📚 Adding and Subtracting Fractions with Unlike Denominators | 异分母分数的加减法
In IGCSE mathematics, you often meet fractions such as 1/2 and 1/3 in the same calculation. You cannot add or subtract them directly before they share the same denominator. This lesson shows how to find a common denominator, rewrite each fraction, simplify the result, and handle mixed numbers confidently.
在 IGCSE 数学中，你经常会遇到 1/2 和 1/3 这样的分数出现在同一个算式中。在它们拥有相同分母之前，你不能直接加减。本节课将讲解如何寻找公分母、改写每个分数、化简结果，并熟练处理带分数。
1. What Are Unlike Denominators? | 什么是异分母？
Fractions have two parts: the numerator (top) and the denominator (bottom). The denominator tells you how many equal parts make up one whole. When two fractions have different denominators, such as 1/4 and 1/6, they are said to have unlike denominators.
分数有两个部分：分子（上方）和分母（下方）。分母表示一个整体被分成多少等份。当两个分数的分母不同，如 1/4 和 1/6，它们就称为异分母分数。
Because the parts are different sizes, adding straight across would be like adding apples and oranges. You must first express both fractions using the same-sized parts.
由于每一份的大小不同，直接把分子相加就像把苹果和橙子相加。你必须先把两个分数表示成同样大小的份数。
- Example: 1/4 + 1/6 cannot be simplified to 2/10.
- 示例：1/4 + 1/6 不能直接等于 2/10。
2. Why We Need a Common Denominator | 为什么需要公分母
A common denominator is a shared multiple of the original denominators. It allows both fractions to be described using the same unit fraction. Once the denominators match, you can add or subtract the numerators directly and keep the denominator unchanged.
公分母是原来几个分母的共同倍数。它让两个分数可以用同一个单位分数表示。一旦分母相同，你就可以直接加减分子，分母保持不变。
For example, 1/4 means one part out of four, while 1/6 means one part out of six. If you rewrite both as twelfths, you get 3/12 and 2/12. Now the parts are identical in size, so addition becomes easy.
例如，1/4 表示四份中的一份，而 1/6 表示六份中的一份。如果把两者都改写为十二分之几，就得到 3/12 和 2/12。此时每一份大小相同，加法就变得简单了。
1/4 = 3/12, 1/6 = 2/12
3. Finding the Least Common Denominator (LCD) | 如何找到最小公分母
The least common denominator is the lowest common multiple (LCM) of the denominators. Using the LCD keeps numbers small and reduces the chance of arithmetic mistakes. You can find it by listing multiples or by prime factorisation.
最小公分母是各分母的最小公倍数。使用最小公分母可以让数字保持较小，减少计算错误。你可以通过列举倍数或质因数分解来找到它。
- Listing method: Multiples of 4 are 4, 8, 12, 16, 20, ... Multiples of 6 are 6, 12, 18, 24, ... The lowest shared multiple is 12.
- 列举法：4 的倍数有 4、8、12、16、20……6 的倍数有 6、12、18、24……最小共同倍数是 12。
- Prime factor method: 4 = 2² and 6 = 2 × 3. Take 2² and 3, so LCM = 2² × 3 = 12.
- 质因数法：4 = 2²，6 = 2 × 3。取 2² 和 3，所以最小公倍数 = 2² × 3 = 12。
In an IGCSE question, you may write the LCD in the working line to show your method clearly.
在 IGCSE 题目中，你可以在计算步骤中写出最小公分母，以清晰展示你的方法。
4. Equivalent Fractions Review | 等值分数回顾
To rewrite a fraction with a new denominator, multiply the numerator and denominator by the same non-zero number. This produces an equivalent fraction because the overall value does not change.
要用新的分母改写一个分数，需要给分子和分母同时乘一个非零数。这样得到的等值分数的整体大小不变。
3/12. When the denominator changes from 6 to 12, the multiplier is 2, so 1/6 becomes 2/12.
当分母从 4 变成 12 时，乘数是 3。因此，1/4 变为 1×3 除以 4×3，等于 3/12。当分母从 6 变成 12 时，乘数是 2，所以 1/6 变为 2/12。
1/4 × 3/3 = 3/12, 1/6 × 2/2 = 2/12
This step is essential before any addition or subtraction of unlike fractions.
这一步是异分母分数加减之前必不可少的一步。
5. Adding Fractions with Unlike Denominators | 异分母分数加法
To add fractions with unlike denominators, follow three steps: find the LCD, write equivalent fractions, then add the numerators. Keep the denominator the same in the final fraction.
异分母分数相加，遵循三步：找到最小公分母，写出等值分数，然后相加分子。最终分数的分母保持不变。
Worked example: Add 1/4 and 1/6.
例题：计算 1/4 + 1/6。
12. Rewrite: 1/4 = 3/12 and 1/6 = 2/12. Add the numerators: 3 + 2 = 5. So the sum is 5/12.
最小公分母是 12。改写：1/4 = 3/12，1/6 = 2/12。分子相加：3 + 2 = 5。所以和是 5/12。
1/4 + 1/6 = 3/12 + 2/12 = 5/12
Always check whether the answer can be simplified. In this case, 5/12 is already in simplest form.
最后一定要检查答案是否可以化简。这里 5/12 已经是最简分数。
6. Subtracting Fractions with Unlike Denominators | 异分母分数减法
Subtraction follows the same preparation as addition: find the LCD and rewrite both fractions. Then subtract the numerators while keeping the shared denominator unchanged.
减法的准备步骤与加法相同：找到最小公分母并改写两个分数。然后分子相减，共同的分母保持不变。
Worked example: Work out 2/3 − 1/4.
例题：计算 2/3 − 1/4。
LCD = 12. Rewrite: 2/3 = 8/12 and 1/4 = 3/12. Subtract: 8 − 3 = 5. The result is 5/12.
最小公分母是 12。改写：2/3 = 8/12，1/4 = 3/12。相减：8 − 3 = 5。结果是 5/12。
2/3 − 1/4 = 8/12 − 3/12 = 5/12
Do not subtract the denominators. The denominator remains 12 because it only names the size of each part.
不要减去分母。分母仍然是 12，因为它只表示每一份的大小。
7. Adding and Subtracting Mixed Numbers | 带分数的加减法
Mixed numbers include a whole-number part and a fractional part, such as 2 1/3. You can either convert them to improper fractions first, or add the whole parts and fractional parts separately. Converting to improper fractions is usually safer when borrowing is needed.
带分数包含整数部分和分数部分，例如 2 1/3。你可以先把它们化为假分数，也可以把整数部分和分数部分分别相加。当需要借位时，化为假分数通常更稳妥。
Worked example: Work out 2 1/3 + 1 1/4.
例题：计算 2 1/3 + 1 1/4。
8/12 and 5/4 = 15/12. Add: 28 + 15 = 43, so 43/12. Convert back to a mixed number: 3 7/12.
转换：2 1/3 = 7/3，1 1/4 = 5/4。3 和 4 的最小公分母是 12。改写：7/3 = 28/12，5/4 = 15/12。相加：28 + 15 = 43，得到 43/12。化回带分数：3 7/12。
2 1/3 + 1 1/4 = 7/3 + 5/4 = 28/12 + 15/12 = 43/12 = 3 7/12
Subtraction with borrowing: 5 1/4 − 2 2/3 = 21/4 − 8/3 = 63/12 − 32/12 = 31/12 = 2 7/12.
带借位的减法：5 1/4 − 2 2/3 = 21/4 − 8/3 = 63/12 − 32/12 = 31/12 = 2 7/12。
8. Simplifying Your Final Answer | 化简最终答案
After adding or subtracting, always simplify the resulting fraction if possible. Divide the numerator and denominator by their greatest common factor. If the numerator is larger than the denominator, convert the improper fraction to a mixed number.
加减之后，如果可能，一定要化简所得分数。用分子和分母的最大公因数约分。如果分子大于分母，要把假分数化为带分数。
- Example: 6/8 simplifies to 3/4 because both 6 and 8 divide by 2.
- 示例：6/8 化简为 3/4，因为 6 和 8 都可以除以 2。
- Example: 13/12 becomes 1 1/12 as a mixed number.
- 示例：13/12 化为带分数是 1 1/12。
IGCSE mark schemes often require answers in simplest form, unless a mixed number or improper fraction is specified.
IGCSE 评分标准通常要求答案是最简形式，除非题目特别说明要带分数或假分数。
9. Common Mistakes to Avoid | 常见错误与避坑
Students sometimes add the denominators when adding fractions. For example, 1/3 + 1/6 is not 2/9. The denominators must be made the same first, and the common denominator stays unchanged.
学生有时会在分数相加时把分母也相加。例如，1/3 + 1/6 不等于 2/9。必须先让分母相同，公分母保持不变。
Another common error is choosing a common multiple that is not the lowest one. This is not wrong mathematically, but it makes simplifying harder. Using 24 instead of 12 is acceptable if you simplify at the end.
另一个常见错误是选择了不是最小的公倍数。这在数学上并不算错，但会让化简更麻烦。使用 24 而不是 12 是可以的，只要最后记得化简。
- Wrong: 1/4 + 1/6 = 2/10.
- 错误：1/4 + 1/6 = 2/10。
- Right: 1/4 + 1/6 = 5/12.
- 正确：1/4 + 1/6 = 5/12。
| Common mistake | 常见错误 | Correct thinking | 正确思路 |
| Adding denominators | 把分母也相加 | Keep the LCD | 保留最小公分母 |
| Forgetting to multiply numerator | 忘记同时乘分子 | Multiply top and bottom by the same number | 分子分母同乘一个数 |
| Not simplifying final answer | 最后忘记化简 | Always divide by the highest common factor | 始终除以最大公因数 |
10. Worked Examples from IGCSE Style Problems | IGCSE 题型精讲
Question 1: Work out 3/8 + 1/6. Give your answer as a fraction in its simplest form.
题 1：计算 3/8 + 1/6。答案用最简分数表示。
LCD of 8 and 6 is 24. Rewrite: 3/8 = 9/24 and 1/6 = 4/24. Add: 9 + 4 = 13. Answer: 13/24.
8 和 6 的最小公分母是 24。改写：3/8 = 9/24，1/6 = 4/24。相加：9 + 4 = 13。答案：13/24。
3/8 + 1/6 = 9/24 + 4/24 = 13/24
Question 2: Work out 7/10 − 2/15.
题 2：计算 7/10 − 2/15。
and 15 is 30. Rewrite: 7/10 = 21/30 and 2/15 = 4/30. Subtract: 21 − 4 = 17. Answer: 17/30.
10 和 15 的最小公分母是 30。改写：7/10 = 21/30，2/15 = 4/30。相减：21 − 4 = 17。答案：17/30。
7/10 − 2/15 = 21/30 − 4/30 = 17/30
Question 3: A recipe needs 2/5 cup of milk for one cake and 1/3 cup for another cake. How much milk is needed in total?
题 3：一个蛋糕需要 2/5 杯牛奶，另一个蛋糕需要 1/3 杯牛奶。总共需要多少牛奶？
LCD of 5 and 3 is 15. 2/5 = 6/15, 1/3 = 5/15. Total = 11/15 cup.
5 和 3 的最小公分母是 15。2/5 = 6/15，1/3 = 5/15。总共 = 11/15 杯。
11. Quick Practice Questions | 速练题
Try these questions to check your understanding. Write each answer in its simplest form.
尝试以下题目来检验你的理解。每个答案都写成最简分数。
- 1. 1/2 + 1/3
- 2. 5/6 − 1/4
- 3. 2 1/5 + 1 1/2
- 4. 4 − 2/3
- 5. 3/7 + 2/5
Check your answers: 1. 5/6, 2. 7/12, 3. 3 7/10, 4. 3 1/3, 5. 29/35.
参考答案：1. 5/6，2. 7/12，3. 3 7/10，4. 3 1/3，5. 29/35。
12. Key Takeaways | 小结
To add or subtract fractions with unlike denominators, always find the least common denominator first. Rewrite each fraction as an equivalent fraction with that denominator. Then add or subtract only the numerators and simplify the result.
异分母分数加减的关键是：始终先找最小公分母。把每个分数改写为该分母下的等值分数。然后只加减分子，最后化简结果。
For mixed numbers, converting to improper fractions helps you handle borrowing and carrying without confusion. With practice, these steps become automatic, and you can solve IGCSE fraction problems quickly and accurately.
对于带分数，化为假分数可以帮助你避免借位和进位时的混乱。只要多加练习，这些步骤就会变得自动化，你就能快速准确地解答 IGCSE 分数问题。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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