Compound Interest and Depreciation | 复利与折旧

📚 Compound Interest and Depreciation | 复利与折旧

In the IGCSE Mathematics syllabus, the topic of compound interest and depreciation appears in both Core and Extended papers. It tests your ability to apply percentage changes repeatedly and to rearrange exponential equations. This skill is directly relevant to real-world finance, including bank savings, loans, and the falling value of cars and equipment.

在 IGCSE 数学考纲中，复利与折旧是 Core 和 Extended 两份试卷都会考查的内容。它考查你反复运用百分比变化以及重排指数方程的能力。这项技能与现实金融直接相关，包括银行存款、贷款，以及汽车和设备的价值下降。

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

Simple interest is calculated only on the original amount (the principal) each year. If you invest 1000 at a simple interest rate of 5% per year for 3 years, the interest each year is 5% of 1000, which is 50. After 3 years, the total interest is 150.

单利只按原始金额（本金）每年计算利息。如果你以每年 5% 的单利投资 1000，为期 3 年，那么每年的利息都是 1000 的 5%，即 50。3 年后总利息为 150。

Compound interest is calculated on the current amount, which includes all interest already added. In the first year, the interest is 5% of 1000 = 50, so the new balance is 1050. In the second year, the interest is 5% of 1050 = 52.50, so the balance becomes 1102.50. The interest grows each year because it is calculated on a larger amount.

复利按当前金额计算利息，而当前金额包含了之前所有已加入的利息。第一年利息为 1000 的 5% = 50，因此新余额为 1050。第二年利息为 1050 的 5% = 52.50，余额变为 1102.50。由于利息是在更大的金额上计算的，所以每年利息都会增长。

The key difference is: simple interest produces a linear growth, while compound interest produces exponential growth. For the same principal, rate, and time, compound interest always gives more interest than simple interest when the time is greater than one year.

关键区别在于：单利产生线性增长，而复利产生指数增长。在相同的本金、利率和时间条件下，只要期限超过一年，复利所得利息总是多于单利。

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2. The Compound Interest Formula | 复利公式

The standard formula for compound interest is shown below. In this formula, A is the final amount, P is the principal, r is the interest rate, and n is the number of compounding periods.

复利的标准公式如下所示。在该公式中，A 是最终金额，P 是本金，r 是利率，n 是复利期数。

A = P (1 + r/100)ⁿ

Here, (1 + r/100) is called the multiplier. If the rate is 5%, the multiplier is 1.05. If the rate is 4.5%, the multiplier is 1.045. Multiplying the principal by this multiplier n times gives the final amount.

这里的 (1 + r/100) 称为乘数。若利率为 5%，乘数为 1.05；若利率为 4.5%，乘数为 1.045。把本金连续乘以该乘数 n 次，即可得到最终金额。

For example, calculate the amount after 6 years if 800 is invested at a compound interest rate of 4% per year. Using the formula: A = 800 × (1.04)⁶ = 800 × 1.2653 = 1012.24 (to the nearest cent).

例如：将 800 以每年 4% 的复利投资 6 年，求到期金额。代入公式：A = 800 × (1.04)⁶ = 800 × 1.2653 = 1012.24（精确到分）。

If a question asks for the interest only, subtract the principal from the final amount: Interest = A − P.

如果题目只要求利息，则用最终金额减去本金：利息 = A − P。

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3. Compounding Periods: Annual, Semi-Annual, Monthly | 复利周期：年、半年、月

In IGCSE questions, interest may be compounded more than once per year. When the compounding period is not annual, you must adjust both the rate and the number of periods.

在 IGCSE 题目中，利息可能每年复利多次。当复利周期不是一年时，你必须同时调整利率和期数。

If the annual rate is 6% and interest is compounded semi-annually, the rate per half-year is 6% ÷ 2 = 3%. For 4 years, the number of half-year periods is 4 × 2 = 8. The multiplier per period is 1.03.

如果年利率为 6%，且利息每半年复利一次，则每半年的利率为 6% ÷ 2 = 3%。对于 4 年，半年期数为 4 × 2 = 8。每期的乘数是 1.03。

If interest is compounded monthly, divide the annual rate by 12 and multiply the number of years by 12. For a monthly rate of 0.5% over 3 years, the number of periods is 36 and the multiplier is 1.005.

如果利息每月复利一次，则将年利率除以 12，并把年数乘以 12。若月利率为 0.5%，期限为 3 年，则期数为 36，乘数为 1.005。

The general rule is: rate per period = annual rate ÷ number of periods per year, and total periods = number of years × number of periods per year. The formula then becomes A = P (1 + r/100)ⁿ, where r is the rate per period and n is the total number of periods.

一般规则是：每期利率 = 年利率 ÷ 每年期数；总期数 = 年数 × 每年期数。公式变为 A = P (1 + r/100)ⁿ，其中 r 是每期利率，n 是总期数。

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4. Finding the Principal | 求本金

Some questions give the final amount and ask you to find the original principal. To do this, rearrange the formula to make P the subject: P = A ÷ (1 + r/100)ⁿ.

有些题目给出最终金额，要求求出原始本金。为此，把公式变形，使 P 成为主项：P = A ÷ (1 + r/100)ⁿ。

Suppose a bank account grows to 2000 after 5 years at a compound rate of 3% per year. Then P = 2000 ÷ (1.03)⁵ = 2000 ÷ 1.1593 = 1725.24.

假设一个银行账户在 5 年后以每年 3% 的复利增长到 2000。则 P = 2000 ÷ (1.03)⁵ = 2000 ÷ 1.1593 = 1725.24。

Always round money to two decimal places unless the question instructs otherwise. If the answer is in dollars, pounds, or another currency, include the correct unit.

除非题目另有要求，金额通常保留两位小数。如果答案以美元、英镑或其他货币为单位，请加上正确的单位。

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5. Finding the Rate | 求利率

To find the annual interest rate, rearrange the formula to isolate (1 + r/100)ⁿ, then take the n-th root. The rearranged form is: (1 + r/100)ⁿ = A ÷ P.

要求年利率，先把公式变形以单独表示 (1 + r/100)ⁿ，然后开 n 次方。变形后的形式为：(1 + r/100)ⁿ = A ÷ P。

For example, if 1500 grows to 1800 in 3 years, then (1 + r/100

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