📚 071 Percentages – Edexcel IGCSE Maths | 071 百分比 – Edexcel IGCSE 数学
Percentages appear throughout the Edexcel IGCSE Mathematics syllabus, from basic number work to compound interest and reverse percentage problems. Understanding how to move fluently between fractions, decimals and percentages is essential for success on both calculator and non-calculator papers. This article revisits every major percentage technique you will need, with clear examples and paired bilingual explanations.
百分比贯穿Edexcel IGCSE数学考纲，从基础的数字运算到复利和反向百分比问题。能否在分数、小数和百分比之间熟练转换，对于在计算器与非计算器试卷中取得好成绩至关重要。本文用清晰的例子和双语对照讲解，带你系统复习每一个重要的百分比技巧。
1. Understanding Percentages | 理解百分比
entage is a way of expressing a number as a fraction of 100. The word comes from the Latin 'per centum', meaning 'by the hundred'. The symbol % tells you to divide by 100, so 7 % = 7/100, 30 % = 30/100 and 120 % = 120/100.
百分比是把一个数表示成100的分数的一种方式。这个词来自拉丁语'per centum'，意为“每一百”。符号%告诉你除以100，因此7% = 7/100，30% = 30/100，而120% = 120/100。
Percentages make it easy to compare proportions on the same scale. They are used in statistics, finance, science and everyday life – from exam grades to discounts. In IGCSE questions, you will often be asked to interpret percentages, calculate a percentage of a quantity, or find what percentage one value is of another.
百分比使得在同样尺度上比较比例变得容易。它们被用于统计学、金融、科学和日常生活——从考试成绩到折扣。在IGCSE题目中，你经常需要解读百分比、计算一个量的百分之几，或者求一个值是另一个值的百分之几。
2. Converting Between Fractions, Decimals and Percentages | 分数、小数和百分比之间的转换
Fluency in converting between the three forms is a core IGCSE skill. The basic rules are: from decimal to percentage, multiply by 100; from percentage to decimal, divide by 100; from fraction to percentage, first convert to a decimal (divide numerator by denominator) and then multiply by 100, or find an equivalent fraction with denominator 100.
在三种形式之间熟练转换是IGCSE的核心技能。基本规则是：从小数到百分比，乘以100；从百分比到小数，除以100；从分数到百分比，可以先转换成小数（分子除以分母）再乘以100，或者求出以100为分母的等值分数。
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/3 | 0.333... (0.3 recurring) | 33.3% or 33 1/3% |
When a question involves mixed formats, convert everything to the same type before comparing. A common exam mistake is to leave an answer in decimal form when a percentage is required – always read the question carefully.
当题目涉及混合形式时，先把所有数转换成同一种类型再进行比较。一个常见的考试错误是题目要求给出百分比答案却留成了小数——一定要仔细审题。
3. Percentage of a Quantity | 求一个数量的百分比
To calculate a percentage of an amount without a calculator, find 1% h a calculator, simply multiply the amount by the percentage as a decimal. For example, 15% of 240 = 240 × 0.15 = 36.
不使用计算器时，可以先除以100求出1%，再乘以所需百分比。使用计算器时，只需将数量乘以百分比对应的小数。例如，240的15% = 240 × 0.15 = 36。
For written methods, consider breaking the percentage into simpler parts: 15% = 10% + 5%. 10% of 240 is 24, and 5% f that, which is 12, so 24 + 12 = 36. This is particularly useful in non-calculator exams.
笔算时，可以把百分比拆分为更简单的部分：15% = 10% + 5%。240的10%是24，5%是它的一半即12，因此24 + 12 = 36。这在非计算器考试中尤其有用。
4. Percentage Increase and Decrease | 百分比增加与减少
The multiplier method is the most efficient way to handle percentage changes. For an increase of r%, the multiplier is (1 + r/100). For a decrease of r%, it is (1 – r/100). The new value is then found by multiplying the original quantity by the multiplier. For instance, a 20% increase uses a multiplier of 1.20; a 5% decrease uses 0.95.
乘数法是处理百分比变化最高效的方法。增加r%时，乘数为(1 + r/100)；减少r%时，乘数为(1 – r/100)。新值就等于原数量乘以乘数。例如，增加20%使用乘数1.20；减少5%使用0.95。
New Value = Original Value × (1 ± r/100)
Always choose the multiplier based on the final intent: a price increase is an upward multiplier, and a discount is a downward multiplier. IGCSE questions often combine several percentage changes, so keeping every step in multiplier form avoids rounding errors.
始终根据最终目的选择乘数：涨价使用大于1的乘数，打折使用小于1的乘数。IGCSE题目经常组合多个百分比变化，所以每一步都保留乘数形式可以避免取整误差。
5. Finding the Original Value – Reverse Percentages | 求原值——反向百分比
he original amount, divide by the multiplier rather than multiplying. Suppose a coat costs £72 after a 20% increase. The multiplier is 1.20, so original price = 72 ÷ 1.20 = £60.
当已知一个值在经过百分比的增加或减少后的结果，需要求原值时，要用除法（除以乘数）而不是乘法。假设一件大衣在涨价20%后售价72英镑，乘数为1.20，那么原价 = 72 ÷ 1.20 = 60英镑。
Many students mistakenly calculate 20% of the new price and subtract it – that is wrong because the 20% applied to the original, smaller amount. The correct reverse method is always to divide by the multiplier corresponding to the change.
很多学生错误地计算新价格的20%然后减去——这是错的，因为20%是针对原价（更小的数额）施加的。正确的反向方法始终是用变化对应的乘数去除。
6. Simple Interest | 单利
l interest rate as a percentage, and t is the number of years. The total amount A = P + I.
单利只基于原始本金计算，不把已经产生的利息计入下一期的本金中。公式为I = (P × r × t) / 100，其中P为本金，r为以百分比表示的年利率，t为年数。总金额A = P + I。
For example, if you invest £500 at 4% simple interest for 3 years, the interest earned is (500 × 4 × 3)/100 = £60, giving a total of £560. Simple interest is arithmetic growth, so the interest is the same every year.
例如，你以4%的单利投资500英镑，为期3年，所赚利息为(500 × 4 × 3)/100 = 60英镑，总额为560英镑。单利是等差增长，所以每年的利息相同。
7. Compound Interest | 复利
nd interest means interest is added to the original amount each period, so the next period' mula given in the exam is A = P(1 + r/100)ⁿ, where n is the number of compounding periods.
复利意味着每个周期利息都加入原金额，因此下一周期的利息是在一个更大的总额上计算的，这带来指数式增长。考试给出的标准公式是A = P(1 + r/100)ⁿ，其中n为复利的周期数。
A = P(1 + r/100)ⁿ
yearly, adjust r and n accordingly. For monthly compounding at a nominal annual rate of r% se (1 + r/1200) as the monthly multiplier and let n be the number of months. For instance, £2000 invested at 5% per annum compounded monthly for 2 years gives A = 2000(1 + 5/1200)²⁴.
如果复利频率高于每年一次，需要相应调整r和n。对于名义年利率r%且每月复利的情况，使用(1 + r/1200)作为月乘数，n取月份数。例如，2000英镑以年利率5%每月复利投资2年，A = 2000(1 + 5/1200)²⁴。
8. Compound Percentage Change – Repeated Changes | 复合百分比变化——反复变化
ms involve repeated application of percentage changes, such as a price that increases by 3% each year or the value of equipment that depreciates by 15% years is (1 ± r/100)ᵏ. For depreciation, use a multiplier less than 1, e.g. 0.85 for a 15% decrease.
很多IGCSE问题涉及反复施加百分比变化，例如价格每年上涨3%，或者设备价值每年折旧15%。经过k年后的整体乘数是(1 ± r/100)ᵏ。对于折旧，使用小于1的乘数，如15%的减少对应乘数0.85。
centage changes in succession, multiply the multipliers in order. Example: first month +10%, second month –5%. The overall multiplier is 1.10 × 0.95 = 1.045, meaning a net increase of 4.5%. This is not the same as simply adding the percentages (10 – 5 = 5%), a trap many students fall into.
对于连续出现不同百分比变化的序列，按顺序把乘数相乘。举例：第一个月+10%，第二个月–5%。整体乘数为1.10 × 0.95 = 1.045，意味着净增长4.5%。这不同于简单地把百分比相加（10 – 5 = 5%），许多学生容易掉进这个陷阱。
9. Working with Multiple Multipliers | 使用多重乘数
When you have to combine several percentage increases and decreases in one question, write each change as a decimal multiplier and multiply them. This technique is essential for chain discounts, successive tax changes, or population models with different growth rates in different years.
当你需要在一道题中组合多个百分比增加和减少时，把每个变化写成小数乘数再相乘。这个技巧对于连环折扣、连续的税率变化或不同年份有不同增长率的人口模型至关重要。
A common structure: a price is increased by 15% and then a 10% d. The final price is Original × 1.15 × 0.9 = Original × 1.035, so there is an overall 3.5% tion is commutative, but the absolute amounts differ if the base value changes in between.
常见结构：价格先上涨15%，再打九折。最终价格为原价 × 1.15 × 0.9 = 原价 × 1.035，因此整体上涨3.5%。如果先打折再涨价，结果一样（乘法交换律），但如果中间基数发生变化，绝对数额会不同。
10. Exam Tips and Common Mistakes | 应试策略与常见错误
1. Always set up a clear multiplier early in your working – it reduces the chance of calculator errors and makes reverse percentages obvious. 2. When using the compound interest formula, check whether the question gives the number of years or months, and adjust n and r accordingly. 3. A percentage profit or loss is always calculated on the original cost price, not on the selling price.
1. 在解题步骤中尽早明确乘数——这可以降低计算器出错风险，并让反向百分比变得一目了然。2. 使用复利公式时，检查题目给出的是年数还是月数，并相应调整n和r。3. 利润或亏损的百分比总是以原始成本价为基准计算，而不是以售价为基准。
4. Never treat a percentage increase followed by an equal percentage decrease as returning to the original. A 20% rise then a 20% fall leaves 96% of the original amount. 5. When you see 'write as a percentage', ensure the final answer includes the % symbol. 6. In reverse percentage questions, draw a simple flow diagram to remind yourself whether to multiply or divide by the multiplier.
4. 不要把先涨后跌相同的百分比当作回到原值。先涨20%再跌20%只剩下原值的96%。5. 看到“写成百分比”时，一定要在最终答案中带上%符号。6. 在反向百分比题目中，画一个简单的流程图可以帮助你记住该用乘数乘还是除以乘数。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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