📚 IGCSE Maths: Fractions, Decimals, Percentages and Ratio | IGCSE 数学：分数、小数、百分比与比例
Fractions, decimals, percentages and ratio are the building blocks of almost every IGCSE Maths paper. A strong command of these topics allows you to convert fluently between forms, solve real-life problems, and avoid the small slips that cost marks in both the non-calculator and calculator papers.
分数、小数、百分比和比例是几乎所有 IGCSE 数学试卷的基础模块。熟练掌握这些主题，你就能在三种形式之间灵活转换，解决现实生活中的问题，并在非计算器和计算器试卷中避免那些导致失分的小错误。
This revision guide works through the subject in the order the skills build up: fractions first, then decimals, then percentages, then ratio, and finally the exam habits that turn correct thinking into awarded marks. Every worked example uses the notation you will see in the examination, and each section ends with the mistakes examiners report most often.
本篇复习指南按照技能递进的顺序展开：先分数，再小数，然后百分比，接着比例，最后是那些能把"想对了"变成"拿到分"的考试习惯。每一个例题都使用你在考卷上会看到的写法，每一节末尾都会点出考官最常提到的错误。
1. Why These Topics Decide Your Grade | 为什么这些主题决定你的成绩
These four ideas sit inside the Number strand of every IGCSE Mathematics specification, including Edexcel 4MA1 and Cambridge 0580 and 0980. Their influence, however, reaches far beyond Number questions. Percentages appear inside statistics and finance questions, ratio appears inside geometry through similar shapes, scale drawings and enlarged triangles, and fractions appear inside probability and algebra through algebraic fractions and equations with fractional coefficients.
这四个主题位于所有 IGCSE 数学大纲的"数"板块，包括 Edexcel 4MA1 与 Cambridge 0580、0980。但它们的影响远不止"数"的题目：百分比会藏在统计与金融题中，比例会通过相似图形、比例图和放大三角形出现在几何题中，分数则会通过分式代数式和含分数系数的方程出现在概率与代数题中。
The non-calculator paper removes your safety net, so you must be able to simplify, convert and compare without a machine. The calculator paper hides the same skills inside multi-step word problems, so you must also be able to read a question and decide which form is most useful. In both papers the arithmetic is rarely the point; the marks are for choosing a method and applying it accurately.
非计算器试卷拿走了你的"安全网"，因此你必须能不用机器就完成化简、转换和比较。计算器试卷则把同样的技能藏进多步骤应用题里，所以你还要会读题，并判断用哪种形式最方便。在两份试卷中，计算本身很少是考点，分数给的是"选对方法并准确执行"。
- Fluently converting between fractions, decimals and percentages | 在分数、小数、百分比之间自如转换
- Choosing the multiplier method for percentage problems | 在百分比问题中选择乘数法
- Reading ratio notation and sharing quantities in a given ratio | 读懂比例记号并按给定比例分配数量
- Estimating an answer before calculating it exactly | 在精确计算之前先估算答案
2. Understanding Fractions: Simplify, Compare and Convert | 理解分数：化简、比较与转换
A fraction records a part of a whole: the denominator tells you how many equal parts the whole is divided into, and the numerator tells you how many of those parts you have. Two fractions are equivalent when one can be turned into the other by multiplying or dividing the top and the bottom by the same non-zero number, and simplifying is simply the reverse of that process.
分数记录的是整体的一个部分：分母表示整体被分成多少个等份，分子表示你取了多少份。如果一个分数可以通过分子分母同时乘以或除以同一个非零数变成另一个，那么这两个分数就是等值的，而化简正是这个过程的逆运算。
24) = 6, to obtain 3/4. Always check at the end that no number greater than 1 divides both parts exactly, because an unsimplified answer can lose a mark even when the value is correct.
像 18/24 这样的分数首先应写成最简形式。将分子和分母同时除以它们的最大公因数 HCF(18, 24) = 6，得到 3/4。最后一定要检查是否还有大于 1 的数能同时整除分子和分母，因为即使数值正确，未化简的答案也可能被扣分。
into a mixed number, divide 17 by 5: the quotient is 3 and the remainder is 2, so 17/5 = 3⅖ reverse the process, multiply the whole number by the denominator and add the numerator: 3⅖ = (3 × 5 + 2)/5 = 17/5.
假分数的分子大于分母，例如 17/5。把它化成带分数时，用 17 除以 5：商为 3，余数为 2，因此 17/5 = 3⅖。反向操作时，把整数部分乘以分母再加上分子：3⅖ = (3 × 5 + 2)/5 = 17/5。
is greater than 3/5. Because 5 × 5 = 25 and 3 × 8 = 24, cross-multiplication gives the same conclusion in one step, and comparing each fraction with 1/2 is a quick mental check that catches careless errors.
要比较 5/8 和 3/5，需要把两个分数改写为同分母。8 和 5 的最小公倍数是 40，所以 5/8 = 25/40，3/5 = 24/40；因此 5/8 大于 3/5。由于 5 × 5 = 25 而 3 × 8 = 24，交叉相乘一步就能得到同样结论，而把每个分数与 1/2 比较则是快速的心算检查，能抓住粗心造成的错误。
3. The Four Operations with Fractions | 分数的四则运算
cting works in exactly the same way: 5/6 − 1/4 = 10/12 − 3/12 = 7/12. A common denominator can always be found by multiplying the two denominators, but the lowest common multiple keeps the numbers small and reduces the risk of arithmetic slips.
分数加减时，必须先找到公分母。对于 2/3 + 1/4，3 和 4 的最小公倍数是 12，因此 8/12 + 3/12 = 11/12。减法完全相同：5/6 − 1/4 = 10/12 − 3/12 = 7/12。公分母总可以用两个分母相乘得到，但用最小公倍数能让数字更小，降低算错的风险。
fractions first. For example, 1½ + 2⅓ becomes 3/2 + 7/3, which is 9/6 + 14/6 = 23/6, or 3⅚ the whole-number parts and the fractional parts separately and losing track of the borrow.
涉及带分数时，先将其化为假分数。例如，1½ + 2⅓ 可化为 3/2 + 7/3，即 9/6 + 14/6 = 23/6，也就是 3⅚。先转换可以避免经典错误：把整数部分和分数部分分开处理时忘记借位。
ancel before multiplying to make the numbers smaller, for example 4/9 × 3/8 cancels to 1/3 × 1/2 = 1/6. Remember that the word "of" also means multiply, so 2/3 of 45 is 2/3 × 45 = 30.
乘法更简单：分子相乘，分母相乘。对于 2/5 × 3/7，乘积为 (2 × 3)/(5 × 7) = 6/35。你可以在相乘前交叉约分，使数字更小，例如 4/9 × 3/8 约分后为 1/3 × 1/2 = 1/6。记住 "of" 也表示相乘，所以 45 的 2/3 就是 2/3 × 45 = 30。
be written as the mixed number 1⅞. Never attempt to divide numerators by numerators and denominators by denominators; the reciprocal method is the one examiners expect to see.
分数除法要乘以倒数。计算 3/4 ÷ 2/5 可转化为 3/4 × 5/2 = 15/8，也可以写成带分数 1⅞。千万不要尝试"分子除分子、分母除分母"，倒数法才是考官期望看到的方法。
4. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分比的互化
Any fraction can be written as a decimal by dividing the numerator by the
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