📚 Mastering Number: Fractions, Decimals, Percentages, Ratio and Proportion | 掌握数字：分数、小数、百分比、比率与比例
Number is the cornerstone of IGCSE Edexcel Mathematics, appearing in nearly every topic and real‑world application. From converting fractions to decimals, calculating percentage increase, to splitting quantities in a given ratio, these skills underpin both core and extended papers. This article provides a complete revision guide with clear methods, worked examples and key pitfalls to avoid.
数字是IGCSE Edexcel数学的基石，几乎出现在每个专题和现实应用中。从分数与小数的互化、计算百分比增减，到按给定比例分配数量，这些技能是基础卷和进阶卷的核心。本文提供完整的复习指南，包含清晰的方法、详细例题和需要避免的常见错误。
1. Understanding Fractions, Decimals and Percentages (FDP) | 理解分数、小数和百分比 (FDP)
Fractions, decimals and percentages are three different ways of expressing the same proportion. A fraction like 3/4 means 3 parts out of 4; as a decimal it is 0.75; as a percentage it is 75%. Being able to switch fluently between these forms is essential for solving IGCSE problems.
分数、小数和百分比是表达同一比例的三种不同形式。分数如3/4表示4份中的3份；小数形式为0.75；百分比形式为75%。能够在这些形式之间自由切换是解决IGCSE问题的关键。
3/4 = 3 ÷ 4 = 0.75. To change a decimal to a percentage, multiply by 100: 0.75 × 100 = 75%. To convert a percentage to a fraction, write it over 100 and simplify: 75% = 75/100 = 3/4.
将分数化为小数，直接用分子除以分母：3/4 = 3 ÷ 4 = 0.75。将小数化为百分比，乘以100：0.75 × 100 = 75%。将百分比化为分数，先写成百分数/100，然后约分：75% = 75/100 = 3/4。
Recurring decimals require special care. For example, 1/3 = 0.333…, written as 0.3̇. To convert a recurring decimal to a fraction, use algebra: let x = 0.3̇, then 10x = 3.3̇, subtract to get 9x = 3, so x = 3/9 = 1/3.
循环小数需要特别处理。例如，1/3 = 0.333…，记作0.3̇。要将循环小数化为分数，可使用代数方法：设 x = 0.3̇，则10x = 3.3̇，相减得9x = 3，因此x = 3/9 = 1/3。
2. Ordering and Comparing FDP | 比较和排序 FDP
When faced with a mix of fractions, decimals and percentages, convert all numbers to the same form. The easiest is to turn everything into decimals or percentages. Ordering them then becomes straightforward.
当遇到混合了分数、小数和百分比的比较题时，将所有数字转化为同一种形式。最简单的方法是都化成小数或百分比，然后排序就变得直观了。
For example, arrange 3/5, 0.58, 56% and 7/12 in ascending order. Convert: 3/5 = 0.6, 56% = 0.56, 7/12 ≈ 0.5833. The order from smallest to largest is: 0.56 (56%), 0.58, 0.5833... (7/12), 0.6 (3/5). Always show the original forms in your final answer.
例如，将3/5、0.58、56% 和7/12按升序排列。转化：3/5 = 0.6，56% = 0.56，7/12 ≈ 0.5833。从小到大的顺序为：0.56 (56%)，0.58，0.5833... (7/12)，0.6 (3/5)。最终答案中要保留原始形式。
Common mistake: forgetting that a larger denominator does not necessarily mean a smaller fraction. Use equivalent fractions or decimal conversion to compare accurately.
常见错误：认为分母越大分数越小，这不总是正确。应使用等值分数或小数转换来准确比较。
3. Percentage Calculations | 百分比计算
IGCSE percentage problems fall into three main types: finding a percentage of a quantity, expressing one quantity as a percentage of another, and percentage increase or decrease. The multiplier method is strongly recommended for growth and decay.
IGCSE 百分数问题主要分为三类：求一个数量的百分比、将一个数量表示为另一个数量的百分比，以及百分比增减。强烈推荐使用乘数法处理增长和衰减问题。
To find 15% multiply 200 by 0.15, giving 30. To express 30 as a percentage of 200: (30/200) × 100 = 15%. For a 20% increase, multiply the original amount by 1.2; for a 15% decrease, multiply by 0.85. Never use the mistaken method of adding 20% then subtracting 20% to get back to the original – the net effect is a 4% decrease.
求200的15%：200 × 0.15 = 30。将30表示为200的百分比：(30/200) × 100 = 15%。对于20%的增长，将原数乘以1.2；对于15%的减少，乘以0.85。绝对不要先增加20%再减少20%来试图回到原数，最终结果是减少了4%。
Compound interest and depreciation use the formula: Amount = P × (1 ± r/100)n, where P is principal, r is rate, n is number of time periods. A positive sign gives growth, negative gives decay.
复利和贬值使用公式：总额 = 本金 × (1 ± r/100)n，其中P为本金，r为利率，n为时间周期数。正号表示增长，负号表示衰减。
4. Reverse Percentages | 逆向百分比
Reverse percentage questions ask: 'After a 10% increase, a price is £44. What was the original price?' Many pupils wrongly calculate 10% of £44 and subtract. The correct approach is to recognise that £44 represents 110% of the original, so divide by 1.10 to get £40.
逆向百分比问题通常是：“增长10%后价格为44英镑，原价是多少？”许多学生错误地计算44的10%再减去。正确方法是认识到44英镑代表原价的110%，因此除以1.10得到40英镑。
Set up an equation: original × multiplier = new value. For a 25% sale reduction, the multiplier is 0.75. If the sale price is £60, original = £60 ÷ 0.75 = £80. Always check by working forward: 80 reduced by 25% is 80 × 0.75 = 60.
建立方程：原值 × 乘数 = 新值。对于25%的折扣，乘数为0.75。如果折后价为60英镑，原价 = 60 ÷ 0.75 = 80英镑。务必正向验证：80减去25%为 80 × 0.75 = 60。
5. Ratio and Proportion Fundamentals | 比和比例基础
A ratio compares parts of a whole, while a proportion compares a part to the whole. In IGCSE, ratios are written in the form a : b, and they can be simplified just like fractions by dividing both sides by a common factor.
比率是比较整体中的部分，而比例是部分与整体的比较。在IGCSE中，比率写成 a : b 的形式，可以像分数一样通过除法约简。
to bananas is 12 : 8 = 3 : 2 (dividing by 4). The proportion of apples is 12/20 = 3/5 = 60%. Always write ratios in their simplest integer form.
例如，一个果盆有12个苹果和8个香蕉。苹果与香蕉的比率为12 : 8 = 3 : 2（除以4）。苹果的比例为12/20 = 3/5 = 60%。始终将比率写成最简整数比。
lit in a ratio, add the parts to get the total number of shares. For a ratio 3 : 5 sharing £120, total shares = 8. One share = £120 ÷ 8 = £15. The amounts are 3 × 15 = £45 and 5 × 15 = £75.
当数量按比例分配时，将各部分相加得到总份额。对于比率3 : 5分配120英镑，总份额为8。每份 = 120 ÷ 8 = 15英镑。分配金额为3 × 15 = 45英镑和5 × 15 = 75英镑。
6. Direct and Inverse Proportion | 正比例与反比例
together at the same rate. The graph is a straight line through the origin. In symbols, y ∝ x means y = kx, where k is the constant of proportionality.
如果两个量以相同速率同时增加或减少，则它们成正比例。图像是一条通过原点的直线。符号表示 y ∝ x 即 y = kx，其中k为比例常数。
erse proportion, as one quantity doubles, the other halves. This relationship is written y ∝ a. To find k, substitute a known pair of values and solve. Always state the formula connecting the variables before answering further parts.
在反比例中，一个量翻倍，另一个减半。关系写作 y ∝ 1/x，即 y = k/x。图像为双曲线。求k时，代入已知的一对值并解方程。在回答后续问题前，务必写出连接变量的公式。
Exam tip: check the question wording. 'y is proportional to the square of x' means y = kx². 'y is inversely proportional to the square root of x' means y = k/√x. The constant k remains the same for all data points.
考试提示：注意题目措辞。“y 与 x 的平方成正比”意味着 y = kx²。“y 与 x 的平方根成反比”意味着 y = k/√x。常数k对所有数据点保持不变。
7. Standard Form | 标准形式
form is used to write very large or very small numbers concisely. It is written as a × 10n a < 10 and n is an integer. For example, 4 500 000 = 4.5 × 10⁶, and 0.00032 = 3.2 × 10⁻⁴.
标准形式用于简洁地书写极大或极小的数字。它写成 a × 10ⁿ 的形式，其中 1 ≤ a < 10，n为整数。例如，4 500 000 = 4.5 × 10⁶，0.00032 = 3.2 × 10⁻⁴。
. To divide, divide the a values and subtract the powers: (6 × 10⁷) ÷ (3 × 10²) = 2 × 10⁵.
进行标准形式数字的加减时，首先统一10的指数。乘法：将a值相乘，指数相加：(3 × 10⁴) × (2 × 10⁵) = 6 × 10⁹。除法：将a值相除，指数相减：(6 × 10⁷) ÷ (3 × 10²) = 2 × 10⁵。
ndard form. If a becomes 10 or more, adjust by increasing the power: 12 × 10³ = 1.2 × 10⁴.
始终检查最终答案是否符合标准形式。如果a达到10或更大，需调整：12 × 10³ = 1.2 × 10⁴。
8. Estimation and Rounding | 估算与舍入
3200 / 0.2 = 16 000. The exact answer is around 14 963, showing the estimate is sensible.
估算是强大的检查工具。将每个数舍入到一位有效数字，然后心算。例如，估算 (412 × 7.9) / 0.218。舍入：400 × 8 / 0.2 = 3200 / 0.2 = 16 000。精确值约为14 963，说明估算是合理的。
Significant figures (s.f.) and decimal places (d.p.) are common rounding requirements. 0.04567 to 3 d.p. is 0.046; to 2 s.f. is 0.046. Keep zero placeholders where needed and avoid premature rounding during multi‑step calculations.
有效数字（s.f.）和小数位数（d.p.）是常见的舍入要求。0.04567 精确到3位小数为0.046；精确到2位有效数字为0.046。在需要时保留占位零，并避免多步骤计算中的过早舍入。
cm. This is crucial for error intervals, compound measures and area / perimeter problems.
上限与下限源于舍入。若长度记作6.4 cm（精确到1位小数），实际长度在区间6.35 cm ≤ L < 6.45 cm。这对误差区间、复合量度和面积/周长问题至关重要。
9. Fractions in Operations | 分数运算
be converted to improper fractions first: 1½ + 2⅓ = 3/2 + 7/3 = 9/6 + 14/6 = 23/6 = 3 5/6.
分数的加减需要公分母。对于2/3 + 1/4，3和4的最小公倍数是12，改写为8/12 + 3/12 = 11/12。带分数应先化为假分数：1½ + 2⅓ = 3/2 + 7/3 = 9/6 + 14/6 = 23/6 = 3 5/6。
or division, invert the second fraction and multiply: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.
乘法直接：分子乘分子，分母乘分母。2/3 × 3/8 = 6/24 = 1/4。除法：将第二个分数翻转然后相乘：2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6。
fore multiplying to make calculations easier. In the above, 2/3 × 3/8 could first be cross‑ancelled: the 2 and 8 become 1 and 4, and the 3s cancel completely, leaving 1/4 instantly.
始终约简最终答案。在乘法前先约分可使计算更简便。上例中，2/3 × 3/8 可以先交叉约分：2和8变成1和4，3完全约去，立即得到1/4。
10. Problem Solving with Ratios and Fractions | 比例与分数的问题求解
IGCSE often embeds ratios within word problems. For instance, 'e ratio of boys to girls in a class is 3 : 4. If there are 28 students, how many are girls?' irls = 4 parts × 4 = 16. Alternatively, fraction of girls = 4/7, so girls = 4/7 × 28 = 16.
IGCSE经常在文字题中嵌套比率。例如，“某班级男生与女生的比例是3 : 4。如果共有28名学生，女生有多少？”总份数为7，每份 = 28 ÷ 7 = 4。女生 = 4份 × 4 = 16。或者，女生所占分数为4/7，所以女生 = 4/7 × 28 = 16。
When a ratio changes, a common approach is to use a variable. Example: ' adding 6 red counters, the ratio becomes 7 : 2. Find the original number of red counters.' the original numbers be 5x and 2x. New red = 5x + 6. New ratio: (5x + 6) / 2x = 7/2. Cross‑multiply: 2(5x + 6) = 14x → 10x + 12 = 14x → 4x = 12 → x = 3. Original red = 5x = 15.
当比率发生变化时，常用设元法。例如：“红球与蓝球的比例为5 : 2。加入6个红球后，比例变为7 : 2。求原来红球的数量。”设原来数量为5x和2x。新红球 = 5x + 6。新比例：(5x + 6) / 2x = 7/2。交叉相乘：2(5x + 6) = 14x → 10x + 12 = 14x → 4x = 12 → x = 3。原来红球 = 5x = 15。
Harder problems combine ratios, percentages and algebra. Stay methodical, define your unknowns clearly, and verify your answer using the context.
更复杂的题目会结合比率、百分比和代数。要保持条理性，明确定义未知数，并在上下文中验证答案。
11. Key Exam Tips and Common Pitfalls | 核心应试技巧与常见陷阱
10 = 2/5. In IGCSE Edexcel, marks are given for method, even if the final answer is wrong.
1. 始终展示解题步骤。即使像0.4化分数（2/5）这样简单的转换也需要步骤：0.4 = 4/10 = 2/5。在IGCSE Edexcel中，即使最终答案错误，方法分仍然会被给予。
2. Beware of units in proportion questions. If a speed is given in km/h and time in minutes, convert minutes to hours. Graphs of direct proportion must pass through (0,0).
2. 在比例问题中注意单位。如果速度单位是km/h，时间是分钟，应将分钟转换为小时。正比例图像必须经过(0,0)点。
s; dots over two digits mean that block repeats. For 1/7 = 0.142857142857..., we write 0.1̇42857̇, but Edexcel also accepts the full set of recurring digits shown clearly.
3. 循环小数记法：单点表示该位数字循环；两点表示该数字块循环。1/7 = 0.142857142857... 记作 0.1̇42857̇，但Edexcel也接受明确写出的循环节。
4. When working with bounds, for calculating the maximum possible value of a ÷ b, use upper bound of a and lower bound of b, and vice versa for minimum. This logic is frequently tested.
4. 处理上下限时，计算 a ÷ b 的最大可能值，使用a的上限和b的下限，最小值则反之。这一逻辑经常被考查。
n ratio problems, never assume total given is always the sum of the parts. Read carefully: 'the ratio of cats to dogs is 5 : 3, and there are 24 dogs' means dog parts = 3 → 3 parts = 24 → 1 part = 8 → cats = 5 × 8 = 40.
5. 在比率题中，不要假设给出的总数总是份数和。仔细阅读：“猫与狗的比例为5 : 3，狗有24只”意味着狗的份数=3 → 3份 = 24 → 1份 = 8 → 猫 = 5 × 8 = 40。
12. Conclusion: Practice with Purpose | 结语：有目的的练习
Mastering the number topic requires not just memorising rules but understanding why they work. Regular practice of past paper questions will expose you to the Edexcel style and common variations. Use estimation to check your answers, and always re‑read the question to ensure you have answered exactly what was asked.
掌握数字专题不仅仅靠记忆规则，更要理解它们背后的原理。经常练习历年真题能让你熟悉Edexcel的题型和常见变式。用估算来检查答案，并始终重读题目确保回答了所问的内容。
Build confidence by starting with straightforward FDP conversions and working up to multi‑step proportion problems. The skills you develop here will support your work in algebra, geometry and data handling throughout the IGCSE course.
从简单的FDP互化开始，逐步提升到多步骤比例问题，树立信心。你在此处练就的技能将助力IGCSE课程中的代数、几何和数据处理等各个部分。
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