Percentages: Core Techniques for IGCSE Mathematics | 百分数：IGCSE 数学核心方法

📚 Percentages: Core Techniques for IGCSE Mathematics | 百分数：IGCSE 数学核心方法

Percentages appear throughout the IGCSE Mathematics syllabus, from basic conversions to compound interest and reverse percentage problems. This article reviews the essential concepts, formulas, and exam strategies you need to solve percentage questions confidently.

百分数贯穿 IGCSE 数学大纲，从基本互化到复利和逆向百分数问题。本文梳理核心概念、公式和考试策略，帮助你自信解答百分数题目。

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1. What is a Percentage? | 什么是百分数？

A percentage is a fraction with denominator 100. The symbol % means “per hundred”, so 25% = 25/100 = 1/4. Percentages make it easy to compare parts of different sizes.

百分数是分母为 100 的分数。符号 % 表示“每一百”，因此 25% = 25/100 = 1/4。百分数便于比较不同大小的部分。

Percentage = (Part ÷ Whole) × 100%

中文公式：百分数 = (部分量 ÷ 总量) × 100%。

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2. Converting Fractions, Decimals and Percentages | 分数、小数与百分数的互化

To convert a percentage to a fraction, divide by 100 and simplify. To convert a decimal to a percentage, multiply by 100. To convert a percentage to a decimal, divide by 100. The table below shows common equivalents that you should memorise.

将百分数化为分数，先除以 100 再化简；将小数化为百分数，乘以 100；将百分数化为小数，除以 100。下表列出了应熟记的常见等值关系。

Percentage	Fraction	Decimal
1%	1/100	0.01
10%	1/10	0.1
20%	1/5	0.2
25%	1/4	0.25
50%	1/2	0.5
75%	3/4	0.75
100%	1	1.0

Use these conversions to switch quickly between forms. In non-calculator papers, fractions often make multiplication simpler.

利用这些等值关系可以快速切换形式。在非计算器试卷中，使用分数通常能让乘法更简单。

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

To find a percentage of a quantity, multiply the quantity by the percentage expressed as a decimal or fraction. For example, 15% of 240 = 0.15 × 240 = 36.

求一个量的百分数，将该量乘以百分数化成的分数或小数。例如，240 的 15% = 0.15 × 240 = 36。

Part = (Percentage ÷ 100) × Whole

中文公式：部分量 = (百分数 ÷ 100) × 总量。

Example: Find 35% of 480.

示例：求 480 的 35%。

Solution: 0.35 × 480 = 168.

解答：0.35 × 480 = 168。

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4. Percentage Increase and Decrease | 百分数增加与减少

To increase a quantity by r%, multiply by (1 + r/100). To decrease a quantity by r%, multiply by (1 – r/100). These multipliers are essential for percentage change and compound growth.

将某量增加 r%，乘以 (1 + r/100)；减少 r%，乘以 (1 – r/100)。这些乘数是解决百分比变化和复利增长的关键。

New value = Original × (1 ± r/100)

中文公式：新值 = 原值 × (1 ± r/100)。增加用加号，减少用减号。

Example: A jacket priced at $80 is increased by 12%. The new price is 80 × 1.12 = $89.60.

示例：一件夹克标价 80 美元，涨价 12%。新价格为 80 × 1.12 = 89.60 美元。

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5. Percentage Change | 百分数变化

Percentage change compares the increase or decrease to the original value. Always use the original value in the denominator. A positive result means an increase; a negative result means a decrease.

百分数变化将增加量或减少量与原始值进行比较。分母必须使用原始值。结果为正表示增加，结果为负表示减少。

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

中文公式：百分数变化 = (变化量 ÷ 原值) × 100%。

Example: A mobile phone increases from 250 to 300. The change is 50, so the percentage change is (50 ÷ 250) × 100% = 20%.

示例：一部手机从 250 涨到 300。变化量为 50，因此百分数变化为 (50 ÷ 250) × 100% = 20%。

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6. Reverse Percentages | 逆向百分数

Reverse percentage problems give the final value after a percentage increase or decrease and ask for the original amount. Divide the final value by the multiplier rather than subtracting or adding the percentage directly.

逆向百分数问题给出增减后的最终值，要求原值。应将最终值除以乘数，而不是直接减去或加上百分数。

Original value = Final value ÷ (1 ± r/100)

中文公式：原值 = 最终值 ÷ (1 ± r/100)。增加用加号，减少用减号。

Example: After a 20% discount, a pair of shoes costs 96. The original price is 96 ÷ 0.8 = 120.

示例：一双鞋打八折后价格为 96。原价为 96 ÷ 0.8 = 120。

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7. Simple Interest | 单利

Simple interest is calculated only on the initial principal. It is common in IGCSE questions involving bank accounts or investments. The amount of interest earned is the same each year.

单利仅根据初始本金计算。它在 IGCSE 涉及银行存款或投资的题目中很常见。每年获得的利息相同。

I = P × r × t / 100

中文公式：利息 = 本金 × 年利率 × 时间 / 100。

Here I is the interest, P is the principal, r is the annual percentage rate, and t is the number of years. The total amount is A = P + I.

这里 I 是利息，P 是本金，r 是年利率，t 是年数。最终总额 A = P + I。

A = P(1 + r × t / 100)

中文公式：总额 = 本金 × (1 + 年利率 × 时间 / 100)。

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8. Compound Interest | 复利

Compound interest is calculated on both the principal and the interest already earned. Each year, the amount grows by the same percentage multiplier, so the growth is exponential.

复利根据本金和已获利息共同计算。每年总额按相同的百分数乘数增长，因此呈指数增长。

A = P(1 + r/100)ⁿ

中文公式：总额 = 本金 × (1 + 年利率/100)ⁿ，其中 n 为年数或计息期数。

Example: A principal of 1000 is invested at 5% compound interest for 3 years. The amount is A = 1000 × (1.05)³ = 1157.625 ≈ 1157.63.

示例：本金 1000，年利率 5%，复利 3 年。总额为 A = 1000 × (1.05)³ = 1157.625 ≈ 1157.63。

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9. Exam-Style Strategy and Common Mistakes | 考试策略与常见错误

Success in percentage questions depends on careful reading of whether the percentage is applied to the original value or to a new value. Before choosing a formula, identify the original amount and the final amount.

百分数题目的关键在于仔细读题，判断百分数是作用在原值还是新值上。选择公式前，务必先确定原始量和最终量。

Common mistake 1: Using the new value as the denominator in a percentage change calculation. Always divide by the original value.

常见错误 1：百分数变化计算中用新值作分母。应始终除以原始值。

Common mistake 2: Adding a percentage when the question requires subtraction, or subtracting when it requires addition. Use the correct multiplier.

常见错误 2：题目要求减少却使用加法，或要求增加却使用减法。应使用正确的乘数。

Common mistake 3: Confusing simple and compound interest. Simple interest adds the same amount each period; compound interest applies the percentage to the growing total.

常见错误 3：混淆单利和复利。单利每期增加相同利息；复利对不断增长的本利和继续应用百分数。

Use multipliers for successive percentage changes. For example, a 10% increase followed by a 10% decrease does not return to the original value: 100 × 1.1 × 0.9 = 99, not 100.

连续百分数变化使用乘数。例如，先增加 10% 再减少 10% 不会回到原值：100 × 1.1 × 0.9 = 99，而不是 100。

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10. Practice Questions and Answers | 练习题与答案

Q1. Increase 240 by 15%.

题目 1：将 240 增加 15%。

Answer: 240 × 1.15 = 276.

答案：240 × 1.15 = 276。

Q2. A shirt is reduced by 30% to a price of $42. Find the original price.

题目 2：一件衬衫降价 30% 后价格为 42 美元。求原价。

Answer: Original = 42 ÷ 0.7 = 60 dollars.

答案：原价 = 42 ÷ 0.7 = 60 美元。

Q3. A salary increases from 1800 to 1980. Find the percentage increase.

题目 3：工资从 1800 增加到 1980。求增加的百分数。

Answer: Change = 180, percentage increase = (180 ÷ 1800) × 100%

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