📚 Percentage Change, Reverse Percentages and Compound Interest | 百分比变化、反向百分比与复利
Percentages appear throughout IGCSE Mathematics, from calculating discounts in shopping to modelling population growth and bank interest. This revision guide brings together percentage increase, percentage decrease, reverse percentages, repeated percentage change, simple interest and compound interest. You will learn how to use multipliers efficiently and apply the correct method under exam pressure.
百分比在 IGCSE 数学中随处可见，从购物折扣到人口增长和银行利息的建模。本篇复习指南涵盖百分比增加、百分比减少、反向百分比、连续百分比变化、单利与复利。你将学会如何高效使用乘数，并在考试压力下选择正确的方法。
1. Percentage Basics: From Fractions and Decimals | 百分比基础：分数与小数
A percentage is simply a fraction with denominator 100. The symbol % means ‘out of 100’, so 35% = 35/100 = 0.35. Being able to move quickly between percentages, fractions and decimals is essential because IGCSE questions often switch between all three forms.
百分比就是分母为 100 的分数。符号 % 表示“每 100”，因此 35% = 35/100 = 0.35。能够在百分比、分数和小数之间快速转换至关重要，因为 IGCSE 题目经常在三种形式之间切换。
- To change a percentage to a decimal, divide by 100: 7% = 0.07.
将百分比转换为小数，除以 100：7% = 0.07。 - To change a decimal to a percentage, multiply by 100: 0.45 = 45%.
将小数转换为百分比，乘以 100：0.45 = 45%。 - To change a fraction to a percentage, divide numerator by denominator and multiply by 100: 3/8 = 37.5%.
将分数转换为百分比，用分子除以分母再乘以 100：3/8 = 37.5%。
| Percentage | Fraction | Decimal |
|---|---|---|
| 50% | 1/2 | 0.5 |
| 25% | 1/4 | 0.25 |
| 10% | 1/10 | 0.1 |
| 5% | 1/20 | 0.05 |
2. The Percentage Multiplier | 百分比乘数
The fastest way to increase or decrease a quantity by a percentage is to use a multiplier. For an increase of r%, add r to 100 and divide by 100. For a decrease of r%, subtract r from 100 and divide by 100.
将一个量按某个百分比增加或减少的最快方法是使用乘数。对于增加 r%，将 r 加到 100 再除以 100。对于减少 r%，用 100 减去 r 再除以 100。
increase multiplier = (100 + r) / 100
decrease multiplier = (100 − r) / 100
For example, increasing by 15% gives a multiplier of 1.15, and decreasing by 15% gives a multiplier of 0.85. This method avoids writing separate addition or subtraction steps, which reduces mistakes in multi-step problems.
例如，增加 15% 得到乘数 1.15，减少 15% 得到乘数 0.85。这种方法避免了单独写加法或减法步骤，从而减少多步问题中的错误。
3. Percentage Increase | 百分比增加
To increase an amount by r%, multiply the original amount by (100 + r) / 100. Suppose a laptop costs $800 and its price increases by 12%. The new price is 800 × 1.12 = $896.
要将一个量增加 r%，用原始量乘以 (100 + r) / 100。假设一台笔记本电脑价格为 800 美元，价格上涨 12%。新价格为 800 × 1.12 = 896 美元。
The same method works for increases in measurements, populations, prices and lengths. Always check whether the question asks for the new amount or only the increase. If only the increase is needed, subtract the original amount from the new amount.
同样的方法适用于测量、人口、价格和长度的增加。始终检查题目要求的是新量还是仅增加量。如果只需要增加量，用新量减去原始量。
new amount = original amount × (100 + r) / 100
4. Percentage Decrease | 百分比减少
To decrease an amount by r%, multiply by (100 − r) / 100. If a jacket is reduced by 30% in a sale and its original price is $90, the sale price is 90 × 0.70 = $63.
要将一个量减少 r%，乘以 (100 − r) / 100。如果一件夹克在促销中降价 30%，原价为 90 美元，则促销价为 90 × 0.70 = 63 美元。
Percentage decrease is very common in discounts, depreciation and loss questions. Be careful: a 30% decrease followed by a 30% increase does not return the original value, because the second percentage is calculated on a different amount.
百分比减少在折扣、折旧和损失题中非常常见。注意：先减少 30% 再增加 30% 不会回到原始值，因为第二个百分比是在不同的量上计算的。
5. Percentage Change: Comparing Two Values | 百分比变化：比较两个值
When you know an original value and a new value, the percentage change is the ratio of the change to the original value, multiplied by 100. The denominator is always the original value, not the new value.
当你知道原始值和新值时，百分比变化等于变化量除以原始值再乘以 100。分母始终是原始值，而不是新值。
percentage change = (new value − original value) / original value × 100%
If the result is positive, it is a percentage increase. If negative, it is a percentage decrease. For example, if a salary rises from $2400 to $2700, the percentage increase is (2700 − 2400) / 2400 × 100% = 12.5%.
如果结果为正，则是百分比增加。如果为负，则是百分比减少。例如，如果工资从 2400 美元升到 2700 美元，百分比增加为 (2700 − 2400) / 2400 × 100% = 12.5%。
IGCSE questions often give two data points such as sales in two different years. Identify which value is the original and which is the new value before substituting into the formula.
IGCSE 题目经常给出两个数据点，例如两个不同年份的销售额。在代入公式之前，先确定哪个值是原始值，哪个是新值。
6. Reverse Percentages: Finding the Original Amount | 反向百分比：求原始量
Reverse percentage problems give you the final amount after a percentage change and ask you to find the original amount. To do this, divide the final amount by the relevant multiplier. Do not multiply by the percentage.
反向百分比问题给出百分比变化后的最终量，要求你求原始量。方法是：用最终量除以相应的乘数。不要乘以百分数。
original amount = new amount / (100 ± r) / 100
For example, after a 20% reduction, a bicycle costs $320. The original price was 320 ÷ 0.80 = $400. If you multiply 320 × 1.20 instead, you get $384, which is incorrect because the 20% was taken from the original, not from the sale price.
例如，降价 20% 后，一辆自行车售价 320 美元。原价为 320 ÷ 0.80 = 400 美元。如果你错误地用 320 × 1.20，会得到 384 美元，这是不正确的，因为 20% 是从原价中扣除的，而不是从促销价中计算的。
A useful check is to ask: ‘Should the original be larger or smaller than the final amount?’ After a decrease, the original must be larger. After an increase, the original must be smaller.
一个有效的检查方法是问：“原始量应该比最终量大还是小？”在减少之后，原始量必然更大；在增加之后，原始量必然更小。
7. Repeated Percentage Change | 连续百分比变化
When the same percentage change is applied several times, multiply the original amount by the multiplier raised to the number of periods. This is much faster than applying the percentage step by step.
当相同的百分比变化被多次应用时，将原始量乘以乘数的若干次幂。这比逐步应用百分比快得多。
new amount = original amount × mn
Here m is the multiplier and n is the number of periods. For example, a population of 5000 increases by 3% each year. After 4 years, the population is 5000 × 1.034 ≈ 5628. Notice that the increase is not simply 4 × 3% because each year’s 3% is calculated on a growing amount.
其中 m 是乘数，n 是周期数。例如，一个 5000 的人口每年增加 3%。4 年后人口为 5000 × 1.034 ≈ 5628。注意，增长并不是简单的 4 × 3%，因为每年的 3% 是在不断增大的量上计算的。
Repeated percentage change is used for compound growth, depreciation, population models and radioactive decay. If the change is a decrease, the multiplier will be less than 1, so raising it to a power will make the amount smaller over time.
连续百分比变化用于复合增长、折旧、人口模型和放射性衰变。如果变化是减少，乘数将小于 1，因此随着时间增加，量的幂次会使其越来越小。
8. Simple Interest | 单利
Simple interest is calculated only on the original principal. Each year the same amount of interest is added. The formula is:
单利仅根据原始本金计算。每年增加相同的利息额。公式为：
A = P(1 + rt / 100)
Where A is the final amount, P is the principal, r is the annual interest rate and t is the time in years. If $2000 is invested at 4% simple interest for 5 years, the interest is 2000 × 4/100 × 5 = $400, so the final amount is $2400.
其中 A 为最终金额，P 为本金，r 为年利率，t 为年数。如果 2000 美元以 4% 的单利投资 5 年，利息为 2000 × 4/100 × 5 = 400 美元，因此最终金额为 2400 美元。
Simple interest is straightforward, but IGCSE questions often test whether you can distinguish it from compound interest. The key point is that simple interest never adds previous interest to the principal.
单利很直接，但 IGCSE 题目经常考查你能否区分单利与复利。关键点是单利从不会将之前的利息加入本金。
9. Compound Interest | 复利
Compound interest calculates interest on the principal plus all interest already earned. The amount grows more quickly than under simple interest. The standard annual compound interest formula is:
复利是根据本金加上已经获得的所有利息来计算利息。金额增长比单利更快。标准的年复利公式为：
A = P(1 + r / 100)n
Where n is the number of compounding periods. If $2000 is invested at 4% compound interest for 5 years, the final amount is 2000 × 1.045 ≈ $2433.31. The extra $33.31 compared with simple interest is the effect of compounding.
其中 n 为复利周期数。如果 2000 美元以 4% 的复利投资 5 年，最终金额为 2000 × 1.045 ≈ 2433.31 美元。与单利相比多出的 33.31 美元就是复利作用的结果。
If interest is compounded more than once per year, adjust the rate and the number of periods. For r% per year compounded k times per year for t years, use r/k per period and n = kt. Many IGCSE questions use annual compounding, but some extended papers may include monthly or quarterly compounding.
如果利息一年复利多次，则要调整利率和周期数。对于年利率 r% 每年复利 k 次，持续 t 年，每期利率用 r/k，周期数 n = kt。许多 IGCSE 题目使用年复利，但一些扩展试卷可能包含月复利或季度复利。
10. Comparing Simple and Compound Interest | 单利与复利比较
The table below shows how $1000 grows at 5% per year under simple and compound interest over three years. Compound interest produces more total interest because the interest itself earns interest.
下表展示 1000 美元在 5% 年利率下三年中单利与复利的增长情况。复利产生更多总利息，因为利息本身也在产生利息。
| Year | Simple Interest Amount | Compound Interest Amount |
|---|---|---|
| 0 | $1000.00 | $1000.00 |
| 1 | $1050.00 | $1050.00 |
| 2 | $1100.00 | $1102.50 |
| 3 | $1150.00 | $1157.63 |
The two amounts are equal after the first period because 5% of the original principal is the same in both
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