📚 Percentage Change & Compound Interest | 百分比变化与复利
Percentage change and compound interest are among the most frequently tested topics in IGCSE Mathematics. They appear not only in the Number section but also as the foundation for problems in Algebra, Statistics, and Financial Mathematics. A solid grasp of these concepts will help you solve both routine and problem-solving questions with confidence.
百分比变化与复利是 IGCSE 数学中最常考的内容之一。它们不仅出现在数论（Number）部分，也是代数、统计与金融数学问题的基础。扎实掌握这些概念，将帮助你在常规题与应用题中均能自信作答。
1. Basic Percentage Change | 基础百分比变化
The percentage change compares the difference between the new value and the original value relative to the original value. The formula is:
Percentage Change = (New Value − Original Value) ÷ Original Value × 100%
If the result is positive, the change is an increase. If it is negative, the change is a decrease. For example, a price rising from $50 to $65 gives a change of (65 − 50) ÷ 50 × 100% = 30%, which is a 30% increase.
百分比的变更是将新值与原始值之差，除以原始值，再乘以 100%。公式为：
百分比变化 =（新值 − 原始值）÷ 原始值 × 100%
若结果为正，则为增加；若结果为负，则为减少。例如，价格从 50 美元涨至 65 美元，变化为 (65 − 50) ÷ 50 × 100% = 30%，即上涨了 30%。
2. Finding the New Value After a Percentage Change | 求百分比变化后的新值
To find the new value after a percentage increase, multiply the original value by the multiplier (1 + p/100), where p is the percentage increase. For a decrease, multiply by (1 − p/100).
求百分比增加后的新值，需将原始值乘以乘数 (1 + p/100)，其中 p 为增加的百分数；若是减少，则乘以 (1 − p/100)。
For example, increasing 200 by 15% gives 200 × 1.15 = 230. Similarly, decreasing 200 by 15% gives 200 × 0.85 = 170. Always identify the multiplier first; this will also be the key to handling reverse problems and compound changes.
例如，将 200 增加 15% 得 200 × 1.15 = 230；将 200 减少 15% 得 200 × 0.85 = 170。一定要先确定乘数，这也是解决反向问题与连续变化问题的关键。
3. Reverse Percentage Change | 反向百分比变化
A reverse percentage problem asks for the original value before a change. If a value after an increase of 20% is 480, we write: Original × 1.20 = 480, so Original = 480 ÷ 1.20 = 400. Note that the original value is not found by subtracting 20% of 480.
反向百分比问题要求的是变化前的原始值。若某值增加 20% 后为 480，则写作：原始值 × 1.20 = 480，所以原始值 = 480 ÷ 1.20 = 400。注意，不能直接用 480 减去 480 的 20% 来求原值。
To succeed here, convert the percentage change to a multiplier, then divide the final value by that multiplier. This skill is frequently tested in exam questions involving discounts, taxes, and profit margins.
在这类题目中，应先将百分比变化转化为乘数，再用最终值除以该乘数。这一技能在涉及折扣、税费与利润率的问题中经常出现。
4. Successive Percentage Change | 连续百分比变化
When two or more percentage changes are applied one after another, they cannot be simply added. For instance, increasing by 10% then decreasing by 10% does not return to the original value.
当两次或多次百分比变化依次发生时，不能简单地将百分比相加。例如，先增加 10% 再减少 10%，并不会回到原始值。
Take 100 as the starting value: after a 10% increase, it becomes 110; then a 10% decrease gives 110 × 0.90 = 99. The overall change is a 1% decrease. This is a common trap in multiple-choice questions.
以 100 为初始值：增加 10% 后变为 110；再减少 10% 得 110 × 0.90 = 99。整体变化为减少 1%。这是选择题中常见的陷阱。
5. Introduction to Simple Interest | 单利入门
Simple interest is calculated only on the original principal amount. The formula is I = P × r × t, where I is the interest, P is the principal, r is the annual interest rate written as a decimal, and t is the time in years.
单利仅基于原始本金计算。公式为 I = P × r × t，其中 I 为利息，P 为本金，r 为以小数表示的年利率，t 为以年为单位的时间。
For example, if $500 is invested at 4% simple interest per year for 3 years, the interest is 500 × 0.04 × 3 = $60. The total amount is 500 + 60 = $560.
例如，将 500 美元以年利率 4% 的单利存入 3 年，利息为 500 × 0.04 × 3 = 60 美元。本息总额为 500 + 60 = 560 美元。
6. Compound Interest Formula | 复利公式
Compound interest is calculated on the principal plus any interest already earned. The total amount A is given by the formula:
A = P × (1 + r/n)ⁿᵗ
Here P is the principal, r is the annual rate as a decimal, n is the number of compounding periods per year, and t is the time in years. If interest is compounded annually, n = 1 and the formula simplifies to A = P × (1 + r)ᵗ.
复利的计算基于本金与已产生利息之和。本息总额 A 的公式为：
A = P × (1 + r/n)ⁿᵗ
其中 P 为本金，r 为小数形式的年利率，n 为每年复利次数，t 为年数。若按年复利，n = 1，则公式简化为 A = P × (1 + r)ᵗ。
7. Common Compounding Frequencies | 常见复利频率
In IGCSE problems, the most common compounding frequencies are annually, semi-annually, quarterly, and monthly. The table below summarises the values of n and the corresponding multiplier.
在 IGCSE 题目中，最常见的复利频率为按年、每半年、每季度与每月。下表总结了 n 的取值及相应乘数。
| Frequency | n (per year) | Multiplier per period |
| Annually | 1 | 1 + r |
| Semi-annually | 2 | 1 + r/2 |
| Quarterly | 4 | 1 + r/4 |
| Monthly | 12 | 1 + r/12 |
For example, with a 6% annual rate compounded quarterly, each quarter uses the multiplier 1 + 0.06/4 = 1.015. Over two years, there are 8 quarters, so A = P × (1.015)⁸.
例如，年利率 6% 按季度复利，每季度的乘数为 1 + 0.06/4 = 1.015。两年共 8 个季度，因此 A = P × (1.015)⁸。
8. Worked Example: Compound Interest | 复利例题精讲
Example: A student invests $2,000 in a savings account that pays 5% annual interest compounded annually. Find the total amount after 3 years.
例题：某学生将 2,000 美元存入年利率 5%、按年复利的储蓄账户，求 3 年后的总额。
Solution: Using A = P × (1 + r)ᵗ, with P = 2000, r = 0.05, and t = 3:
解答：使用 A = P × (1 + r)ᵗ，其中 P = 2000，r = 0.05，t = 3：
A = 2000 × (1.05)³ = 2000 × 1.157625 = 2315.25
Therefore, the total amount after 3 years is $2,315.25. Notice that the interest earned is $315.25, which is greater than the simple interest of $300 because interest is earned on interest in the second and third years.
因此，3 年后总额为 2,315.25 美元。注意利息为 315.25 美元，大于单利的 300 美元，因为在第二、三年中，利息也参与了计息。
9. Worked Example: Reverse Compound Interest | 复利反求本金例题
Example: After 4 years, an investment grown at 6% per annum compounded annually has a value of $1,262.48. What was the original principal?
例题：一笔投资以年利率 6% 按年复利增长，4 年后的价值为 1,262.48 美元，求本金。
Solution: We are solving P × (1.06)⁴ = 1262.48. Divide both sides by (1.06)⁴:
解答：解方程 P × (1.06)⁴ = 1262.48。两边同除以 (1.06)⁴：
P = 1262.48 ÷ (1.06)⁴ = 1262.48 ÷ 1.262477 = 1000.00
The original principal was $1,000. This reverse process is exactly the same skill as reverse percentage change, but applied over multiple periods.
原始本金为 1,000 美元。这一反向求解过程与反向百分比变化完全相同，只是应用于多个周期。
10. Exponential Growth and Decay | 指数增长与衰减
In IGCSE Mathematics, compound interest is a special case of exponential growth. The general model is A = P × rᵗ, where r is the growth factor per unit time. When 0 < r < 1, the model describes exponential decay.
在 IGCSE 数学中，复利是指数增长的特例。一般模型为 A = P × rᵗ，其中 r 是每单位时间的增长因子。当 0 < r < 1 时，该模型描述指数衰减。
For example, if a radioactive substance decays by 12% per year, its decay factor is 0.88. The remaining mass after t years is P × (0.88)ᵗ. These problems are common in both Mathematics and Science contexts.
例如，若一种放射性物质每年衰减 12%，则其衰减因子为 0.88。t 年后的剩余质量为 P × (0.88)ᵗ。这类问题在数学与科学情境中都十分常见。
11. Comparing Simple and Compound Interest | 单利与复利的比较
The difference between simple and compound interest becomes more significant over longer periods. In simple interest, growth is linear: the interest added each year is constant. In compound interest, growth is exponential: each year’s interest increases because it is calculated on a growing balance.
单利与复利的差异在长期投资中更加显著。单利增长是线性的，每年新增的利息恒定；而复利增长是指数的，由于利息基于不断增长的余额计算，每年的利息也在增加。
For the same principal, rate, and time, compound interest always produces a greater total amount except for t = 1, where they are equal. Examiners often ask students to explain this difference in words, so be prepared to describe it clearly.
在本金、利率与时间相同的情况下，复利所得的总额总是高于单利，除非时间 t = 1 时二者相等。考试中常要求学生用文字解释这一差异，请务必能够清晰地表述。
12. Exam Tips and Common Mistakes | 考试技巧与常见错误
Tip 1: Always convert percentages to decimals or multipliers before doing calculations. A rate of 8% becomes 0.08 or a multiplier of 1.08.
技巧一：计算前务必将百分数转化为小数或乘数。8% 应化为 0.08 或乘数 1.08。
Tip 2: Read whether the interest is simple or compound. If the word ‘simple’ is used, use I = P × r × t. If ‘compound’ is used, use the power formula.
技巧二：看清题目问的是单利还是复利。若有 simple（单利），使用 I = P × r × t；若为 compound（复利），使用乘幂公式。
Tip 3: For reverse problems, divide by the multiplier. Never subtract the percentage directly. This is the most common source of lost marks.
技巧三：反向问题要用除法，即除以乘数，切勿直接减去百分比。这是最常见的失分原因。
Tip 4: For successive percentage changes, never simply add the percentages. Instead, multiply the multipliers together step by step.
技巧四：连续百分比变化时，绝不能简单相加，而要逐步将乘数相乘。
Tip 5: Round your final answer to 2 decimal places for currency problems. Use the unrounded values in intermediate steps to avoid rounding errors.
技巧五：涉及货币的问题，最终答案四舍五入到两位小数；中间步骤保留未舍入数值，以免累积误差。
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