📚 Percentages and Percentage Change for IGCSE Mathematics | IGCSE 数学中的百分数与百分比变化
Percentages are one of the most practical topics in IGCSE Mathematics. They appear in financial problems, data comparisons, percentage change, profit and loss, and repeated growth. This guide covers the key methods, worked examples and examiner-friendly techniques you need for Core and Extended levels.
百分数是 IGCSE 数学中最实用的主题之一。它出现在金融问题、数据比较、百分比变化、利润与亏损以及重复增长等问题中。本指南涵盖核心与扩展级别所需的关键方法、例题和适合考官的解题技巧。
1. Understanding Percentages and Key Equivalences | 理解百分数与关键等价关系
A percentage is a way of representing a number as a fraction of 100. The symbol % means 'out of 100'. For example, 35% means 35 hundredths, which can be written as 35/100, 0.35 or 7/20 in simplest form. Understanding this equivalence is essential before using percentages in calculations.
百分数是一种把数字表示为分母为 100 的分数的表示方式。符号 % 表示 '每一百' 或 '占 100 中的多少'。例如，35% 表示 35/100，也可以写成 0.35 或最简分数 7/20。在进行百分数计算之前，理解这种等价关系非常重要。
You should memorise common conversions to save time in the examination.
你应当熟记常见换算，以便在考试中节省时间。
| Fraction | Decimal | Percentage |
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/10 | 0.1 | 10% |
| 1/100 | 0.01 | 1% |
2. Converting Fractions, Decimals and Percentages | 分数、小数和百分数的互化
To convert a fraction into a percentage, multiply by 100 and add the % sign. To convert a decimal into a percentage, multiply by 100. To convert a percentage into a decimal, divide by 100. To convert a percentage into a fraction, write it over 100 and simplify if possible.
将分数转换为百分数时，先乘以 100，再加上 % 符号。将小数转换为百分数时，乘以 100。将百分数转换为小数时，除以 100。将百分数转换为分数时，先写成以 100 为分母的分数，再尽可能化简。
Example: 0.625 = 0.625 × 100 = 62.5%. Also, 45% = 45 ÷ 100 = 0.45 = 9/20.
例如：0.625 = 0.625 × 100 = 62.5%。同样，45% = 45 ÷ 100 = 0.45 = 9/20。
3. Finding a Percentage of an Amount | 求一个数的百分数
On a calculator, find r% of a quantity by multiplying the quantity by r ÷ 100. Without a calculator, build the percentage from known facts such as 10%, 5% and 1%.
使用计算器时，要求一个量的 r%，可以用该量乘以 r ÷ 100。不使用计算器时，可以利用 10%、5% 和 1% 等已知百分比来组合计算。
Example: Find 35% of 240. Since 10% = 24, 30% = 72, 5% = 12, so 35% = 72 + 12 = 84. Alternatively, 0.35 × 240 = 84.
例如：求 240 的 35%。因为 10% = 24，30% = 72，5% = 12，所以 35% = 72 + 12 = 84。也可以直接用 0.35 × 240 = 84。
4. Expressing One Quantity as a Percentage of Another | 求一个量占另一个量的百分比
To write one quantity as a percentage of another, divide the given part by the whole and multiply by 100. Always check that both quantities are in the same units.
要写出一个量占另一个量的百分比，用已知部分除以整体，再乘以 100。务必检查两个量的单位是否相同。
percentage = (part ÷ whole) × 100%
Example: A student scores 27 marks out of 45. The percentage score is (27 ÷ 45) × 100 = 60%.
例如：一名学生得到 45 分中的 27 分。其百分比成绩为 (27 ÷ 45) × 100 = 60%。
5. Percentage Increase and Decrease | 百分数增减
Percentage increase and decrease are best handled with multipliers. An increase of r% uses the multiplier 1 + r ÷ 100. A decrease of r% uses the multiplier 1 − r ÷ 100.
百分数增减最好用乘数来处理。增加 r% 时，乘数为 1 + r ÷ 100；减少 r% 时，乘数为 1 − r ÷ 100。
increase multiplier = 1 + r ÷ 100
decrease multiplier = 1 − r ÷ 100
Example: Increase 320 by 15% multiplier is 1 + 15 ÷ 100 = 1.15, so the new value is 320 × 1.15 = 368. A decrease of 15% would give 320 × 0.85 = 272.
例如：把 320 增加 15%。乘数为 1 + 15 ÷ 100 = 1.15，因此新值为 320 × 1.15 = 368。如果减少 15%，则得到 320 × 0.85 = 272。
6. Repeated Percentage Changes and Net Percentage Effect | 连续百分数变化与净百分比效果
percentage changes, multiply all the multipliers together. For example, an increase of 10% followed by a decrease of 10% gives 1.10 × 0.90 = 0.99. The final value is 99% of the original, so the overall change is a 1% decrease.
当一个量经历多次百分数变化时，应将所有乘数相乘。例如，先增加 10%，再减少 10%，得到 1.10 × 0.90 = 0.99。最终值是原值的 99%，因此整体变化是减少 1%。
Example: After two successive decreases of 20% and 10%, the combined multiplier is 0.80 × 0.90 = 0.72, which means a 28% decrease overall.
例如：连续两次减少 20% 和 10% 后，合并乘数为 0.80 × 0.90 = 0.72，这意味着总体减少 28%。
7. Reverse Percentages | 逆百分数
Reverse percentage problems give you the final amount after a percentage change and ask for the original amount. To solve them, divide the final amount by the multiplier that produced it.
逆百分数问题给出百分数变化后的最终量，要求求出原始量。解决方法是：用最终量除以产生该变化的乘数。
original amount = final amount ÷ multiplier
Example: After a 20% increase, a jacket costs £96. The original price is 96 ÷ 1.20 = £80.
例如：一件夹克在增加 20% 后售价为 96 英镑。原价为 96 ÷ 1.20 = 80 英镑。
8. Simple and Compound Interest | 单利与复利
Simple interest is calculated only on the original principal. Compound interest is calculated on the total amount, so interest is earned on previous interest as well.
单利仅根据原始本金计算。复利则根据总金额计算，因此利息也会产生利息。
simple interest: I = P × r ÷ 100 × t
compound amount: A = P × (1 + r ÷ 100)ᵗ
Example: £500 is invested at 3% compound interest for 4 years. The final amount is 500 × 1.03⁴ = 500 × 1.12550881 ≈ £562.75.
例如：将 500 英镑以 3% 的复利投资 4 年。最终金额为 500 × 1.03⁴ = 500 × 1.12550881 ≈ 562.75 英镑。
9. Profit, Loss, Discount and Percentage Change in Context | 利润、亏损、折扣与实际百分数变化
Profit or loss is always expressed as a percentage of the cost price, not the selling price. Discount is usually expressed as a percentage of the marked or original price.
利润或亏损总是表示为成本价的百分比，而不是售价的百分比。折扣通常表示为标价或原价的百分比。
profit % = (profit ÷ cost price) × 100%
discount % = (discount ÷ original price) × 100%
Example: A shop buys a bag for $40 and sells it for $52. The profit is $12, so the profit percentage is (12 ÷ 40) × 100 = 30%.
例如：一家商店以 40 美元买入一个包，以 52 美元卖出。利润为 12 美元，因此利润百分比为 (12 ÷ 40) × 100 = 30%。
10. Exam Tips and Common Mistakes | 考试技巧与常见错误
Many errors come from using the wrong original amount. In a reverse percentage, do not subtract 20% from the final amount; divide by the multiplier instead. Also remember that 5% is 0.05, not 0.5, and that percentage changes cannot simply be added when they act on different base amounts.
许多错误源于使用了错误的原始量。在逆百分数问题中，不要从最终量直接减去 20%，而应该除以相应的乘数。还要记住 5% 是 0.05，而不是 0.5；当百分数变化作用于不同基数时，不能简单地相加。
Use multipliers instead of repeated addition and subtraction. Always state the base amount for a percentage. Check whether interest is simple or compound. Write clear working to gain method marks.
使用乘数而不是反复加减。始终明确百分数所对应的基数。检查题目是单利还是复利。书写清晰的计算过程以获得方法分。
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