Compound Interest and Depreciation | 复利与折旧

📚 Compound Interest and Depreciation | 复利与折旧

In IGCSE Mathematics, you often meet problems where money grows or an asset loses value. These ideas are modelled by repeated percentage changes. The two most important applications are compound interest and depreciation. This article explains the formulas, shows worked examples, and points out the common mistakes to avoid in exams.

在 IGCSE 数学中，你经常会遇到钱增长或资产贬值的题目。这些情况可以用重复百分比变化来建模。两个最重要的应用是复利和折旧。本文解释相关公式、展示例题，并指出考试中需要避免的常见错误。

1. What is Compound Interest? | 什么是复利？

Compound interest means that interest is added to the original amount at the end of each year or period. The next period’s interest is then calculated on the new, larger balance. This is different from simple interest, where interest is always calculated on the original principal.

复利是指在每年或每期结束时将利息加入原始金额。下一期的利息将根据新的、更大的余额计算。这与单利不同，单利始终按照原始本金计算利息。

Simple interest	Interest is paid only on the original principal.
Compound interest	Interest is added to the principal, so interest earns interest.

单利仅按原始本金支付利息；复利将利息加入本金，因此利息也能产生利息。

2. The Simple Interest Formula | 单利公式

Before using compound interest, it is useful to recall simple interest. The simple interest earned is given by I = P × r × n / 100, where P is the principal, r is the annual interest rate, and n is the number of years. The final amount is A = P + I.

在使用复利之前，先回忆单利公式很有帮助。单利利息为 I = P × r × n ÷ 100，其中 P 为本金，r 为年利率，n 为年数。最终金额为 A = P + I。

I = P × r × n ÷ 100

A = P + I

3. The Compound Interest Formula | 复利公式

For compound interest, the amount after n years is given by A = P(1 + r/100)ⁿ. The multiplier (1 + r/100) is applied once per year, and raising it to the power n accounts for n years of growth.

复利的计算公式为 A = P(1 + r/100)ⁿ。乘数 (1 + r/100) 每年应用一次，将其提高到 n 次幂就表示 n 年的增长。

A = P(1 + r/100)ⁿ

Here, A is the final amount, P is the principal, r is the annual interest rate, and n is the number of years.

这里 A 是最终金额，P 是本金，r 是年利率，n 是年数。

4. Worked Example: Compound Interest | 例题：复利

$2000 is invested at 5% per year compound interest. Find the amount after 3 years. Substitute P = 2000, r = 5, n = 3 into the formula.

将 $2000 以 5% 的年复利投资。求 3 年后的金额。把 P = 2000，r = 5，n = 3 代入公式。

A = 2000 × (1 + 5/100)³ = 2000 × 1.05³ = 2000 × 1.157625 = $2315.25

The interest earned is the final amount minus the original principal: $2315.25 − $2000 = $315.25.

所得利息为最终金额减去原始本金：$2315.25 − $2000 = $315.25。

5. What is Depreciation? | 什么是折旧？

Depreciation is a decrease in value, such as a car losing value each year. The formula is very similar to compound interest, but the multiplier is less than 1 because the value falls. The formula is A = P(1 − r/100)ⁿ.

折旧是价值的减少，比如汽车每年贬值。公式与复利非常相似，但乘数小于 1，因为价值在下降。公式为 A = P(1 − r/100)ⁿ。

A = P(1 − r/100)ⁿ

In this formula, r is the percentage decrease per year.

在这个公式中，r 是每年下降的百分比。

6. Worked Example: Depreciation | 例题：折旧

A car costs $12,000 and depreciates by 15% each year. Find its value after 4 years. Use P = 12000, r = 15, n = 4.

一辆汽车价格为 $12,000，每年折旧 15%。求 4 年后的价值。使用 P = 12000，r = 15，n = 4。

A = 12000 × (1 − 15/100)⁴ = 12000 × 0.85⁴ = 12000 × 0.52200625 = $6264.08

The value is given to the nearest cent as $6264.08.

该价值精确到分为 $6264.08。

7. Percentage Multipliers | 百分比乘数

A percentage multiplier is a quick way to apply a percentage change. To increase by r%, multiply by (1 + r/100). To decrease by r%, multiply by (1 − r/100). For repeated changes, raise the multiplier to the power n.

百分比乘数是应用百分比变化的快捷方式。增加 r% 时，乘以 (1 + r/100)；减少 r% 时，乘以 (1 − r/100)。对于重复变化，将乘数提高到 n 次幂。

An 8% increase gives multiplier 1 + 0.08 = 1.08.
A 12% decrease gives multiplier 1 − 0.12 = 0.88.
Compound interest: multiply by (1 + r/100)ⁿ.
Depreciation: multiply by (1 − r/100)ⁿ.

增加 8% 得到乘数 1 + 0.08 = 1.08；减少 12% 得到乘数 1 − 0.12 = 0.88。复利使用 (1 + r/100)ⁿ，折旧使用 (1 − r/100)ⁿ。

8. Finding Time or Rate | 求时间或利率

If n or r is unknown, you may need to use trial and improvement, logarithms, or solve from given values. Most IGCSE core questions ask for the final amount, while extended questions may ask you to find the number of years or the annual rate.

如果 n 或 r 未知，你可能需要使用试错法、对数或根据已知值求解。大多数 IGCSE 核心考试题目要求求最终金额，而扩展考试题目可能要求你求年数或年利率。

For example, if A = 2000(1.05)ⁿ, and you know A = 2552.56, you can test values of n until the right amount appears.

例如，如果 A = 2000(1.05)ⁿ，且你知道 A = 2552.56，你可以尝试不同的 n 值，直到得到正确的金额。

9. Common Mistakes | 常见错误

Do not use simple interest when the question says compound. Do not write the multiplier as just r/100; always add 1 or subtract from 1 first. Round only at the final answer, not after each step, because this can make the final result inaccurate.

如果题目说是复利，不要使用单利。不要只把 r/100 当成乘数；必须首先加上 1 或从 1 中减去。只对最终答案取近似值，不要在每一步取近似，因为这会使最终结果不准确。

For 5% compound interest, the multiplier is 1.05, not 0.05.
For 15% depreciation, the multiplier is 0.85, not 1.15.
Keep at least 4 significant figures during intermediate calculations.

对于 5% 的复利，乘数是 1.05，而不是 0.05；对于 15% 的折旧，乘数是 0.85，而不是 1.15。在中间计算中至少保留 4 位有效数字。

10. Exam Tips | 考试技巧

Identify P, r, and n before you start. Underline the key words ‘compound’, ‘depreciates’, ‘increases’, or ‘decreases’. Show the substituted formula clearly, then calculate the final amount step by step. When money is involved, give your final answer to 2 decimal places unless the question says otherwise.

开始做题前先确定 P、r 和 n。在题目中标出 ‘compound’、’depreciates’、’increases’ 或 ‘decreases’ 这样的关键词。清楚地写出代入公式，然后逐步计算最终金额。涉及金额时，除非题目另有说明，最终答案保留两位小数。

If a question asks for interest earned, remember to subtract the original principal from the final amount.

如果题目要求所得利息，记住从最终金额中减去原始本金。

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Compound Interest and Depreciation | 复利与折旧