📚 Percentages | 百分比
Percentages are a fundamental part of IGCSE Mathematics, appearing in pure number work, statistics, probability and everyday financial contexts. A percentage literally means 'per hundred' and gives us a consistent way to compare proportions, express growth and calculate discounts, interest and probabilities. Mastering percentages requires fluency in converting between fractions, decimals and percentages, applying multipliers efficiently and interpreting word problems, especially those involving reverse percentages and compound changes.
百分比是 IGCSE 数学的基础内容，广泛出现在纯数、统计、概率以及日常财务情境中。百分比字面意思是“每一百份”，为比较比例、表达增长以及计算折扣、利息和概率提供了一致的方法。掌握百分比需要熟练地在分数、小数和百分比之间转换、灵活运用乘数，并准确解读应用题，尤其是涉及反向百分比和复合变化的问题。
1. What is a Percentage? | 什么是百分比？
A percentage is a fraction with denominator 100. The symbol % means 1/100, so 47% always relate to the same base, they allow us to compare quantities that might be measured in different units or come from different sized groups. When a test is marked out of 80 and a student scores 60, saying they got 75 % instantly communicates their performance relative to the whole.
百分比是分母为 100 的分数。符号 % 表示 1/100，因此 47% = 47/100 = 0.47。由于百分比总是与同一基准相关联，我们可以用它比较不同单位或不同总体的数量。当一次测验满分 80 分而一名学生得到 60 分时，说他的成绩是 75 % 能立刻反映出相对于整体的表现。
In IGCSE you will meet percentages as simple conversions but also as operators – for example, 15% of a number means multiply by 0.15. It is vital to understand the difference between an absolute change and a relative (percentage) change. A £5 increase might be large on a £10 item (50 % increase) but negligible on a £500 item (1 % increase).
在 IGCSE 中，你既会遇到简单的百分比转换，也会遇到作为“算子”的百分比——例如，某个数的 15% 意味着乘以 0.15。理解绝对变化与相对（百分比）变化的区别至关重要。涨价 £5 在 £10 的商品上可能是大幅增长（50 % 增加），而在 £500 的商品上则微不足道（1 % 增加）。
2. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分比之间的转换
ion → percent : first convert the fraction to a decimal by division, then multiply by 100.
熟练地在三种形式之间转换是解决几乎所有百分比问题的基础。关键转换如下：百分比 → 小数：除以 100（小数点向左移动两位）；小数 → 百分比：乘以 100（小数点向右移动两位）；分数 → 百分比：先用除法将分数转化为小数，再乘以 100。
| Fraction | Decimal | Percentage |
| 1/2 | 0.5 | 50 % |
| 1/4 | 0.25 | 25 % |
| 3/5 | 0.6 | 60 % |
| 2/3 | 0.666... | 66.7 % (to 1 d.p.) |
表格：分数、小数与百分比的常见换算（分数、小数、百分比）
ficant figures. On the non-calculator paper, knowing common equivalents such as 1/8 = 12.5 % and 1/3 ≈ 33.3 % saves time.
当分数的分母不能转化为整齐的小数时，你需要将百分比表达至规定的精确度，通常精确到 1 位小数或 3 位有效数字。在非计算器试卷中，熟记常见等值如 1/8 = 12.5 %、1/3 ≈ 33.3 % 可以节省时间。
Modern IGCSE questions often involve ordering fractions, decimals and percentages, so you should be able to place them on a number line quickly by converting everything to the same form, usually decimals.
现代 IGCSE 试题经常要求对分数、小数和百分比进行排序，因此你应能通过将所有数转化为同一形式（通常为小数）快速将它们放到数轴上。
3. Finding a Percentage of a Quantity | 求某个数量的百分比
e of an amount, write the percentage as a decimal multiplier and multiply. For example, 28 % = 0.28 × 650. For non-calculator methods, break the percentage into manageable chunks: 10 % first, then 5 %, then 1 % and so on. To find 37 % of 200: 10 % = 20, 30 % = 60; 5 % = 10, so 35 % = 70; 1 % = 2, so 37 % = 70 + 4 = 74.
求某个数量的百分比，可将百分比写成小数乘数并相乘。例如，650 的 28 % = 0.28 × 650。在没有计算器的方法中，可将百分比拆分成容易计算的部分：先求 10 %、再求 5 %、然后求 1 %，以此类推。求 200 的 37 %：10 % = 20，30 % = 60；5 % = 10，因此 35 % = 70；1 % = 2，因此 37 % = 70 + 4 = 74。
When the percentage is greater than 100, the quantity increases. Earning 130 % of a target means you have produced 1.3 times the original figure. The multiplier approach works for all percentages, including fractional and decimal percentages such as 0.5 % (multiply by 0.005) and 120.5 % (multiply by 1.205).
当百分比大于 100 时，数量会增加。完成目标的 130 % 意味着你的产出是原数据的 1.3 倍。乘数法适用于所有百分比，包括小数百分比如 0.5 %（乘以 0.005）和 120.5 %（乘以 1.205）。
In word problems, always identify the 'whole' first. If a sweater originally costs £42 and a 15 % discount is applied, the discount is 0.15 × 42 = £6.30 and the sale price is 42 − 6.30 = £35.70. Alternatively, recognise that a 15 % discount means you pay 85 %, so sale price = 0.85 × 42.
在文字题中，务必首先确定“整体”。如果一件毛衣原价 £42，打 15% 折扣，折扣额为 0.15 × 42 = £6.30，售价为 42 − 6.30 = £35.70。另外，要知道 15 % 折扣意味着你支付 85 %，因此售价 = 0.85 × 42。
4. Expressing One Quantity as a Percentage of Another | 用一个量表示另一个量的百分比
To write A as a percentage of B, form the fraction A/B and multiply by 100. The formula is:
Percentage = (A ÷ B) × 100
将 A 表示为 B 的百分比，写出分数 A/B 再乘以 100。公式为：
百分比 = (A ÷ B) × 100
tball player scores 18 out of 25 free throws, the percentage success is (18/25) × 100 = 72 % from 15 cm to 21 cm, the new height as a percentage of the original is (21/15) × 100 = 140 %, showing a 40 % increase.
计算前要确保两个量单位一致。如果一名篮球运动员 25 次罚球命中 18 次，成功率为 (18/25) × 100 = 72 %。如果一株植物从 15 cm 长到 21 cm，新高度是原高度的 (21/15) × 100 = 140 %，显示出 40 % 的增长。
IGCSE exam questions often embed this skill in contexts like test scores, budget allocations and survey data. Being able to interpret a statement such as '28 out of 70 students walk to school' as 40 % is essential for statistical and probability sections.
IGCSE 考试常将这种技能嵌入到测验成绩、预算分配和调查数据等情境中。能将“70 名学生中有 28 人步行上学”解读为 40 % 是统计与概率部分的关键能力。
5. Percentage Increase and Decrease | 百分比增加与减少
Percentage change compares the difference between a new and an original value to the original value.
Percentage change = (change ÷ original value) × 100
百分比变化将新值与原始值之间的差异与原始值进行比较。
百分比变化 = (变化量 ÷ 原始值) × 100
If a price rises from £80 to £92, the increase is £12, so percentage increase = (12/80) × 100 = 15 % alls from 6000 to 5700, the decrease is 300, so percentage decrease = (300/6000) × 100 = 5 %.
如果价格从 £80 上涨到 £92，涨幅为 £12，因此增长百分比 = (12/80) × 100 = 15 %。如果人口从 6000 下降到 5700，减少量为 300，因此减少百分比 = (300/6000) × 100 = 5 %。
For successive percentage changes, do not add percentages directly. A 10 % increase followed by a 20 % increase on an original value of 200 works like this: after 10 % increase → 200 × 1.1 = 220; after another 20 % increase → 220 × 1.2 = 264, giving an overall increase of 32 %, not 30 %. This is because the second increase operates on a larger base.
对于连续百分比变化，不要直接将百分比相加。以原始值 200 为例，先增加 10 % 再增加 20 %：增加 10 % 后 → 200 × 1.1 = 220；再增加 20 % 后 → 220 × 1.2 = 264，总体增加了 32 %，而不是 30 %。这是因为第二次增长是在更大的基数上进行的。
a value increases by a percentage, the multiplier is (1 + percentage as decimal). For a 6 % increase, use ×1.06. For a decrease, use (1 − percentage as decimal). A 6 % useful in compound growth and decay problems, which appear frequently on the Higher Tier.
当数值按百分比增加时，乘数为 (1 + 百分比的小数形式)。增加 6 % 则用 ×1.06。减少时，用 (1 − 百分比的小数形式)。减少 6 % 则用 ×0.94。这在高级别考试中常见的复利增长与衰减问题中尤其实用。
6. Reverse Percentages | 反向百分比
l amount. The key is to identify what multiplier has been applied. If a price including 20 % VAT is £144, the multiplier was 1.2, so original price = 144 ÷ 1.2 = £120. Never simply subtract 20 % of the final value, because 20 % of 144 is different from 20 % of the original.
反向百分比问题给出百分比变化之后的最终值，要求找出原始数值。关键在于确定已经应用了哪个乘数。如果含 20 % 增值税的价格为 £144，乘数为 1.2，因此原价 = 144 ÷ 1.2 = £120。千万不要简单地从最终值中减去它的 20 %，因为 144 的 20 % 与原值的 20 % 不同。
linking the original amount (let it be x) to the final amount via the multiplier. For a 15 % reduction that results in £51, the multiplier is 0.85, so 0.85x = 51, giving x = 51 ÷ 0.85 = £60. Always check your answer by applying the percentage change forwards: 60 − 15 % of 60 = 60 − 9 = 51.
一种稳妥的方法是写出等式，将原始量（设为 x）通过乘数与最终量关联起来。若减少 15 % 后得到 £51，乘数为 0.85，因此 0.85x = 51，得 x = 51 ÷ 0.85 = £60。务必通过正向应用百分比变化来检验答案：60 − 60 的 15 % = 60 − 9 = 51。
These questions often describe discounts, depreciation or population decline. Identifying whether the given value represents the 'before' or 'after' state is vital. Underline phrases like “after a 12 % increase” or “including tax” to decide which operation to use.
这类题目常描述折扣、折旧或人口下降。识别已知数值代表“变化前”还是“变化后”的状态至关重要。在“增加 12 % 后”或“含税”等短语下面划线，有助于决定使用哪种运算。
7. Simple and Compound Interest | 单利与复利
Interest calculations are direct applications of percentages. Simple interest is calculated only on the initial principal. For a principal P, rate r % per year, time t years, the amount A is:
A = P + (P × r/100 × t)
利息计算是百分比的直接应用。单利仅根据初始本金计算。对于本金 P、年利率 r %、时间 t 年，总金额 A 为：
A = P + (P × r/100 × t)
Compound interest, however, calculates interest on the accumulated amount each year. The formula given in Edexcel IGCSE is:
A = P (1 + r/100)t
而复利则是每年按累计金额计算利息。Edexcel IGCSE 给出的公式为：
A = P (1 + r/100)t
Be careful with units: t must be in years, and r is the annual percentage rate. If interest is compounded quarterly, divide r by 4 and multiply t by 4, using n periods. The same structure applies for depreciation (negative growth), where the multiplier becomes (1 − r/100).
注意单位：t 必须以年为单位，r 为年利率百分数。如果按季度复利，需将 r 除以 4，t 乘以 4，使用 n 个周期。同样的结构适用于折旧（负增长），此时乘数变为 (1 − r/100)。
l. Realistic IGCSE questions often present a table where you must calculate the final amount after three years of compound interest and compare it with a simple interest bank account.
比较题常问在相同利率下，单利和复利哪个收益更高。当 t > 1 时，复利总比单利更高。当 t = 1 时，二者相等。实际的 IGCSE 题目通常会给出表格，要求计算三年复利后的最终金额，并与单利银行账户比较。
8. Percentage Change and Profit/Loss | 百分比变化与利润/损失
Profit and loss are expressed as percentages of the cost price unless stated otherwise. The formulas are:
Percentage profit = (profit ÷ cost price) × 100
Percentage loss = (loss ÷ cost price) × 100
除非另有说明，利润和损失均以成本价的百分比表示。公式为：
利润率 = (利润 ÷ 成本价) × 100
亏损率 = (亏损额 ÷ 成本价) × 100
If a shop buys a bicycle for £150 and sells it for £210, profit = 210 − 150 = £60, so percentage profit = (60/150) × 100 = 40 %. If the selling price is £120, loss = 30, percentage loss = (30/150) × 100 = 20 % ice is given and you must work backwards to find the cost price using reverse percentages.
如果商店以 £150 购进一辆自行车并以 £210 售出，利润 = 210 − 150 = £60，因此利润率 = (60/150) × 100 = 40 %。若售价为 £120，亏损 30，亏损率 = (30/150) × 100 = 20 %。务必仔细审题：有时已知售价，需用反向百分比求出成本价。
Discount problems also use the same concept. A shop offers a 20 % discount on an item marked at £90 and still makes a 10 % e cost price. You can calculate the cost price by first finding the sale price (90 × 0.8 = £72), then letting the cost price be C and setting up 1.1C = 72, giving C = £65.45 (to 2 d.p.).
折扣问题也采用相同的概念。某商品标价 £90，商店给予 20 % 折扣后仍可获得成本价 10 % 的利润。你可以先求出售价 (90 × 0.8 = £72)，然后设成本价为 C，建立等式 1.1C = 72，得到 C = £65.45（保留两位小数）。
9. Applying Percentages in Real-Life Contexts | 现实生活中百分比的应用
re a tax-free allowance and ask for net income after income tax. For example, an income of £34 000 with a tax-free allowance of £12 570 and a basic rate of 20 % gives taxable income = 34 000 − 12 570 = 21 430, tax = 0.2 × 21 430 = £4 286, net income = 34 000 − 4 286 = £29 714.
Edexcel IGCSE 经常将百分比嵌入金融、科学和日常场景中。税务问题通常给出免税额度前的总收入，要求计算缴纳所得税后的净收入。例如，收入 £34 000，免税额 £12 570，基本税率 20 %，则应纳税所得额 = 34 000 − 12 570 = 21 430，税款 = 0.2 × 21 430 = £4 286，净收入 = 34 000 − 4 286 = £29 714。
Inflation, wage rises and retail markups are also standard contexts. A salary increase of 3.5 % followed by a one-off bonus of £500 combines a percentage multiplier with an addition. Similarly, mixtures and alloy problems may involve percentage purity, where 18-carat gold is 75 % pure gold, and you must find the mass of pure gold in a given mass of alloy.
通货膨胀、工资上涨和零售加价也是常见情境。工资先上涨 3.5 %，再加发一次性 £500 奖金，就需要将百分比乘数与加法相结合。类似地，混合物和合金问题可能涉及纯度百分比，例如 18K 金含 75 % 的纯金，要求计算给定合金质量中纯金的质量。
s links percentages with rounding and bounds, which are high-mark topics on the IGCSE papers.
另一个重要应用是百分比误差，主要出现在上下界情境中。测量值的百分比误差 = (最大误差 ÷ 测量值) × 100。这将百分比与舍入和误差界联系起来，属于 IGCSE 试卷中的高分主题。
10. Common Mistakes and Tips | 常见错误与提示
Many students lose marks by confusing the original and final values in percentage change, especially in reverse percentage questions. Always identify whether the given quantity is before or after the change. A simple test: If the quantity has been increased by 20 % and is now 48, then 48 must be 120 % of the original – it is not the 20 % or the 100 % alone.
许多学生在百分比变化中混淆原始值与最终值，尤其在反向百分比问题中失分。务必判断已知量是在变化前还是变化后。一个简单的检验法：如果数量增加了 20 % 后变成 48，那么 48 必须是原始值的 120 %——它既不是单独的 20 %，也不是 100 %。
or is treating a sequence of percentage changes as additive. If a value is increased by 10 % and then decreased by 10 %, the overall effect is not zero; the final value is original × 1.10 × 0.90 = 0.99, a 1 % crease. This concept appears in compound change and is often tested with graphs or tables.
另一个经典错误是把连续的百分比变化当作加减法。如果一个值先增加 10 % 再减少 10 %，总效果并不是零；最终值为原值 × 1.10 × 0.90 = 0.99，即下降了 1 %。这一概念出现在复合变化中，常结合图表或表格进行考查。
Also, be precise with rounding. When converting 2/3 to a percentage, state 66.7 % or 66.67 % as required, not just 66 %. In financial questions, money answers must be given to 2 decimal places unless the context dictates otherwise. Use the exact multiplier throughout the calculation and round only at the final answer to avoid cumulative rounding errors.
此外，舍入要精确。将 2/3 转化为百分比时，应根据要求写为 66.7 % 或 66.67 %，而不仅仅是 66 %。在财务题中，金额答案必须保留两位小数，除非题目另有说明。整个计算过程中应使用精确的乘数，仅在最后答案中进行舍入，以避免累积舍入误差。
- Always underline the key quantity in the question: original, new, discount, tax, profit.
- Write the multiplier clearly before applying it.
- For reverse percentages, set up an equation with the unknown original amount and the known multiplier.
- Double-check by applying the percentage forwards once you have an answer.
- 务必在题目中划出关键词：原始、新值、折扣、税款、利润。
- 在使用乘数前先把乘数写清楚。
- 对于反向百分比，用未知的原始量和已知的乘数建立方程。
- 得出答案后，正向应用百分比进行检验。
11. Practice Strategy and Exam Technique | 练习策略与应试技巧
he non-calculator paper, expect fractions labelled as percentages, finding multiples of 10 %, 5 % and 1 %. Mental arithmetic with the most common equivalents (1/3 ≈ 33.33 %, 1/6 ≈ 16.7 %) gives a time advantage.
IGCSE 百分比题目的分值分布很广，从 1 分的选择题到 5 分的结构化问题。在非计算器试卷中，会看到标注为百分比的分数、求 10 %、5 % 和 1 % 的倍数。能心算最常见等值（1/3 ≈ 33.33 %、1/6 ≈ 16.7 %）可以赢得时间优势。
On the calculator paper, become proficient with the percentage button (often marked % st, using the power button (^) or repeated multiplication is faster than a percentage key.
在计算器试卷中，要熟练使用设备上的百分比键（通常标有 %），但不要盲目依赖。要理解该键执行的运算逻辑，以便在需要时能手算重现。对于复利，使用乘方键（^）或重复乘法比百分比键更快。
Time management: allocate roughly one mark per minute. Read the whole problem before computing; many students do unnecessary work by finding an original amount when the question only asked for the percentage itself. Present your method clearly, because IGCSE awards method marks even if the final answer has a slip.
时间管理：大约每题分配每分钟 1 分的时间。计算前先通读全题；许多学生会做额外工作，比如题目只要求求出百分比本身，他们却去找原始值。清晰地展示你的解题过程，因为 IGCSE 会在方法正确但最终答案有小错时给予方法分。
12. Summary and Key Takeaways | 总结与核心要点
Percentages are a connecting thread across the Edexcel IGCSE Mathematics syllabus. They link number, proportion, statistics and real-life finance. Master the decimal multiplier approach and the three core relationships:
- Part = percentage as decimal × whole
- Percentage = (part ÷ whole) × 100
- Original amount = final amount ÷ multiplier
百分比是贯穿 Edexcel IGCSE 数学课程的一条纽带，将数字、比例、统计和现实金融连接起来。要掌握小数乘数法以及三个核心关系：
- 部分 = 百分比小数形式 × 整体
- 百分比 = (部分 ÷ 整体) × 100
- 原始量 = 最终量 ÷ 乘数
Whether calculating the discount on a jacket, predicting population growth or determining the effect of tax, percentages enable clear, comparative reasoning. Regular practice with past paper questions, paying close attention to the exact wording (‘before’, ‘after’, ‘of’, ‘more than’, ‘less than’), will build the accuracy and confidence needed to secure full marks on this topic.
无论是计算夹克的折扣、预测人口增长还是确定税款的影响，百分比都能让你进行清晰、可比较的推理。通过定期练习历年真题，并密切注意准确措辞（“之前”/“之后”、“的”、“比……多”、“比……少”），你就能培养出准确度和信心，在这一主题上稳获满分。
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