📚 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
ge 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。
00. 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
age 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
st means interest is added to the principal at the end of each period, and the next period' here 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。
re 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
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。
veral 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。
e 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。
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。
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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