Multiplying and Dividing Fractions | 分数乘法与除法

📚 Multiplying and Dividing Fractions | 分数乘法与除法

Multiplying and dividing fractions are foundational skills in Grade 5 mathematics. Once you understand these operations, you can solve a huge range of problems involving parts of a whole, ratios, and proportional reasoning. This article breaks down the rules step by step, with clear examples and bilingual explanations to help you master the topic.

分数的乘法和除法是五年级数学的基础技能。一旦你掌握了这些运算，就可以解决大量涉及部分整体、比率和比例推理的问题。本文逐步拆解规则，配有清晰的示例和双语讲解，帮助你精通这一知识点。

1. Multiplying Fractions: The Basic Rule | 分数乘法的基本规则

When multiplying two fractions, simply multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. Then simplify the result if possible. In symbols: (a/b) × (c/d) = (a×c) / (b×d).

两个分数相乘时，只需将分子相乘得到新的分子，分母相乘得到新的分母，然后尽可能化简。用符号表示：(a/b) × (c/d) = (a×c) / (b×d)。

For example, 2/5 × 3/4 = (2×3) / (5×4) = 6/20. This simplifies to 3/10 by dividing both numerator and denominator by 2.

例如，2/5 × 3/4 = (2×3) / (5×4) = 6/20。将分子分母同时除以 2 即化简为 3/10。

Always check for cross-cancelling before multiplying to make simplification easier. / 在相乘之前先进行交叉约分，可以简化计算。

2. Multiplying a Fraction by a Whole Number | 分数乘以整数

A whole number can be written as a fraction with denominator 1. So multiplying a fraction by a whole number follows the same rule. For instance, 3 × 2/7 = (3/1) × (2/7) = (3×2)/(1×7) = 6/7.

整数可以写成分母为 1 的分数。因此，分数乘以整数遵循相同的规则。例如，3 × 2/7 = (3/1) × (2/7) = (3×2)/(1×7) = 6/7。

This means you simply multiply the whole number by the numerator of the fraction, keeping the denominator unchanged.

这意味着只需将整数乘以分数的分子，分母保持不变。

3. Multiplying Mixed Numbers | 带分数相乘

To multiply mixed numbers, first convert each mixed number to an improper fraction. For example, 1½ × 2⅓ becomes 3/2 × 7/3. Then multiply as usual: (3×7)/(2×3) = 21/6 = 3½ or 7/2.

带分数相乘时，首先将每个带分数转化为假分数。例如，1½ × 2⅓ 转化为 3/2 × 7/3。然后照常相乘：(3×7)/(2×3) = 21/6 = 3½ 或 7/2。

After finding the product, change the improper fraction back to a mixed number if the answer needs to be in simplest form.

求出乘积后，如果需要最简形式，将假分数转回带分数。

4. Dividing Fractions: Reciprocals and ‘Multiply by the Reciprocal’ | 分数除法：倒数与“乘以倒数”

To divide by a fraction, we multiply by its reciprocal. The reciprocal of a fraction a/b is b/a (swap the numerator and denominator). The rule is: a/b ÷ c/d = a/b × d/c.

除以一个分数，等于乘以它的倒数。分数 a/b 的倒数是 b/a（交换分子和分母）。规则是：a/b ÷ c/d = a/b × d/c。

For example, 3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10 = 2 1/10. This simple trick turns every division problem into a multiplication problem.

例如，3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10 = 2 1/10。这个简单的技巧将每个除法问题转化为乘法问题。

5. Dividing a Fraction by a Whole Number | 分数除以整数

Write the whole number as a fraction with denominator 1, then take its reciprocal. For instance, 2/3 ÷ 4 becomes 2/3 ÷ 4/1, which equals 2/3 × 1/4 = 2/12 = 1/6.

将整数写成分母为 1 的分数，然后取它的倒数。例如，2/3 ÷ 4 转化为 2/3 ÷ 4/1，等于 2/3 × 1/4 = 2/12 = 1/6。

Another way to think about it: dividing by a whole number is the same as multiplying by the fraction 1 over that number.

另一种思考方式：除以一个整数，等同于乘以该整数分之 1 的分数。

6. Dividing a Whole Number by a Fraction | 整数除以分数

Again, put the whole number over 1, then multiply by the reciprocal of the fraction. Example: 5 ÷ 2/3 = 5/1 × 3/2 = 15/2 = 7½.

同样，将整数放在 1 之上，然后乘以除数的倒数。例如：5 ÷ 2/3 = 5/1 × 3/2 = 15/2 = 7½。

This often results in an answer larger than the original whole number, which makes sense: you are finding how many fractional parts fit into the whole.

结果通常比原整数大，这是合理的：你正在求出该整数包含多少个分数部分。

7. Combined Operations with Multiplication and Division | 乘除混合运算

When an expression involves both multiplication and division of fractions, work from left to right, just as with whole numbers. Convert any division to multiplication by the reciprocal first.

当一个表达式同时包含分数的乘法和除法时，按从左到右的顺序计算，就像整数运算一样。先将所有除法转换为乘以其倒数。

Example: 2/3 ÷ 4/5 × 3/10 → first change ÷ to × reciprocal: 2/3 × 5/4 × 3/10. Then multiply across: (2×5×3)/(3×4×10) = 30/120 = 1/4. Cross-cancelling can speed this up.

例如：2/3 ÷ 4/5 × 3/10 → 先将 ÷ 改为 × 倒数：2/3 × 5/4 × 3/10。然后分子分母各自相乘：(2×5×3)/(3×4×10) = 30/120 = 1/4。交叉约分可以加快计算。

8. Real-life Word Problems | 实际应用问题

Fractions appear in many everyday situations: cooking, measuring, sharing. For instance, if a recipe needs 2/3 cup of sugar and you want to make half the recipe, you multiply: 1/2 × 2/3 = 2/6 = 1/3 cup.

分数出现在许多日常生活场景中：烹饪、测量、分享。例如，如果一个食谱需要 2/3 杯糖，而你只想做一半的量，就需要相乘：1/2 × 2/3 = 2/6 = 1/3 杯。

Division problems: if you have 3/4 of a pizza left and each person eats 1/8 of a whole pizza, how many people can eat? 3/4 ÷ 1/8 = 3/4 × 8/1 = 24/4 = 6 people.

除法问题：如果你剩下 3/4 个披萨，每人吃 1/8 个披萨，可以供多少人吃？ 3/4 ÷ 1/8 = 3/4 × 8/1 = 24/4 = 6 人。

9. Common Mistakes and How to Avoid Them | 常见错误与规避方法

One common error is forgetting to take the reciprocal before multiplying in division. Always flip the second fraction (the divisor), not the first.

一个常见错误是在除法相乘前忘记取倒数。始终翻转第二个分数（除数），而不是第一个。

Another mistake is forgetting to simplify the final answer or not simplifying before multiplying. Cross-cancelling reduces the numbers and makes calculation easier.

另一个错误是忘记化简最终答案，或者在相乘前未进行约分。交叉约分能减小数字，使计算更简单。

Never multiply mixed numbers directly without converting them to improper fractions first.

切勿不经转化为假分数就直接对带分数进行乘法运算。

10. Practice Problems with Step-by-Step Solutions | 练习题与逐步解答

Try these problems to test your understanding:

试试这些问题来检验你的理解：

Problem / 题目	Solution / 解答
3/8 × 4/9	(3×4)/(8×9) = 12/72 = 1/6 (or cross-cancel 3 and 9, 4 and 8) / (3×4)/(8×9)=12/72=1/6
2⅕ × 1¼	11/5 × 5/4 = 55/20 = 2 15/20 = 2¾ / 11/5 × 5/4 = 55/20 = 2 15/20 = 2¾
7/10 ÷ 2/5	7/10 × 5/2 = 35/20 = 1 15/20 = 1¾ / 7/10 × 5/2 = 35/20 = 1 15/20 = 1¾
6 ÷ 3/4	6/1 × 4/3 = 24/3 = 8 / 6/1 × 4/3 = 24/3 = 8
2/3 × 9 ÷ 1/2	2/3 × 9/1 × 2/1 = (2×9×2)/(3×1×1) = 36/3 = 12 / (2×9×2)/3=36/3=12

Review each step to see where the reciprocal is used and where simplification occurs. / 回顾每一步，看哪里使用了倒数，哪里进行了化简。

11. Visual Models for Multiplication and Division | 乘除法的可视化模型

Area models can help you understand multiplication. For instance, 2/3 × 3/4 can be shown as a rectangle divided into thirds one way and quarters the other. The overlapping area shows 6 out of 12 parts shaded, giving 6/12 = 1/2.

面积模型可以帮助理解乘法。例如，2/3 × 3/4 可以表示为一个长方形，按一个方向分成三等份，另一个方向分成四等份，重叠区域显示 12 份中的 6 份被涂色，得到 6/12 = 1/2。

For division, think of the question ‘how many groups of one fraction fit into another?’ This interpretation is extremely useful in word problems.

对于除法，可以思考“一个分数中包含多少组另一个分数？”这种解释在应用题中非常有用。

12. Summary and Key Takeaways | 总结与关键要点

Multiplying fractions is straightforward: numerator times numerator, denominator times denominator. Dividing fractions is equally easy once you remember ‘invert and multiply’. Always simplify your answers and convert mixed numbers before operating. With practice, these operations become second nature.

分数乘法很简单：分子乘分子，分母乘分母。分数除法一旦记住“颠倒相乘”也同样容易。始终化简答案，并在运算前转化带分数。经过练习，这些运算会变得自然而然。

Key rule: Multiply fractions by multiplying straight across. / 关键规则：分数相乘直接用分子分母分别相乘。
Key rule: Divide fractions by multiplying by the reciprocal of the divisor. / 关键规则：分数相除等于乘以除数的倒数。
Remember: Convert mixed numbers to improper fractions first. / 记住：先将带分数转化为假分数。

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Multiplying and Dividing Fractions | 分数乘法与除法