📚 Mastering Percentages, Compound Interest and Reverse Percentages for IGCSE Mathematics | 掌握百分数、复利与逆向百分数——IGCSE 数学
Percentages are one of the most practical and frequently examined topics in IGCSE Mathematics. They link fractions, decimals, ratios and real-life financial situations such as interest, discounts, tax, profit and population growth. A strong understanding of percentage multipliers, reverse percentages and repeated change is essential for both Core and Extended papers.
百分数是 IGCSE 数学中最实用、最常考的主题之一。它把分数、小数、比以及利息、折扣、税收、利润和人口增长等现实情境联系在一起。扎实掌握百分数乘数、逆向百分数和连续变化，对 Core 和 Extended 试卷都至关重要。
1. Percentages, Fractions and Decimals | 百分数、分数与小数的互化
A percentage is simply a number expressed as a fraction of 100. To change a percentage to a fraction, write it over 100 and simplify if possible. For example, 45% = 45/100 = 9/20.
百分数本质上就是以 100 为分母的分数。把百分数化为分数时，先写成以 100 为分母的分数，再尽可能化简。例如 45% = 45/100 = 9/20。
To convert a percentage to a decimal, divide by 100. This is the same as moving the decimal point two places to the left. For instance, 7.5% = 0.075 and 130% = 1.30.
把百分数化为小数时，除以 100，也就是把小数点向左移动两位。例如 7.5% = 0.075，130% = 1.30。
To convert a decimal or fraction to a percentage, multiply by 100. For example, 0.68 = 68% and 3/8 = 3 ÷ 8 × 100 = 37.5%.
把小数或分数化为百分数时，乘以 100。例如 0.68 = 68%，3/8 = 3 ÷ 8 × 100 = 37.5%。
Percentage = value ÷ total × 100
百分数 = 数值 ÷ 总量 × 100
2. Percentage Increase and Decrease | 百分数增减
To increase an amount by r%, multiply the original amount by the multiplier (1 + r/100). To decrease an amount by r%, multiply by (1 – r/100). This method is faster and less error-prone than finding the change and adding or subtracting separately.
把一个量增加 r% 时，用乘数 (1 + r/100) 去乘原量；减少 r% 时，用乘数 (1 – r/100) 去乘。这个方法比先求变化量再加或减更快，也更不容易出错。
For example, increasing 80 by 15% gives 80 × 1.15 = 92. Decreasing 250 by 12% gives 250 × 0.88 = 220.
例如，80 增加 15%，得到 80 × 1.15 = 92。250 减少 12%，得到 250 × 0.88 = 220。
The multiplier must always be positive. A 100% decrease gives a multiplier of 0, while an increase of more than 100% gives a multiplier greater than 2.
乘数必须始终为正。减少 100% 时乘数为 0，而增加超过 100% 时乘数大于 2。
Increase: New = Original × (1 + r/100)
增加：新值 = 原值 × (1 + r/100)
Decrease: New = Original × (1 – r/100)
减少：新值 = 原值 × (1 – r/100)
3. Finding the Original Amount: Reverse Percentages | 逆向百分数求原值
Reverse percentage problems give the final amount after a percentage change and ask for the original amount. The correct method is to divide the final amount by the multiplier, not to subtract the percentage from the final value.
逆向百分数问题会给出百分数变化后的最终量，要求原值。正确方法是用最终量除以乘数，而不是从最终值中直接减去百分数。
For example, a shirt increases in price by 20% and now costs $240. The original price is 240 ÷ 1.20 = $200. A common mistake is to calculate 240 × 0.80 = $192, which is incorrect because 20% of 240 is not the same as 20% of the original price.
例如，一件衬衫涨价 20% 后售价为 240 美元。原价是 240 ÷ 1.20 = 200 美元。常见错误是计算 240 × 0.80 = 192 美元，这是不正确的，因为 240 的 20% 不等于原价的 20%。
If a value is decreased by r%, divide by (1 – r/100). For instance, after a 30% discount a jacket costs $63. The original price is 63 ÷ 0.70 = $90.
如果数值减少了 r%，则除以 (1 – r/100)。例如，一件夹克打七折后售价为 63 美元。原价是 63 ÷ 0.70 = 90 美元。
Original = Final ÷ (1 ± r/100)
原值 = 最终值 ÷ (1 ± r/100)
4. Repeated Percentage Change | 连续百分数变化
When the same percentage change happens several times, use powers. For example, if an investment increases by 6% per year for 5 years, the final value is Initial × (1.06)⁵, not Initial × 1.30. The 6% is applied to a new, larger amount each year, so the growth compounds.
当同一个百分数变化发生多次时，要使用幂。例如，如果一项投资每年增长 6%，持续 5 年，最终价值是 初始值 × (1.06)⁵，而不是 初始值 × 1.30。因为每年 6% 都作用在一个新的、更大的金额上，所以增长会复利累积。
For successive different percentage changes, multiply the multipliers in order. A 10% increase followed by a 20% decrease gives an overall multiplier of 1.10 × 0.80 = 0.88, which is a 12% overall decrease.
对于连续不同的百分数变化，按顺序把乘数相乘。先增加 10% 再减少 20%，整体乘数是 1.10 × 0.80 = 0.88，也就是整体下降 12%。
Never add or subtract percentages when the base has changed. A 15% increase followed by a 15% decrease does not return to the original value, because the 15% decrease is taken from the larger increased amount.
当基数已经改变时，绝不能直接把百分数相加或相减。先增加 15% 再减少 15%，不会回到原值，因为 15% 的减少是从增大后的量中扣除的。
Final after n changes = Initial × (1 + r/100)ⁿ
n 次变化后的值 = 初始值 × (1 + r/100)ⁿ
5. Compound Interest: The Key Formula | 复利核心公式
Compound interest means interest is added to the principal at the end of each period, and the next period’s interest is calculated on the new total. The formula is A = P(1 + r/100)ⁿ, where A is the final amount, P is the principal, r is the interest rate per period, and n is the number of periods.
复利表示每个周期结束时把利息加入本金，下一个周期的利息按新的总额计算。公式为 A = P(1 + r/100)ⁿ，其中 A 是最终金额，P 是本金，r 是每周期利率，n 是周期数。
For example, $2000 is invested at 5% per year compound interest. After 3 years, A = 2000 × (1.05)³ = 2000 × 1.157625 = $2315.25. The compound interest earned is 2315.25 – 2000 = $315.25.
例如，2000 美元按年利率 5% 的复利投资。3 年后，A = 2000 × (1.05)³ = 2000 × 1.157625 = 2315.25 美元。所获复利为 2315.25 – 2000 = 315.25 美元。
If interest is compounded more than once a year, adjust r and n. For monthly compounding at a nominal annual rate of 6%, use r = 6/12 = 0.5% per month and n = number of months.
如果每年复利多次，需要调整 r 和 n。例如名义年利率为 6%，按月复利，则每月利率 r = 6/12 = 0.5%，n 为总月数。
A = P(1 + r/100)ⁿ
最终金额 = 本金 × (1 + 每期利率/100)的期数次方
6. Depreciation and Exponential Decay | 折旧与指数衰减
Depreciation is a decrease in value by a fixed percentage each year. The value of a car, for example, can be modelled by A = P(1 – r/100)ⁿ, where P is the original value, r is the annual rate of depreciation, and n is the number of years.
折旧是指价值每年按固定百分比下降。例如汽车的价值可以用 A = P(1 – r/100)ⁿ 来建模，其中 P 是原始价值，r 是年折旧率，n 是年数。
A car bought for $15,000 depreciates by 18% per year. After 4 years, its value is 15000 × (0.82)⁴ = 15000 × 0.4521 ≈ $6781.50.
一辆汽车以 15000 美元购入，每年折旧 18%。4 年后，其价值为 15000 × (0.82)⁴ = 15000 × 0.4521 ≈ 6781.50 美元。
Exponential decay also applies to radioactive decay, cooling, and reducing populations. The same formula works whenever a quantity falls by a constant percentage per time period.
指数衰减也适用于放射性衰变、冷却和人口减少。只要一个量每个时间段按固定百分比下降，就可以使用同一个公式。
Decay: Final = Initial × (1 – r/100)ⁿ
衰减：最终值 = 初始值 × (1 – r/100)ⁿ
7. Multipliers and Efficient Problem Solving | 乘数与高效解题
A percentage multiplier is a single decimal number that replaces the original amount in one step. For a 25% increase, the multiplier is 1.25; for a 40% decrease, it is 0.60. Using multipliers saves time and reduces arithmetic errors.
百分数乘数是用一个单一步骤代替原值的十进制数。增加 25% 时乘数为 1.25；减少 40% 时乘数为 0.60。使用乘数可以节省时间并减少算术错误。
When several changes occur, multiply the individual multipliers. For example, a value rises by 12% then falls by 8%. The combined multiplier is 1.12 × 0.92 = 1.0304, so the overall change is a 3.04% increase, not a 4% increase.
当发生多次变化时，将各个乘数相乘。例如，一个数值先上升 12% 再下降 8%。组合乘数是 1.12 × 0.92 = 1.0304，因此总体变化是增长 3.04%，而不是增长 4%。
You can also use multipliers to compare offers. A 20% discount followed by a further 10% discount gives an overall multiplier of 0.80 × 0.90 = 0.72, which is a 28% total reduction, not a 30% reduction.
你还可以用乘数来比较优惠。先打八折再打九折，整体乘数为 0.80 × 0.90 = 0.72，即总共降价 28%，而不是 30%。
8. Percentage Error and Bounds | 百分误差与界限
When a measurement is rounded, the true value lies within an interval called the upper and lower bounds. Percentage error compares the maximum possible error with the measured value.
当测量值被四舍五入时，真值位于上限和下限之间。百分误差将最大可能误差与测量值进行比较。
The percentage error is calculated as: percentage error = (maximum error ÷ measured value) × 100. For example, a length is recorded as 25 cm to the nearest cm. The maximum error is 0.5 cm, so the percentage error is (0.5 ÷ 25) × 100 = 2%.
百分误差的计算公式为：百分误差 = (最大误差 ÷ 测量值) × 100。例如，一个长度记录为 25 cm，精确到最近整厘米。最大误差是 0.5 cm，因此百分误差为 (0.5 ÷ 25) × 100 = 2%。
In problems involving bounds, always identify the smallest and largest possible values before calculating area, speed, density or other derived quantities. Use the upper and lower bounds consistently.
在涉及界限的问题中，计算面积、速度、密度或其他导出量之前，要先确定可能的最小值和最大值。要一致地使用上限和下限。
Percentage error = (Maximum absolute error ÷ Measured value) × 100
百分误差 = (最大绝对误差 ÷ 测量值) × 100
9. Real-Life Applications in IGCSE Problems | IGCSE 真题中的实际应用
IGCSE questions often embed percentage work in real-world contexts. You may need to calculate sale prices, VAT, income tax, profit or loss, currency conversion, or population growth.
IGCSE 题目经常把百分数运算嵌入现实情境中。你可能需要计算折扣价、增值税、所得税、利润或亏损、货币兑换或人口增长。
For example, a shop buys a watch for $40 and sells it for $56. The profit is $16, so the percentage profit is (16 ÷ 40) × 100 = 40%. Profit percentage is always based on the cost price unless the question says otherwise.
例如，一家商店以 40 美元买入一块手表，以 56 美元卖出。利润是 16 美元，因此利润率是 (16 ÷ 40) × 100 = 40%。除非题目另有说明，利润率总是以成本价为基础。
Population growth can be modelled with the compound interest formula. If a town has 12,000 people and grows by 3% per year, after 10 years its population is 12000 × (1.03)¹⁰ ≈ 16127 people.
人口增长可以用复利公式建模。如果一个城镇有 12000 人，每年增长 3%，10 年后人口约为 12000 × (1.03)¹⁰ ≈ 16127 人。
Discount and tax questions often combine percentage increase and decrease. A TV has a list price of $500. It is given a 15% discount, then 8% VAT is added. The final price is 500 × 0.85 × 1.08 = $459.
折扣和税收题经常结合百分数增减。一台电视标价 500 美元，先打八五折，再加 8% 增值税。最终价格为 500 × 0.85 × 1.08 = 459 美元。
10. Common Mistakes and Exam Tips | 常见错误与应试技巧
One of the most common errors in percentage problems is adding or subtracting percentages when the base has changed. Always draw a timeline or write the multiplier at each stage to avoid this.
百分数问题中最常见的错误之一，是在基数改变后仍直接加减百分数。要避免这种错误，可以画出时间线或在每一步写出乘数。
Another frequent mistake is using the final amount as the base in reverse percentage questions. Remember: if the final value is after an increase, divide by (1 + r/100); if after a decrease, divide by (1 – r/100).
另一个常见错误是在逆向百分数题中把最终值当作基数。请记住：如果最终值是增加后的，除以 (1 + r/100)；如果是减少后的，除以 (1 – r/100)。
In compound interest, check whether interest is compounded annually, monthly, or quarterly. Adjust the rate and number of periods carefully. Also give money answers to 2 decimal places when required by the question.
在复利题中，要检查利息是按年、按月还是按季度复利。仔细调整利率和期数。当题目要求时，货币答案要保留两位小数。
Use estimation to check if your answer is reasonable. If a price after a 10% increase is roughly 110% of the original, then the original should be slightly less than the final price, not significantly different.
用估算来检查答案是否合理。如果增加 10% 后的价格大约是原价的 110%，那么原价应略低于最终价格，而不是相差很大。
Core multiplier summary: Increase × (1 + r/100) | Decrease × (1 – r/100) | Reverse ÷ (1 ± r/100)
核心乘数总结：增加 × (1 + r/100) | 减少 × (1 – r/100) | 逆向 ÷ (1 ± r/100)
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