中所有的百分数问题。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Master Percentages: A Complete IGCSE Guide | 掌握百分数：IGCSE 完整指南