📚 IGCSE Mathematics: Mastering Percentages and Compound Interest | IGCSE 数学：掌握百分数与复利
Percentages appear throughout the IGCSE Mathematics syllabus, from a simple percentage of an amount to compound interest and reverse percentage problems. This revision guide brings together the essential methods, the standard formulas and the most common exam-style applications so that you can work confidently both with and without a calculator.
百分数贯穿 IGCSE 数学考纲，从基础的“求一个数的百分之几”，到复利与反向百分数问题都会出现。本复习指南整合了核心方法、标准公式与最常见的考试应用，帮助你无论是否使用计算器都能自信作答。
The key idea behind every percentage question is the multiplier: a single number that you multiply by to move from the old value to the new value. Once you are fluent with multipliers, increase, decrease, interest, depreciation and reverse percentage questions all become variations of the same skill.
所有百分数题目背后的核心思想都是乘数：用一个数乘以原值即可得到新值。一旦熟练掌握乘数，增减、利息、折旧与反向百分数问题本质上都是同一技能的变形。
1. Percentages, Fractions and Decimals | 百分数、分数与小数的互化
A percentage is simply a fraction with denominator 100: 35% means 35 out of 100, which is 35/100 or 7/20 in its simplest form. Because the denominator is always 100, converting between percentages and decimals is nothing more than moving the decimal point two places.
百分数本质上是以 100 为分母的分数：35% 表示一百分之三十五，即 35/100，约分后为 7/20。因为分母永远是 100，所以百分数与小数之间的互化只是小数点移动两位而已。
To convert a percentage to a decimal, divide by 100. To convert a decimal to a percentage, multiply by 100%. To convert a fraction to a percentage, divide the numerator by the denominator and then multiply by 100%.
百分数化小数，除以 100；小数化百分数，乘以 100%；分数化百分数，先用分子除以分母，再乘以 100%。
35% = 35 ÷ 100 = 0.35 | 0.175 × 100% = 17.5% | 7/20 = (7 ÷ 20) × 100% = 35%
Small percentages need care with place value. For example, 8% = 0.08 and 0.4% = 0.004, not 0.4. A common exam slip is to write 6% as 0.6 instead of 0.06.
较小的百分数要特别注意数位。例如 8% = 0.08，0.4% = 0.004，而不是 0.4。考试中常见的失误就是把 6% 写成 0.6 而不是 0.06。
- 1/2 = 50%, 1/4 = 25%, 3/4 = 75%, 1/5 = 20%, 1/8 = 12.5%, 1/10 = 10%
1/2 = 50%，1/4 = 25%，3/4 = 75%，1/5 = 20%，1/8 = 12.5%，1/10 = 10% - 1/3 = 33⅓% = 33.3% (3 s.f.), 2/3 = 66⅔% = 66.7% (3 s.f.)
1/3 = 33⅓% ≈ 33.3%（三位有效数字），2/3 = 66⅔% ≈ 66.7%（三位有效数字） - Memorising these pairs saves valuable time in non-calculator papers.
记住这些对应关系，能在非计算器试卷中节省大量时间。
2. Finding a Percentage of an Amount | 求一个数的百分之几
is works for every percentage, calculator or not, and it is the safest method to show in your working.
标准做法是把百分数写成小数再相乘：结果 = (P ÷ 100) × 原数。这个方法适用于任何百分数，无论是否使用计算器，也是最稳妥的书写步骤。
P% of A = (P ÷ 100) × A
For example, 15% 0.15 × 240 = 36. Without a calculator, build the percentage from easy building blocks: 10% is the amount divided by 10, 5% is half of the 10% value, and 1% is the amount divided by 100.
例如，240 的 15% = 0.15 × 240 = 36。若不用计算器，就用容易的“积木块”拼出来：10% 是原数除以 10，5% 是 10% 的一半，1% 是原数除以 100。
Worked example without a calculator: find 17.5% of 400. Take 10% = 40, 5% = 20 and 2.5% = 10, so the total is 40 + 20 + 10 = 70. The same answer comes from 0.175 × 400 = 70.
非计算器例题：求 400 的 17.5%。10% = 40，5% = 20，2.5% = 10，合计 40 + 20 + 10 = 70。用 0.175 × 400 = 70 结果一致。
- 10% → divide by 10; 1% → divide by 100; 50% → divide by 2
10% → 除以 10；1% → 除以 100；50% → 除以 2 - 5% → half of 10%; 20% → divide by 5; 25% → divide by 4
5% → 10% 的一半；20% → 除以 5；25% → 除以 4 - Application: a school has 320 students and 45% are girls, so 320 × 0.45 = 144 girls.
应用：某校有 320 名学生，45% 是女生，则 320 × 0.45 = 144 名女生。
3. Percentage Increase and Decrease | 百分数的增加与减少
Rather than calculating the increase and adding it on, use a single multiplier. An increase of p% gives the multiplier (1 + p ÷ 100); a decrease of p% gives the multiplier (1 − p ÷ 100).
不要先算增加量再加回去，而应使用单一乘数。增加 p% 的乘数是 (1 + p ÷ 100)；减少 p% 的乘数是 (1 − p ÷ 100)。
= Original × (1 + p ÷ 100) or New value = Original × (1 − p ÷ 100)
Example: a coat costs £84 and the price rises by 15%, so the new price is 84 × 1.15 = £96.60. Example: 250 kg of flour is reduced by 12%, giving 250 × 0.88 = 220 kg.
例：一件外套原价 £84，涨价 15%，新价为 84 × 1.15 = £96.60。例：250 kg 面粉减少 12%，得 250 × 0.88 = 220 kg。
Read the wording carefully. "Increase by 15%" means the new value is 115% of the original, so the multiplier is 1.15. An increase of 15 percentage points is a different idea and is not the same as a 15% increase when the starting value is not 100.
读题要仔细。“增加 15%”意味着新值是原值的 115%，乘数为 1.15。而“增加 15 个百分点”是另一个概念，当起始值不是 100 时，它与增加 15% 并不相同。
- +8% → ×1.08; +100% → ×2; +0.5% → ×1.005
+8% → ×1.08；+100% → ×2；+0.5% → ×1.005 - −8% → ×0.92; −100% → ×0; −0.5% → ×0.995
−8% → ×0.92；−100% → ×0；−0.5% → ×0.995 - Sanity check: an increase must give a multiplier greater than 1, a decrease less than 1.
合理性检验：增加得到大于 1 的乘数，减少得到小于 1 的乘数。
4. Repeated Percentage Change | 连续百分数变化
When several percentage changes happen one after another, multiply the multipliers together in order. Never add the percentages. Each change is applied to the value produced by the previous change.
当多次百分数变化接连发生时，把各乘数依次相乘，绝不能把百分数相加。每一次变化都作用在上一次变化后的数值上。
Final value = Original × m₁ × m₂ × m₃ × …
Example: £500 rises by 20% and then falls by 20%. The calculation is 500 × 1.2 × 0.9 = 540, so the final value is £540, a net fall of 4%, not a return to £500.
例：£500 先涨 20%，再跌 20%。计算为 500 × 1.2 × 0.9 = 540，最终为 £540，净跌 4%，而不是回到 £500。
This is the thinking behind compound interest: a value that grows by the same percentage each year for n years is multiplied by the same multiplier n times, which is written as a power.
这正是复利背后的思路：若一个数值每年按相同百分比增长，连续 n 年，就相当于把同一个乘数相乘 n 次，写成幂的形式。
- 500 × 1.2 × 0.9 = 540
500 × 1.2 × 0.9 = 540 - Net change = 540 ÷ 500 = 1.08, i.e. +8%
净变化 = 540 ÷ 500 = 1.08，即 +8% - Three years of 6% growth: P × 1.06 × 1.06 × 1.06 = P × 1.06³
连续三年增长 6%：P × 1.06 × 1.06 × 1.06 = P × 1.06³
5. Simple Interest | 单利
Simple interest is paid only on the original amount, so the interest is the same every year. The formula uses the principal P, the annual rate R per cent and the time T in years.
单利只按最初的本金计算利息，因此每年的利息相同。公式涉及本金 P、年利率 R（百分数）和时间 T（年）。
I = P × R × T ÷ 100 Total = P + I
Example: £1500 is invested at 4% simple interest for 3 years. The interest is 1500 × 4 × 3 ÷ 100 = £180, so the total value is 1500 + 180 = £1680. Each year adds exactly £60.
例：£1500 以 4% 单利存 3 年。利息为 1500 × 4 × 3 ÷ 100 = £180，最终总额 1500 + 180 = £1680，每年恰好增加 £60。
Recognise simple interest questions from wording such as "simple interest" or "the same amount of interest each year". If the question says the interest is added to the account and then earns interest itself, it is compound interest instead.
看到“单利”或“每年利息相同”这样的表述，就判断为单利题。若题目说利息计入账户并继续产生利息，那就是复利。
- Interest per year = P × R ÷ 100
每年利息 = P × R ÷ 100 - Total interest = interest per year × T
总利息 = 每年利息 × T - Simple interest grows linearly, compound interest grows exponentially.
单利线性增长，复利指数增长。
6. Compound Interest | 复利
With compound interest, the interest earned each year is added to the balance, and the next year's interest is calculated on that larger balance. The result is a repeated percentage increase, so we use a power.
复利是把每年所得利息加入余额，下一年的利息按这个更大的余额计算。本质上就是反复的百分数增加，因此使用幂。
A = P × (1 + R ÷ 100)ⁿ Interest = A − P
Example: £2000 is invested at 5% per year compound interest for 3 years. Then A = 2000 × 1.05³ = 2000 × 1.157625 = £2315.25, so the interest earned is £315.25. Compare this with simple interest, which would give only £300.
例：£2000 以年利率 5% 复利投资 3 年。A = 2000 × 1.05³ = 2000 × 1.157625 = £2315.25，利息为 £315.25。若按单利计算，利息只有 £300。
If you are unsure, a year-by-year table is an accepted method and often earns method marks. Show the working clearly and round only at the final stage.
如果不确定，逐年列表是被接受的解法，通常也能拿到方法分。步骤要写清楚，只在最后一步取近似值。
| Year 年 | Start 年初 | ×1.05 乘以 1.05 | End 年末 |
| 1 | 2000.00 | 2100.00 | 2100.00 |
| 2 | 2100.00 | 2205.00 | 2205.00 |
| 3 | 2205.00 | 2315.25 | 2315.25 |
7. Depreciation | 折旧
Depreciation is compound interest in reverse: the value falls by the same percentage each year. The multiplier is less than 1, so the power produces a decreasing sequence.
折旧是“反向的复利”：价值每年按相同百分比下降。乘数小于 1，因此幂运算产生递减数列。
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