📚 Number 6: Percentages | 数字6：百分比
Percentages are a cornerstone of IGCSE Number, appearing in every exam. They allow us to express parts of a whole as fractions of 100, compare data, calculate financial growth, and interpret real-world changes. Mastering percentages means understanding conversions, increase-decrease multipliers, reverse calculations, compound interest, and multi-step applications.
百分比是IGCSE数字部分的核心，每次考试都会出现。它让我们能以百分数为分母表达整体的一部分，比较数据，计算财务增长，解读现实变化。掌握百分比意味着理解转换、增减乘数、逆向运算、复利以及多步骤应用题。
1. Understanding Percentages | 理解百分比
A percentage is a way of expressing a number as a fraction of 100. The word 'percent' comes from Latin 'per centum', meaning 'by the hundred'. The symbol % is used. For instance, 25% means 25 out of every 100 and can be written as 25/100 or 0.25. Percentages are useful for comparing quantities and describing proportions in everyday contexts such as discounts, interest rates, and statistics.
百分比是将一个数表示为100的分数的一种方式。“percent”一词源于拉丁文“per centum”，意为“每一百”。符号%用于表示。例如，25%表示每一百份中的二十五份，可以写为25/100或0.25。百分比在比较数量和描述比例时非常有用，常用于折扣、利率和统计等日常情境。
2. Converting Fractions, Decimals and Percentages | 分数、小数与百分比转换
tage to a decimal, divide by 100 (move the decimal point two places left). For example, 85% fraction to a decimal by division, then multiply by 100. For instance, 3/8 = 0.375 = 37.5%. Common conversions such as 50% = 1/2, 25% = 1/4, 20% = 1/5, and 10% = 1/10 should be memorised.
将百分比转换为小数，除以100（小数点左移两位）。例如85% = 0.85。将小数转换为百分比，乘以100。将分数转换为百分比，先通过除法将分数化为小数，再乘以100。例如3/8 = 0.375 = 37.5%。常见转换如50% = 1/2、25% = 1/4、20% = 1/5和10% = 1/10应熟记。
3. Percentage of a Quantity | 求数量的百分比
ultiply the quantity by the percentage expressed as a decimal or fraction. For example, 15% y 15/100 × 240 = 36 g or 0.15 × 240 = 36 g. On a calculator, you can simply enter 240 × 15%. Always include the units in the answer and check that your result is reasonable.
求一个数量的百分比，用该数量乘以对应的小数或分数。例如，求240克的15%：可用15/100 × 240 = 36克，或0.15 × 240 = 36克。使用计算器时，可直接输入240 × 15%。结果一定要带上单位，并检查是否合理。
4. Percentage Increase and Decrease | 百分比增加与减少
To increase a quantity by r%, multiply the original value by (1 + r/100). To decrease by r%, multiply by (1 – r/100). These multipliers are efficient. For example, increasing £80 by 5% gives £80 × 1.05 = £84. Decreasing £60 by 20% gives £60 × 0.8 = £ When applying multiple changes, multiply the successive multipliers in the correct order.
将某量增加r%，原值乘以(1 + r/100)。减少r%，乘以(1 – r/100)。这些乘数非常高效。例如，80英镑增加5%得80 × 1.05 = 84英镑；60英镑减少20%得60 × 0.8 = 48英镑。当连续应用变化时，应按正确顺序将各个乘数相乘。
5. Expressing One Quantity as a Percentage of Another | 一个量是另一个量的百分之几
Write a fraction with the 'part' as numerator and the 'whole' example, in a class of 30 students, 18 are girls. The percentage of girls is (18/30) × 100% = 60% h quantities to the same units before calculating; otherwise the answer will be incorrect.
先写出“部分”除以“整体”的分数，再乘以100。例如，一个班30名学生中有18名女生，女生百分比为(18/30) × 100% = 60%。计算前务必统一两个量的单位，否则答案会出错。
6. Percentage Change | 百分比变化
Percentage change measures how much a value has changed relative to its original value. The change is new value minus original value. The formula is:
Percentage change = (new – original) / original × 100%
For example, a shirt's price rises from £25 to £30. The change is £5, and the percentage increase is (5/25) × 100% = 20%. A negative result means a decrease. Always use the original value as the denominator.
百分比变化衡量一个值相对于原值的变化程度。变化量是新值减去原值。公式为：
Percentage change = (new – original) / original × 100%
例如，一件衬衫价格从25英镑涨到30英镑，变化量为5英镑，百分比增加为(5/25) × 100% = 20%。若结果为负则表示减少。分母一定要用原值。
7. Reverse Percentages | 逆向百分比
mount before a percentage increase or decrease. If the final amount after an increase of r% is known, original = final / (1 + r/100). If after a decrease, original = final / (1 – r/100).
Original = Final ÷ (1 ± r/100)
Example: a jacket's price after a 15% rise is £92. Original = 92 ÷ 1.15 = £80. A common mistake is to subtract 15% from £92, which gives £78.20, an incorrect result.
逆向百分比用于求百分比增减前的原始量。若已知增加r%后的最终值，原值 = 最终值 / (1 + r/100)。若为减少，原值 = 最终值 / (1 – r/100)。
原值 = 最终值 ÷ (1 ± r/100)
例如，一件夹克涨价15%后为92英镑，原价 = 92 ÷ 1.15 = 80英镑。常见错误是从92英镑直接减去15%得到78.20英镑，这是不正确的。
8. Compound Interest and Depreciation | 复利与贬值
Compound interest is calculated on the initial principal and the accumulated interest. The formula is:
A = P(1 + r/100)ⁿ
, n the number of periods, and A the final amount. Depreciation uses a minus sign: A = P(1 – r/100)ⁿ. Example: invest £500 at 4% per year compound for 3 years. A = 500 × 1.04³ = 500 × 1.12486 ≈ £562.43. Use the superscript ⁿ for n and ², ³ for powers.
复利基于初始本金和累积利息计算。公式为：
A = P(1 + r/100)ⁿ
其中P为本金，r为每期利率，n为期数，A为最终金额。贬值使用减号：A = P(1 – r/100)ⁿ。例如：以年利率4%的复利投资500英镑，3年后A = 500 × 1.04³ = 500 × 1.12486 ≈ 562.43英镑。可使用上标字符ⁿ表示n次幂，²、³表示平方和立方。
9. Simple Interest | 简单利息
inearly. IGCSE problems might ask you to compare or decide which yields more. Example: P = £200, r = 5%, t = 3 years. I = 200 × 5 × 3 / 100 = £30, total A = £230.
简单利息仅按原始本金计算。利息I = P × r × t / 100，其中t为年数。总金额A = P + I。复利呈指数增长，而简单利息呈线性增长。IGCSE考题可能会要求比较或判断哪种利息获利更多。例如：P = 200英镑，r = 5%，t = 3年。I = 200 × 5 × 3 / 100 = 30英镑，总金额A = 230英镑。
10. Real-life Applications and Multi-step Problems | 实际应用与多步问题
commission. Many exam questions require two or more steps. For example, an item priced at £45 is given a 20% discount, then 10% tax is added to the discounted price. First, discounted price = 45 × 0.8 = £36. Then, with tax = 36 × 1.1 = £ ply multipliers in the order described, as tax is usually charged on the discounted price.
百分比出现在折扣、税、盈亏和佣金问题中。许多考题需要两步或更多步骤。例如，一件商品原价45英镑，先打八折，再按折后价加收10%的税。首先，折后价 = 45 × 0.8 = 36英镑；然后含税价 = 36 × 1.1 = 39.60英镑。务必按题目描述的顺序应用乘数，因为税款通常按折后价计算。
11. Percentage Error | 百分误差
Percentage error expresses the accuracy of an estimate or measurement compared to a true value. The formula is:
Percentage error = |measured – true| / true × 100%
Example: you estimate a length as 98 cm, but the true length is 100 cm. Error = |98 – 100| / 100 × 100% = 2% d rounding. Always take the absolute value of the difference to get a positive percentage.
百分误差表示估计值或测量值相较于真实值的准确度。公式为：
百分误差 = |测量值 – 真实值| / 真实值 × 100%
例如，你估计长度为98厘米，真实长度为100厘米，误差 = |98 – 100| / 100 × 100% = 2%。这个概念常结合界限和舍入考查。对差值取绝对值，确保得到正的百分误差。
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