Mastering Compound Interest & Growth | 精通复利与增长

📚 Mastering Compound Interest & Growth | 精通复利与增长

Compound interest is one of the most powerful ideas in mathematics and finance. It is the interest earned on both the original amount of money (the principal) and any interest that has already been added. This means the total balance grows faster and faster over time, creating an exponential pattern. In the IGCSE Edexcel Mathematics syllabus, understanding compound interest is essential for solving problems about savings, loans, population growth, and depreciation.

复利是数学和金融中最强大的概念之一。它不仅对本金产生利息，还对已经累积的利息产生利息。这意味着总余额会随时间以越来越快的速度增长，形成指数模式。在IGCSE Edexcel数学课程中，理解复利对于解决储蓄、贷款、人口增长和折旧等问题至关重要。

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1. Simple Interest vs Compound Interest | 单利与复利对比

Before diving into compound interest, it is useful to compare it with simple interest. Simple interest is calculated only on the original principal for the entire period. Compound interest is calculated on the current balance, which includes previously earned interest. The table below shows the key differences.

在深入研究复利之前，先将其与单利进行对比会很有帮助。单利只在原始本金上按整个时期计算利息。复利则在当前余额上计算利息，而当前余额包含之前已赚取的利息。下表列出了主要区别。

Feature 特点	Simple Interest 单利	Compound Interest 复利
Interest earned on 利息计算基础	Original principal only 仅原始本金	Principal + accumulated interest 本金 + 累积利息
Growth pattern 增长模式	Linear 线性	Exponential 指数
Formula after n years n年后的公式	I = Pnr/100; A = P + I I = Pnr/100；A = P + I	A = P(1 + r/100)ⁿ A = P(1 + r/100)ⁿ

For example, if you invest £1000 at 5% per year for 3 years, simple interest gives £1000 × 0.05 × 3 = £150, so the total is £1150. Compound interest gives £1000 × (1.05)³ = £1157.63, which is slightly higher because interest is earned on interest.

例如，如果你以每年5%的利率投资1000英镑3年，单利为1000 × 0.05 × 3 = 150英镑，因此总额为1150英镑。复利为1000 × (1.05)³ = 1157.63英镑，略高一些，因为利息也会产生利息。

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2. The Compound Interest Formula | 复利公式

The standard formula for compound interest when interest is compounded once per year is shown below. A represents the final amount, P is the initial principal, r is the annual interest rate as a percentage, and n is the number of years.

当利息每年复利一次时，标准复利公式如下。A表示最终金额，P是初始本金，r是年利率（用百分比表示），n是年数。

A = P(1 + r/100)ⁿ

If interest is compounded more than once per year, the formula becomes A = P(1 + r/(100m))^(mn), where m is the number of compounding periods per year. For example, if interest is compounded monthly, then m = 12 and n is the number of years.

如果一年内多次计息，公式变为 A = P(1 + r/(100m))^(mn)，其中m是每年计息次数。例如，如果按月计息，则m = 12，n为年数。

A = P(1 + r/(100m))^(mn)

In IGCSE exams, you may be asked to use either form. Always check whether the interest is compounded annually or more frequently.

在IGCSE考试中，你可能会被要求使用其中任一形式。请务必检查利息是按年复利还是更频繁地复利。

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3. Worked Example: Calculating Compound Interest | 例题：计算复利

Suppose you invest £5000 at an annual interest rate of 4% compounded yearly. How much money will you have after 3 years? What is the total interest earned?

假设你以每年4%的年利率存入5000英镑，按年复利。3年后你会有多少钱？总共获得多少利息？

Step 1: Identify P, r and n. Here P = 5000, r = 4, n = 3.

第一步：确定P、r和n。此处P = 5000，r = 4，n = 3。

Step 2: Substitute into the formula A = P(1 + r/100)ⁿ.

第二步：代入公式 A = P(1 + r/100)ⁿ。

A = 5000(1 + 4/100)³ = 5000 × 1.04³

Step 3: Calculate 1.04³ = 1.124864. Then A = 5000 × 1.124864 = £5624.32.

第三步：计算1.04³ = 1.124864。于是A = 5000 × 1.124864 = 5624.32英镑。

Step 4: The interest earned is A − P = 5624.32 − 5000 = £624.32.

第四步：获得的利息为 A − P = 5624.32 − 5000 = 624.32英镑。

Notice that the same amount under simple interest would earn only 5000 × 0.04 × 3 = £600, so compound interest gives £24.32 more over three years.

请注意，同样的金额在单利下只能获得5000 × 0.04 × 3 = 600英镑，因此复利在三年内多获得24.32英镑。

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4. Reverse Calculations: Finding Principal, Rate or Time | 反向计算：求本金、利率或时间

Sometimes you are given the final amount and need to find the original principal, the interest rate, or the number of years. Rearranging the compound interest formula allows us to solve each of these problems.

有时题目给出最终金额，需要你求原始本金、利率或年数。重新整理复利公式即可解决这些问题。

Finding the principal P: from A = P(1 + r/100)ⁿ, we get

求本金P：由 A = P(1 + r/100)ⁿ，得到

P = A / (1 + r/100)ⁿ

Finding the rate r: first find (A/P)^(1/n) = 1 + r/100, then r = 100[(A/P)^(1/n) − 1].

求利率r：先求 (A/P)^(1/n) = 1 + r/100，然后 r = 100[(A/P)^(1/n) − 1]。

r = 100[(A/P)^(1/n) − 1]

Finding the time n: take logarithms of both sides: n = log(A/P) / log(1 + r/100).

求时间n：对方程两边取对数：n = log(A/P) / log(1 + r/100)。

n = log(A/P) ÷ log(1 + r/100)

For example, if an investment grows from £800 to £1000 in 5 years at compound interest, the rate is found as follows: (1 + r/100)⁵ = 1000/800 = 1.25, so 1 + r/100 = 1.25^(1/5) ≈ 1.04564, giving r ≈ 4.56%.

例如，如果一笔投资在5年内从800英镑增长到1000英镑，则利率计算如下：(1 + r/100)⁵ = 1000/800 = 1.25，因此1 + r/100 = 1.25^(1/5) ≈ 1.04564，所以r ≈ 4.56%。

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5. Exponential Growth and Decay | 指数增长与衰减

Compound interest is just one example of exponential growth. Many real-world quantities increase by a fixed percentage each year: populations, incomes, and the value of some investments. The general growth formula is the same as compound interest.

复利只是指数增长的一个例子。许多现实世界的量会每年按固定百分比增加：人口、收入以及某些投资的价值。一般增长公式与复利相同。

A = P(1 + r/100)ᵗ

Exponential decay occurs when a quantity decreases by a fixed percentage each year. The formula is identical except the sign changes.

指数衰减发生在量每年按固定百分比减少时。公式相同，只是符号发生变化。

A = P(1 − r/100)ᵗ

For example, a town has a population of 20,000 that grows at 2% per year. After 5 years, the population is 20,000 × (1.02)⁵ ≈ 22,081. If a car is bought for £15,000 and depreciates at 12% per year, its value after 3 years is 15,000 × (0.88)³ ≈ £10,222.

例如，一个小镇有20,000人口，每年增长2%。5年后，人口为20,000 × (1.02)⁵ ≈ 22,081。如果一辆汽车以15,000英镑购买，每年贬值12%，则3年后的价值为15,000 × (0.88)³ ≈ 10,222英镑。

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6. Repeated Percentage Change | 重复百分比变化

When a quantity undergoes a series of percentage changes, we can use multipliers to combine them. Increasing by 10% means multiplying by 1.10; decreasing by 10% means multiplying by 0.90. If both changes happen one after another, the overall multiplier is the product of the individual multipliers.

当一个量经历一系列百分比变化时，我们可以使用乘数将它们合并。增加10%意味着乘以1.10；减少10%意味着乘以0.90。如果两个变化接连发生，总乘数是各个乘数的乘积。

overall multiplier = (1 + r₁/100) × (1 + r₂/100)

For instance, a price rises by 10% and then falls by 10%. The overall multiplier is 1.10 × 0.90 = 0.99, which means the final price is 99% of the original — a net decrease of 1%. This surprises many students because the two percentage changes do not cancel out.

例如，某价格上涨10%，然后下跌10%。总乘数为1.10 × 0.90 = 0.99，这意味着最终价格是原价的99%——净减少1%。许多学生对此感到惊讶，因为两个百分比变化并不相互抵消。

When the same percentage change repeats n times, use the power rule: multiplier = (1 + r/100)ⁿ for an increase, or (1 − r/100)ⁿ for a decrease. This is exactly the compound interest formula applied to any quantity.

当相同的百分比变化重复n次时，使用幂法则：增加时乘数为 (1 + r/100)ⁿ，减少时为 (1 − r/100)ⁿ。这正是复利公式应用于任意量的形式。

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7. Comparing Compound Interest Options | 比较不同复利方案

Banks may offer the same nominal interest rate but compound with different frequencies. The more frequently interest is compounded, the higher the effective annual rate. For example, 4% compounded quarterly actually gives 1% each quarter, so the effective annual rate is (1.01)⁴ − 1 = 4.0604%.

银行可能提供相同的名义利率，但复利频率不同。利息复利越频繁，有效年利率就越高。例如，4%按季度复利实际上是每季度1%，因此有效年利率为 (1.01)⁴ − 1 = 4.0604%。

Compounding Frequency 计息频率	m (times per year) m（每年次数）	Effective Annual Rate 有效年利率
Annually 每年	1	4.000%
Semi-annually 每半年	2	(1.02)² − 1 = 4.040%
Quarterly 每季度	4	(1.01)⁴ − 1 = 4.060%
Monthly 每月	12	(1 + 0.04/12)¹² − 1 ≈ 4.074%

The formula for

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