📚 Adding and Subtracting Fractions with Unlike Denominators | 异分母分数的加减法
In IGCSE Mathematics, you are expected to add and subtract fractions with different denominators accurately, often without a calculator. This topic, introduced in earlier grades and used throughout IGCSE Number skills, depends on equivalent fractions and lowest common multiples (LCM). This article explains the complete method, from finding the lowest common denominator to simplifying the final answer, with worked examples and common exam traps.
在 IGCSE 数学中，你需要不借助计算器准确地进行异分母分数的加减法。这一知识点在低年级引入，并贯穿 IGCSE 数与代数部分，依赖于等价分数和最小公倍数（LCM）的概念。本文将完整讲解方法，从寻找最小公分母到化简最终答案，并提供例题和常见考试陷阱。
1. Why Unlike Denominators Cannot Be Added Directly | 为什么异分母不能直接相加
A fraction represents a number of equal parts of a whole. The denominator tells you the size of each part. If two fractions have different denominators, their parts are different sizes, so you cannot simply add the numerators. For example, 1/2 and 1/3 represent pieces of different sizes, so 1/2 + 1/3 is not 2/5.
分数表示整体被等分成若干份后取出的份数。分母表示每一份的大小。如果两个分数的分母不同，每一份的大小就不同，因此不能直接把分子相加。例如，1/2 和 1/3 代表的每一份大小不同，所以 1/2 + 1/3 不等于 2/5。
You can only add or subtract numerators directly when the denominators are the same. Therefore, the first step is always to rewrite both fractions with a common denominator. This does not change their values because you are using equivalent fractions.
只有当两个分数的分母相同时，才能直接对分子进行加减。因此第一步总是把两个分数改写成具有相同分母的形式。由于使用的是等价分数，这不会改变分数的值。
2. Key Vocabulary | 关键术语
Before working through the method, make sure you know these terms.
开始学习方法之前，先确保你认识以下术语。
- Numerator | 分子: the top number of a fraction, showing how many parts are taken.
- Denominator | 分母: the bottom number, showing the total number of equal parts.
- Equivalent fractions | 等价分数: fractions that have the same value but different numerators and denominators.
- Lowest common multiple (LCM) | 最小公倍数: the smallest number that is a multiple of two or more numbers.
- Lowest common denominator (LCD) | 最小公分母: the LCM of the denominators, used to rewrite fractions.
- Improper fraction | 假分数: a fraction where the numerator is greater than or equal to the denominator.
- Mixed number | 带分数: a whole number plus a proper fraction, such as 2 1/3.
3. Finding the Lowest Common Denominator Using LCM | 用最小公倍数找最小公分母
The lowest common denominator is the LCM of the denominators. Using the LCD keeps the numbers small and makes simplifying easier. To find it, list the multiples of each denominator and choose the smallest number that appears in both lists.
最小公分母就是两个分母的最小公倍数。使用最小公分母可以使数字保持较小，化简也更容易。要求它，分别列出两个分母的倍数，并选择两个列表中都出现的最小数字。
Example: Find the LCD of 1/4 and 1/6.
例子：求 1/4 和 1/6 的最小公分母。
Multiples of 4: 4, 8, 12, 16, 20, 24 ...
Multiples of 6: 6, 12, 18, 24, 30 ...
LCD = 12
So 12 is the lowest common denominator because it is the smallest multiple shared by
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