📚 Multiplying Fractions | 分数乘法
In IGCSE Mathematics, multiplying fractions is a core number skill that appears in topics such as fractions, percentages, probability and algebra. This article develops a clear understanding of the rule and its visual meaning, together with worked examples and common errors.
分数乘法是 IGCSE 数学中的核心数字运算技能，在分数、百分数、概率和代数等主题中都会出现。本文帮助你真正理解分数乘法的法则和图形含义，并通过例题与常见错误进行巩固。
1. Meaning of Fraction Multiplication | 分数乘法的含义
Multiplying fractions can be understood as taking a part of a part. For example, 1/2 × 3/4 means “one half of three quarters”. The word “of” often signals multiplication in fraction problems.
分数乘法可以理解为“求一个部分的若干等份”。例如 1/2 × 3/4 表示“四分之三的一半”。在分数应用题中，英文单词 “of” 通常表示乘法。
You can model this with two shadings. Draw one rectangle to show 3/4 by dividing it into 4 equal columns and shading 3 columns. Then divide the same rectangle into 2 equal rows and shade 1 row to show 1/2. The overlap covers 3 out of 8 equal parts, so 1/2 × 3/4 = 3/8.
你可以用两次涂色来建立图形模型。画一个长方形，先把它平均分成 4 列并涂出 3 列表示 3/4；再把同一个长方形平均分成 2 行并涂出 1 行表示 1/2。两次涂色重叠的区域是 8 等份中的 3 份，所以 1/2 × 3/4 = 3/8。
2. The Core Multiplication Rule | 核心乘法法则
To multiply two fractions, multiply the numerators to obtain the new numerator, and multiply the denominators to obtain the new denominator.
两个分数相乘时，用分子相乘的积作为新分子，用分母相乘的积作为新分母。
a/b × c/d = (a × c)/(b × d)
For example, 2/3 × 4/5 = (2 × 4)/(3 × 5) = 8/15. This rule works for all fractions, including improper fractions and fractions with negative signs.
例如 2/3 × 4/5 = (2 × 4)/(3 × 5) = 8/15。这个法则适用于所有分数，包括假分数和带有负号的分数。
3. Simplifying Before You Multiply | 先约分再相乘
It is usually easier to cancel common factors before multiplying rather than simplifying a large product at the end. You may cancel any numerator with any denominator.
计算时通常先约分再相乘，这样会比先乘出大数再化简更简单。任意一个分子都可以和任意一个分母进行约分。
Example: 3/8 × 4/9. Cancel 3 and 9 by 3, then cancel 4 and 8 by 4. This gives 1/2 × 1/3 = 1/6. Without cancelling, you would get 12/72 = 1/6, but with larger numbers.
例如：3/8 × 4/9。先把 3 和 9 同时除以 3，再把 4 和 8 同时除以 4，得到 1/2 × 1/3 = 1/6。如果不约分，你会得到 12/72 = 1/6，但数字会更大。
4. Multiplying by a Whole Number | 分数乘整数
To multiply a fraction by a whole number, write the whole number as a fraction with denominator 1. Then apply the usual rule.
分数乘整数时，先把整数写成分母为 1 的分数，然后使用普通的分数乘法法则。
n × a/b = n/1 × a/b = (n × a)/b
Example: 5 × 3/10 = 5/1 × 3/10 = 15/10 = 3/2 = 1 1/2. A faster method is to cancel 5 and 10 by 5 first, giving 1/1 × 3/2 = 3/2.
例如：5 × 3/10 = 5/1 × 3/10 = 15/10 = 3/2 = 1 1/2。更快的方法是先把 5 和 10 同时除以 5，得到 1/1 × 3/2 = 3/2。
5. Multiplying Mixed Numbers | 带分数的乘法
When multiplying mixed numbers, convert each mixed number to an improper fraction first. Then multiply straight across and simplify the result.
计算带分数乘法时，先把每个带分数化为假分数，然后再按分数乘法法则计算并化简结果。
Example: 1 1/2 × 2 2/3. Convert to 3/2 × 8/3. Multiply to get (3 × 8)/(2 × 3) = 24/6 = 4.
例如：1 1/2 × 2 2/3。先化为 3/2 × 8/3，相乘得到 (3 × 8)/(2 × 3) = 24/6 = 4。
Do not multiply the whole number parts and fraction parts separately. Conversion to improper fractions avoids this mistake.
不要把整数部分和分数部分分别相乘。先化为假分数可以避免这种错误。
6. Area Models and Visual Proof | 面积模型与直观证明
An area model gives a visual reason for the multiplication rule. To show 2/3 × 4/5, divide a unit square into 3 equal rows and 5 equal columns, producing 15 equal small rectangles.
面积模型可以直观解释分数乘法法则。为了表示 2/3 × 4/5，把一个单位正方形平均分成 3 行和 5 列，得到 15 个相等的小长方形。
Shade 2 of the 3 rows to represent 2/3, and shade 4 of the 5 columns to represent 4/5. The double-shaded region has 2 × 4 = 8 small rectangles out of 3 × 5 = 15, so the result is 8/15.
给 3 行中的 2 行涂色表示 2/3，再给 5 列中的 4 列涂色表示 4/5。双重涂色区域有 2 × 4 = 8 个小长方形，占总份数 3 × 5 = 15，因此结果是 8/15。
This shows why numerators and denominators multiply separately: the total number of parts comes from multiplying the denominators, and the shaded parts come from multiplying the numerators.
这解释了分子和分母为什么分别相乘：总份数来自分母相乘，涂色份数来自分子相乘。
7. Common Misconceptions | 常见错误辨析
- English: Adding denominators instead of multiplying: 1/2 × 1/3 is not 1/5. The correct answer is 1/6.
中文：把分母相加而不是相乘：1/2 × 1/3 不等于 1/5，正确答案是 1/6。 - English: Forgetting to simplify: 4/6 should be written as 2/3 in its simplest form.
中文：忘记化简：4/6 应写成最简分数 2/3。 - English: Multiplying mixed numbers without converting to improper fractions often gives an incorrect answer.
中文：带分数没有先化为假分数就直接相乘，经常导致错误答案。 - English: Cancelling incorrectly between two numerators or two denominators; you can only cancel across a numerator and a denominator.
中文：约分时错误地在两个分子之间或两个分母之间约分；只能在一个分子和一个分母之间约分。
8. Worked Examples | 例题精讲
Example 1: Work out 2/3 × 3/4.
例题 1：计算 2/3 × 3/4。
2/3 × 3/4 = (2 × 3)/(3 × 4) = 6/12 = 1/2
Cancel the common factor 3 before multiplying: 2/3 × 3/4 becomes 2/1 × 1/4 = 2/4 = 1/2.
相乘前先约去公因数 3：2/3 × 3/4 变成 2/1 × 1/4 = 2/4 = 1/2。
Example 2: Work out 6 × 5/12.
例题 2：计算 6 × 5/12。
6 × 5/12 = 6/1 × 5/12 = 30/12 = 5/2 = 2 1/2
Alternatively, cancel 6 and 12 by 6 to get 1/1 × 5/2 = 5/2.
也可以先把 6 和 12 同时除以 6，得到 1/1 × 5/2 = 5/2。
Example 3: Work out 2 1/4 × 1 2/3.
例题 3：计算 2 1/4 × 1 2/3。
2 1/4 × 1 2/3 = 9/4 × 5/3 = 45/12 = 15/4 = 3 3/4
You can also cancel 9 and 3 by 3 before multiplying: 9/4 × 5/3 becomes 3/4 × 5/1 = 15/4.
也可以在相乘前先把 9 和 3 同时除以 3：9/4 × 5/3 变成 3/4 × 5/1 = 15/4。
9. Real-Life Applications | 生活实际应用
Recipes often require fraction multiplication. If a recipe needs 3/4 cup of sugar and you make half the recipe, you need 1/2 × 3/4 = 3/8 cup of sugar.
食谱中经常用到分数乘法。如果一份食谱需要 3/4 杯糖，而你只做一半的份量，就需要 1/2 × 3/4 = 3/8 杯糖。
Distance problems also use fraction multiplication. If a walking trail is 20 km long and you complete 3/5 of it, you walk 3/5 × 20 = 12 km.
路程问题也会用到分数乘法。如果一条步行路线长 20 km，你走了全程的 3/5，那么你走了 3/5 × 20 = 12 km。
In probability, the chance that two independent events both happen can be found by multiplying their fractions. For example, 1/2 × 1/3 = 1/6.
在概率中，两个独立事件同时发生的概率可以通过分数相乘得到。例如 1/2 × 1/3 = 1/6。
10. Quick Practice | 快速练习
Try these four problems and check each answer in simplest form.
尝试完成下面四道题，并将答案化为最简分数。
- English: 3/4 × 5/6
中文：3/4 × 5/6 - English: 4 × 7/8
中文：4 × 7/8 - English: 1 2/5 × 2 1/2
中文：1 2/5 × 2 1/2 - English: 2/9 × 3/4
中文：2/9 × 3/4
Answers: 3/4 × 5/6 = 15/24 = 5/8; 4 × 7/8 = 7/2 = 3 1/2; 1 2/5 × 2 1/2 = 7/5 × 5/2 = 7/2 = 3 1/2; 2/9 × 3/4 = 6/36 = 1
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