Percentages, Ratios & Proportions — Full Revision Guide | 百分比、比率与比例

📚 Percentages, Ratios & Proportions — Full Revision Guide | 百分比、比率与比例 — 完整复习指南

Percentages, ratios and proportions form the backbone of IGCSE Mathematics. They appear in nearly every topic — from simple interest to probability, from map scales to chemical concentrations. This guide covers everything you need for the Edexcel IGCSE syllabus, with worked examples and exam-style practice.

百分比、比率与比例是 IGCSE 数学的基石。它们出现在几乎所有专题中——从单利到概率，从地图比例尺到化学浓度。本指南涵盖 Edexcel IGCSE 教学大纲所需的全部内容，配有典型例题和考试风格练习。

1. Understanding Percentages | 理解百分比

A percentage is a fraction out of 100. The word ‘percent’ literally means ‘per hundred’. To convert a fraction to a percentage, multiply by 100%. For example, 3/4 = 3/4 × 100% = 75%.

百分比是以 100 为分母的分数。”percent” 一词字面意思是”每一百”。将分数转换为百分比，乘以 100%。例如，3/4 = 3/4 × 100% = 75%。

Percentage = (Part ÷ Whole) × 100%

To find 15% of 240, convert 15% to 0.15 and multiply: 0.15 × 240 = 36. Alternatively, find 10% (24), then 5% (12), and add: 24 + 12 = 36.

求 240 的 15%，将 15% 转换为 0.15 并相乘：0.15 × 240 = 36。或者先求 10%（24），再求 5%（12），然后相加：24 + 12 = 36。

2. Percentage Change | 百分比变化

Percentage change measures how much a quantity has increased or decreased, expressed as a percentage of the original value. The formula is:

百分比变化衡量一个量增加或减少了多少，以原始值的百分比表示。公式为：

Percentage Change = (New Value − Original Value) ÷ Original Value × 100%

If a price rises from £80 to £96, the change is 96 − 80 = 16. The percentage change is (16 ÷ 80) × 100% = 20%. A positive result means an increase; a negative result means a decrease.

如果价格从 80 英镑涨到 96 英镑，变化量为 96 − 80 = 16。百分比变化为 (16 ÷ 80) × 100% = 20%。正结果表示增加；负结果表示减少。

For a decrease, suppose a shirt drops from £40 to £30. The change is 30 − 40 = −10. Percentage change = (−10 ÷ 40) × 100% = −25%, meaning a 25% decrease.

对于减少的情况，假设一件衬衫从 40 英镑降至 30 英镑。变化量为 30 − 40 = −10。百分比变化 = (−10 ÷ 40) × 100% = −25%，即减少 25%。

3. Reverse Percentages | 反向百分比

Reverse percentage problems ask you to find the original value when you know the final value after a percentage change. The key is to identify the multiplier.

反向百分比问题是在已知百分比变化后的最终值的情况下，要求你找出原始值。关键在于识别乘数。

If a price increased by 12%, the original value was multiplied by 1.12. To reverse this, divide the new value by 1.12. For example, if a 12% increase results in £560, the original price is 560 ÷ 1.12 = £500.

如果价格上涨 12%，原始值乘以了 1.12。要反推，将新值除以 1.12。例如，如果 12% 的涨幅导致结果为 560 英镑，则原始价格为 560 ÷ 1.12 = 500 英镑。

For a decrease, if an item is reduced by 20%, the multiplier is 0.80. If the sale price is £64, the original price is 64 ÷ 0.80 = £80. Always ask: ‘What multiplier was applied?’

对于降价，如果商品降价 20%，乘数为 0.80。如果售价为 64 英镑，原始价格为 64 ÷ 0.80 = 80 英镑。始终问自己：”应用了什么乘数？”

4. Simple and Compound Interest | 单利与复利

Simple interest is calculated only on the principal amount. If you invest £500 at 4% per year for 3 years, the interest each year is 4% of £500 = £20. Total interest = £20 × 3 = £60.

单利仅根据本金计算。如果你以每年 4% 的利率投资 500 英镑 3 年，每年的利息为 500 英镑的 4% = 20 英镑。总利息 = 20 英镑 × 3 = 60 英镑。

Simple Interest = Principal × Rate × Time

Compound interest is calculated on the principal plus previously earned interest. After 3 years at 4% per annum: Amount = 500 × (1.04)³ = 500 × 1.124864 = £562.43 (rounded to the nearest penny).

复利基于本金加此前已赚取的利息计算。按年利率 4% 存 3 年后：本息和 = 500 × (1.04)³ = 500 × 1.124864 = 562.43 英镑（四舍五入到分）。

For compound interest, use the formula A = P(1 + r)ⁿ, where P is the principal, r is the interest rate as a decimal, and n is the number of periods.

计算复利时使用公式 A = P(1 + r)ⁿ，其中 P 为本金，r 为小数形式的利率，n 为期数。

5. Repeated Percentage Change | 连续百分比变化

Repeated percentage change problems involve multiple percentage changes applied in sequence. The overall multiplier is the product of the individual multipliers.

连续百分比变化问题涉及按顺序应用的多个百分比变化。总乘数是个体乘数的乘积。

Example: A car depreciates by 15% each year. Its value after 4 years is found by multiplying the original value by 0.85⁴. If the car was worth £20,000, its value after 4 years = 20,000 × 0.85⁴ ≈ 20,000 × 0.522 = £10,440.

示例：一辆汽车每年贬值 15%。其 4 年后的价值通过将原始值乘以 0.85⁴ 求得。如果这辆车价值 20,000 英镑，4 年后价值 = 20,000 × 0.85⁴ ≈ 20,000 × 0.522 = 10,440 英镑。

Careful: a 10% increase followed by a 10% decrease does NOT return to the original value. The multipliers are 1.10 × 0.90 = 0.99, which is a 1% overall decrease.

注意：先增加 10% 再减少 10% 并不会回到原始值。乘数为 1.10 × 0.90 = 0.99，总体减少 1%。

6. Ratios — Simplifying and Dividing | 比率 — 化简与分配

A ratio compares two or more quantities of the same kind. To simplify a ratio, divide all parts by their highest common factor. For example, 12:18 simplifies to 2:3.

比率比较两个或多个同类的量。要化简比率，将所有部分除以它们的最大公因数。例如，12:18 化简为 2:3。

To divide an amount in a given ratio, find the total number of parts. For £150 divided in the ratio 2:3, there are 2 + 3 = 5 parts. Each part is 150 ÷ 5 = £30. The two shares are 2 × 30 = £60 and 3 × 30 = £90.

按给定比率分配金额时，先求出总份数。按 2:3 分配 150 英镑，共有 2 + 3 = 5 份。每份为 150 ÷ 5 = 30 英镑。两份分别为 2 × 30 = 60 英镑和 3 × 30 = 90 英镑。

When the ratio of boys to girls in a class is 5:4 and there are 27 students, the parts total 9. One part = 3 students. Boys = 15, girls = 12.

当班级中男生与女生之比为 5:4，学生总数为 27 人时，总份数为 9。每份为 3 名学生。男生 = 15，女生 = 12。

7. Direct Proportion | 正比例

Two quantities are in direct proportion if their ratio remains constant. As one quantity doubles, the other doubles too. This relationship can be written as y = kx, where k is the constant of proportionality.

如果两个量的比值保持恒定，则它们成正比。当一个量翻倍时，另一个量也翻倍。这种关系可以写成 y = kx，其中 k 为比例常数。

Example: The cost of petrol is proportional to the number of litres. If 12 litres cost £18, then 1 litre costs 18 ÷ 12 = £1.50. The constant k = 1.50, and the cost of 20 litres = 1.5 × 20 = £30.

示例：汽油的成本与升数成正比。如果 12 升花费 18 英镑，则 1 升成本为 18 ÷ 12 = 1.50 英镑。常数 k = 1.50，20 升的成本 = 1.5 × 20 = 30 英镑。

When x and y are in direct proportion, the graph is a straight line through the origin. The gradient equals the constant of proportionality.

当 x 和 y 成正比时，图形是一条过原点的直线。斜率等于比例常数。

8. Inverse Proportion | 反比例

Two quantities are inversely proportional when one increases and the other decreases proportionally. As one doubles, the other halves. This is written as y = k ÷ x, or xy = k.

当一个量增大而另一个量按比例减小时，这两个量成反比。一个翻倍，另一个减半。写作 y = k ÷ x，或 xy = k。

Example: The time to build a wall is inversely proportional to the number of workers. If 4 workers take 9 hours, then the total work is 4 × 9 = 36 worker-hours. With 6 workers, the time is 36 ÷ 6 = 6 hours.

示例：砌墙所需时间与工人人数成反比。如果 4 名工人需要 9 小时，总工作量为 4 × 9 = 36 工时。6 名工人时，所需时间为 36 ÷ 6 = 6 小时。

For inverse proportion, the graph is a curve called a hyperbola. It never touches either axis, because k ÷ x is undefined at x = 0 and approaches 0 as x grows large.

对于反比例，图形是称为双曲线的曲线。它永不相交于任一坐标轴，因为在 x = 0 处 k ÷ x 无定义，且当 x 增大时它趋近于 0。

9. Map Scales and Unit Conversion | 地图比例尺与单位换算

Map scales are ratios that relate distances on a map to actual ground distances. A scale of 1:50,000 means that 1 cm on the map represents 50,000 cm = 500 m on the ground.

地图比例尺是将地图上的距离与实际地面距离相关联的比率。1:50,000 的比例尺表示地图上的 1 厘米代表实际地面上的 50,000 厘米 = 500 米。

To convert between units, use the scale as a multiplier. If the scale is 1:25,000 and two towns are 8 cm apart on the map, the real distance is 8 × 25,000 = 200,000 cm = 2 km.

要在单位之间进行换算，将比例尺用作乘数。如果比例尺为 1:25,000，两个城镇在地图上的距离为 8 厘米，则实际距离为 8 × 25,000 = 200,000 厘米 = 2 千米。

Common unit conversions to remember: 1 km = 1,000 m = 100,000 cm; 1 m = 100 cm = 1,000 mm; 1 tonne = 1,000 kg; 1 litre = 1,000 ml.

需要记住的常见单位换算：1 千米 = 1,000 米 = 100,000 厘米；1 米 = 100 厘米 = 1,000 毫米；1 吨 = 1,000 千克；1 升 = 1,000 毫升。

10. Percentage and Ratio in Real-Life Problems | 生活中的百分比与比率应用题

Exam questions often combine percentages and ratios in practical contexts. Profit and loss, discounts, taxes, and recipe scaling all require flexible application of these skills.

考试题目常在实际情况中结合百分比与比率。利润与亏损、折扣、税收和食谱换算都需要灵活运用这些技能。

Example: A shop buys a jacket for £60 and sells it with a 25% profit. Selling price = 60 × 1.25 = £75. During a sale, the jacket is reduced by 20%. Sale price = 75 × 0.80 = £60. The shop makes no profit or loss at the sale price.

示例：一家商店以 60 英镑购入一件夹克，以 25% 的利润售出。售价 = 60 × 1.25 = 75 英镑。在促销中，夹克降价 20%。促销价 = 75 × 0.80 = 60 英镑。以促销价出售，商店既不盈利也不亏损。

For recipe problems: if a recipe for 8 people requires 400 g of flour, the amount per person is 50 g. For 6 people, the flour needed is 6 × 50 = 300 g. Always identify the quantity per person or per unit first.

对于食谱问题：如果供 8 人食用的食谱需要 400 克面粉，则每人份为 50 克。供 6 人食用时，所需面粉为 6 × 50 = 300 克。始终先找出每人或每单位的量。

11. Common Exam Mistakes | 常见考试错误

Students often lose marks by confusing the base quantity in percentage problems. Remember: percentage change is always calculated relative to the original value, not the new value.

学生经常在百分比问题中混淆基准量而失分。记住：百分比变化始终相对于原始值计算，而不是相对于新值。

Another common error is adding percentage changes directly. A 10% increase followed by a 10% increase is NOT a 20% increase. The correct multiplier is 1.10 × 1.10 = 1.21, which is a 21% increase.

另一个常见错误是直接将百分比变化相加。先增加 10% 再增加 10% 不是增加 20%。正确乘数为 1.10 × 1.10 = 1.21，即增加 21%。

When dividing in a ratio, always add the parts first to find the total number of parts. Forgetting to divide by the total parts count and instead dividing by one part alone leads to entirely wrong shares.

按比率分配时，务必先加总各份数求出总份数。忘记除以总份数而只除以其中一份，会导致分配结果完全错误。

Always check whether your answer is reasonable. If a 30% discount is applied, the sale price must be lower than the original. A quick estimate before finalising your answer can catch careless mistakes.

始终检查答案是否合理。如果打了 30% 的折扣，售价必须低于原价。在确定最终答案前快速估算可以避免粗心错误。

12. Exam-Style Practice Questions | 考试风格练习

The following questions reflect the style of Edexcel IGCSE papers. Work through them step by step, showing your working clearly to earn method marks.

以下问题反映 Edexcel IGCSE 试卷的风格。请逐步完成，清晰展示解题过程以获得方法分。

Question 1: A laptop costs £850 in a sale after a 15% reduction. Calculate the original price before the reduction.

问题 1：一台笔记本电脑在打 85 折促销后的价格为 850 英镑。计算降价前的原始价格。

Original Price = 850 ÷ 0.85 = £1,000

Question 2: Three friends share £720 in the ratio 2:3:4. Find the amount each friend receives.

问题 2：三位朋友按 2:3:4 的比率分享 720 英镑。求每位朋友得到的金额。

Total parts = 2 + 3 + 4 = 9. One part = 720 ÷ 9 = £80. The shares are 2 × 80 = £160, 3 × 80 = £240, and 4 × 80 = £320.

总份数 = 2 + 3 + 4 = 9。每份 = 720 ÷ 9 = 80 英镑。各份额为 2 × 80 = 160 英镑、3 × 80 = 240 英镑和 4 × 80 = 320 英镑。

Question 3: y is inversely proportional to x². When x = 2, y = 9. Find y when x = 3.

问题 3：y 与 x² 成反比。当 x = 2 时，y = 9。求 x = 3 时 y 的值。

The constant k = y × x² = 9 × 4 = 36. When x = 3, y = 36 ÷ 9 = 4. Note that squaring x matters — this is not standard inverse proportion.

常数 k = y × x² = 9 × 4 = 36。当 x = 3 时，y = 36 ÷ 9 = 4。注意 x² 的平方非常关键——这不是标准的反比例。

Question 4: A bank pays compound interest of 2.5% per year. How much would £2,000 grow to after 5 years?

问题 4：一家银行每年支付 2.5% 的复利。2,000 英镑 5 年后会增长到多少？

Amount = 2000 × (1.025)⁵ ≈ 2000 × 1.1314 = £2,262.80

Master these techniques and you will approach any percentage, ratio or proportion question in the IGCSE exam with confidence. Practice until each method becomes automatic.

掌握这些技巧后，你将能够自信地应对 IGCSE 考试中任何百分比、比率或比例问题。反复练习直到每种方法都了如指掌。

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Percentages, Ratios & Proportions — Full Revision Guide | 百分比、比率与比例 — 完整复习指南