📚 Mastering Fraction Multiplication and Division | 掌握分数乘法与除法
Multiplying and dividing fractions might seem tricky at first, but once you understand the straightforward rules, they become some of the most satisfying calculations in Key Stage 3 mathematics. This article will guide you through every step, from simple fractions to mixed numbers, and will provide plenty of examples and common pitfalls to watch out for. Whether you are tackling Cambridge checkpoint questions or building your confidence for algebra later on, mastering these operations now will give you a solid foundation.
一开始，分数的乘法和除法可能看起来有些棘手，但一旦你理解了那些直截了当的规则，它们就会变成 KS3 数学中最令人感到满足的计算之一。本文将带你一步步走过每一个环节，从简单分数到带分数，并提供大量示例和需要留意的常见错误。无论你是在攻克 Cambridge checkpoint 题目，还是为以后的代数学习建立信心，现在掌握这些运算都将为你打下坚实的基础。
1. What Are Fractions? | 什么是分数？
A fraction represents a part of a whole. It consists of a numerator (the top number) and a denominator (the bottom number). For example, in the fraction ¾, the numerator is 3 and the denominator is 4, meaning we have 3 parts out of an equal whole divided into 4. Understanding this basic idea is essential before moving on to multiplication and division.
分数表示整体的一部分。它由分子（上面的数）和分母（下面的数）组成。例如，在分数 ¾ 中，分子是 3，分母是 4，表示我们把整体平均分成 4 份，取了其中的 3 份。在继续学习乘法和除法之前，理解这个基本概念至关重要。
Fractions can be proper (numerator smaller than denominator, like ⅔), improper (numerator larger or equal, like 7/4), or mixed numbers (a whole number and a proper fraction combined, like 1½). All of these appear in multiplication and division problems, so it is helpful to be comfortable converting between them.
分数可以是真分数（分子小于分母，如 ⅔）、假分数（分子大于或等于分母，如 7/4）或是带分数（一个整数和一个真分数组合，如 1½）。这些都会在乘法和除法问题中出现，因此熟练地在它们之间进行转换会很有帮助。
2. Multiplying Simple Fractions | 简单分数相乘
To multiply two simple fractions, you simply multiply the numerators together and multiply the denominators together. The rule is:
a/b × c/d = (a×c) / (b×d)
将两个简单分数相乘，只需将分子相乘，分母相乘。规则是：
a/b × c/d = (a×c) / (b×d)
For example, ½ × ⅓ = (1×1)/(2×3) = ⅙. Another example: ⅖ × ³⁄₇ = (2×3)/(5×7) = ⁶⁄₃₅. Notice that you do not need a common denominator here — that rule is for addition and subtraction, not for multiplication.
例如，½ × ⅓ = (1×1)/(2×3) = ⅙。另一个示例：⅖ × ³⁄₇ = (2×3)/(5×7) = ⁶⁄₃₅。请注意，这里你不需要公分母——那是用于加法和减法的规则，乘法则不需要。
ing. Simplifying makes your final answer cleaner and is often required to gain full marks.
始终检查得出的分数是否可以化简。例如，²⁄₄ × ½ = ²⁄₈ = 化简后得到 ¼。化简可以使你的最终答案更简洁，而且通常要得到满分就必须化简。
3. Multiplying Mixed Numbers | 带分数乘法
When you multiply mixed numbers, the safest first step is to convert them into improper fractions. This avoids mistakes that happen when trying to multiply the whole number and fractional parts separately.
当带分数相乘时，最稳妥的第一步是把它们转换为假分数。这样可以避免因试图将整数部分和分数部分分开相乘而带来的错误。
× (8/3) = (3×8)/(2×3) = 24/6 = 4. Notice how the answer simplifies to a neat whole number.
例如，要计算 1½ × 2⅔：先将 1½ 转换为 3/2（因为 1×2+1=3，分母为 2），将 2⅔ 转换为 8/3（因为 2×3+2=8，分母为 3）。然后像之前一样相乘：(3/2) × (8/3) = (3×8)/(2×3) = 24/6 = 4。注意答案简化为了一个整洁的整数。
final answer. For instance, 2½ × ¼ = (5/2) × (1/4) = 5/8, which remains a proper fraction.
乘法结束后，如果假分数不能简化成整数，就把它转换回带分数作为最终答案。例如，2½ × ¼ = (5/2) × (1/4) = 5/8，结果仍是一个真分数。
4. Dividing Fractions: The Reciprocal Rule | 分数除法：倒数规则
Division with fractions uses the concept of the reciprocal (or multiplicative inverse). The reciprocal of a fraction is simply that fraction flipped upside down. For example, the reciprocal of ⅔ is 3/2. The rule for dividing fractions is: multiply the first fraction by the reciprocal of the second.
(a/b) ÷ (c/d) = (a/b) × (d/c) = (a×d) / (b×c)
分数的除法会用到倒数（或乘法逆）的概念。一个分数的倒数就是将该分数上下颠倒。例如，⅔ 的倒数是 3/2。分数除法的规则是：将第一个分数乘以第二个分数的倒数。
(a/b) ÷ (c/d) = (a/b) × (d/c) = (a×d) / (b×c)
× 4/1 = 4/2 = 2. This makes sense because you are finding how many quarters are in a half — there are 2. Another example: ²⁄₇ ÷ ⅗ = (2/7) × (5/3) = 10/21.
例如，½ ÷ ¼ = ½ × 4/1 = 4/2 = 2。这很合理，因为你在求一个一半里有多少个四分之一——有 2 个。另一个示例：²⁄₇ ÷ ⅗ = (2/7) × (5/3) = 10/21。
p, change, flip. Keep the first fraction, change the ÷ to ×, and flip the second fraction.
千万不要不使用倒数就直接去除分数；一个常见的错误是分别把分子相除、分母相除，这是行不通的。记住：保留、变号、翻转。保留第一个分数，将除号变为乘号，再把第二个分数翻转。
5. Dividing Mixed Numbers | 带分数除法
Just as with multiplication, the best way to handle division of mixed numbers is to convert them to improper fractions first. Then apply the keep-change-flip method.
和乘法一样，处理带分数除法的最佳方法是先把它们转换为假分数，然后应用“保留-变号-翻转”的方法。
numerator and denominator by 5: 40/15 = 8/3 = 2 ⅔. So the answer as a mixed number is 2⅔.
例如，3⅓ ÷ 1¼ 转换为 (10/3) ÷ (5/4)。然后 (10/3) × (4/5) = (10×4)/(3×5) = 40/15。再通过将分子分母同时除以 5 来化简：40/15 = 8/3 = 2 ⅔。因此，作为带分数的答案是 2⅔。
In some questions, you might be asked to leave the answer as an improper fraction or a mixed number. Always read the question carefully. If it doesn't specify, either form is usually acceptable as long as it is fully simplified.
在某些题目中，可能会要求你将答案保留为假分数或带分数。要仔细读题。如果没有明确说明，只要已经完全化简，两种形式通常都可以接受。
6. Cross-Cancellation (Simplifying First) | 交叉约分（先化简）
To save time and avoid working with large numbers, you can simplify fractions before multiplying. This is called cross-cancellation. You look for a common factor between any numerator and any denominator across the multiplication, then divide them by that factor.
为了节省时间并避免处理大数字，你可以在乘法前先化简分数。这叫做交叉约分。你要在乘法式子里寻找任何一个分子与任何一个分母之间的公因数，然后用那个因数去除它们。
ithout cross-cancellation, you would multiply to get 12/90 and then simplify, which takes longer.
例如，在 (4/9) × (3/10) 中，分子 4 和分母 10 有公因数 2，把 4 化简为 2，10 化简为 5。同时，分子 3 和分母 9 有公因数 3，化简为 1 和 3。于是乘积变为 (2/3) × (1/5) = 2/15。如果不进行交叉约分，你会先乘得 12/90 然后再化简，花费更多时间。
Cross-cancellation also works in division after you have flipped the second fraction. Perform the flip first, then look for factors to cancel before multiplying. This technique is very useful in exam conditions when time is limited.
在除法中，你翻转了第二个分数后，交叉约分同样适用。先进行翻转，然后在乘法之前寻找可以约分的公因数。在考试时间有限的情况下，这个技巧非常有用。
7. Word Problems Involving Multiplication | 乘法应用题
Word problems often ask you to find a fraction of a quantity, which is simply multiplication by that fraction. The phrase 'of' in mathematics usually means multiply.
应用题经常会让你求一个数量的几分之几，这其实就是乘以那个分数。在数学中，“of”这个词通常意味着乘。
e requires ¾ cup of sugar. If you want to make ½ of the recipe, how much sugar do you need? alising the problem can help you understand why the product is smaller: you are taking half of three-quarters.
示例：一个蛋糕食谱需要 ¾ 杯糖。如果你只想做原食谱一半的量，你需要多少糖？这里，你要计算 ¾ × ½ = 3/8 杯。在脑海中形象化这个问题可以帮助你理解为什么乘积变小了：你是在取四分之三的一半。
Another common type: A class has 30 students. ²⁄₅ of them are girls. How many are girls? 0)/5 = 60/5 = 12. To multiply a fraction by a whole number, write the whole number over 1.
另一种常见类型：一个班级有 30 名学生，其中 ²⁄₅ 是女生。有多少名女生？计算 ²⁄₅ × 30 = (2×30)/5 = 60/5 = 12。将分数乘以整数时，把整数写成分母为 1 的形式。
8. Word Problems Involving Division | 除法应用题
u have 6 metres of ribbon and each piece needs to be ¾ m long, how many pieces can you cut? This is 6 ÷ ¾ = 6 × 4/3 = 24/3 = 8 pieces.
涉及分数的除法应用题通常包括分配或求出可以分成多少组某个大小。例如，如果你有 6 米丝带，每段需要 ¾ 米长，你能剪出多少段？计算 6 ÷ ¾ = 6 × 4/3 = 24/3 = 8 段。
: A pizza is cut into slices that are each ⅛ of the whole pizza. How many slices are there? otice how dividing by a fraction smaller than 1 gives a result larger than the original number.
另一个示例：一个比萨被切成每片占整个比萨 ⅛ 的大小，一共有多少片？计算 1 ÷ ⅛ = 1 × 8/1 = 8 片。注意，除以一个小于 1 的分数，得到的结果会比原来的数大。
When a problem asks something like, 'How many ⅔-litre jugs can be filled from a 5-litre container?' vide 5 by ⅔: 5 ÷ ⅔ = 5 × 3/2 = 15/2 = 7½ jugs. This means 7 full jugs and a half-full one.
当题目问道“一个 5 升的容器可以装满多少个 ⅔ 升的水壶？”时，你将 5 除以 ⅔：5 ÷ ⅔ = 5 × 3/2 = 15/2 = 7½ 壶。这表示可以装满 7 壶，外加半壶。
9. Common Errors and How to Avoid Them | 常见错误及如何避免
One of the most common mistakes is applying the addition/subtraction rule to multiplication and division. Some students incorrectly look for a common denominator before multiplying, which is unnecessary and leads to confusion.
最常见的错误之一是将加减法的规则应用于乘除法。有些学生会在乘法之前错误地寻找公分母，这完全没必要，还会造成混乱。
by multiplying the whole and fraction parts separately gives a wrong result. Always convert first.
另一个常见错误是遗忘在乘除之前将带分数转换为假分数。直接把 1⅓ × 2¼ 的整数部分与分数部分分别相乘，会得到错误的结果。务必先转换。
Students also sometimes forget to simplify the final answer. In many tests, leaving a fraction unsimplified may lose marks. Make it a habit to check if both numerator and denominator can be divided by the same number.
学生有时也会忘记化简最终答案。在许多测试中，留下未化简的分数可能会被扣分。养成检查分子和分母是否能同时被某个数整除的习惯。
A subtle error is misapplying the reciprocal: dividing the first fraction by the second but flipping the first instead of the second. Memorise 'flip the second' or 'change the divisor into its reciprocal'. Writing down each step helps to avoid this.
还有一个不易察觉的错误是误用倒数：用第一个分数除以第二个分数时，翻错对象，把第一个翻转了。要记住“翻转第二个”或“将除数变为其倒数”。把每一步写下来有助于避免这一错误。
10. Practice Questions and Answers | 练习题与答案
Let's consolidate your skills with a few practice questions. Try them yourself before checking the solutions.
让我们用几道练习题来巩固你的技能。先自己尝试，再核对答案。
Question 1: Multiply ²⁄₅ × ⁴⁄₇.
Question 2: Find 1¾ × 2²⁄₃.
Question 3: Divide ⁵⁄₈ ÷ ¼.
Question 4: Work out 3 ÷ ⅖.
Question 5: A bottle contains 1½ litres of juice. How many ¾-litre glasses can be filled?
问题 1：计算 ²⁄₅ × ⁴⁄₇。
问题 2：计算 1¾ × 2²⁄₃。
问题 3：计算 ⁵⁄₈ ÷ ¼。
问题 4：计算 3 ÷ ⅖。
问题 5：一个瓶子里有 1½ 升果汁，可以倒满多少个 ¾ 升的杯子？
Solutions:
1. (2×4)/(5×7) = 8/35 (already simplified).
2. Convert: 7/4 × 8/3 = (7×8)/(4×3) = 56/12 = 14/3 = 4⅔.
3. ⁵⁄₈ ÷ ¼ = ⁵⁄₈ × 4/1 = (5×4)/(8×1) = 20/8 = 5/2 = 2½.
4. 3 ÷ ⅖ = 3 × 5/2 = 15/2 = 7½.
= 3/2 ÷ 3/4 = 3/2 × 4/3 = 12/6 = 2 glasses. This makes sense: ½ litre is half of a ¾-litre?, Actually check: 1.5 ÷ 0.75 = 2, correct.
解答：
1. (2×4)/(5×7) = 8/35（已化简）。
2. 转换：7/4 × 8/3 = (7×8)/(4×3) = 56/12 = 14/3 = 4⅔。
3. ⁵⁄₈ ÷ ¼ = ⁵⁄₈ × 4/1 = (5×4)/(8×1) = 20/8 = 5/2 = 2½。
4. 3 ÷ ⅖ = 3 × 5/2 = 15/2 = 7½。
5. 1½ ÷ ¾ = 3/2 ÷ 3/4 = 3/2 × 4/3 = 12/6 = 2 杯。这是合理的：1½ 是 ¾ 的两倍吗？计算：1.5 ÷ 0.75 = 2，正确。
11. Multiplying and Dividing with Whole Numbers | 整数与分数乘除
omes 5/1. Then you apply the fraction rules. This means 5 × ⅔ = (5/1) × (2/3) = 10/3 = 3⅓.
从练习中可以看到，任何整数都可以写成分母为 1 的分数。因此，5 可以写成 5/1，然后应用分数规则。这意味着 5 × ⅔ = (5/1) × (2/3) = 10/3 = 3⅓。
Viewing whole numbers this way makes them just another fraction and removes any confusion.
除法：4 ÷ ²⁄₃ = (4/1) ÷ (2/3) = (4/1) × (3/2) = 12/2 = 6。以这种方式看待整数，它们就只是另一种分数，这消除了任何困惑。
12. Linking Fractions, Decimals and Percentages | 联系分数、小数和百分数
ivision also strengthens your work with decimals and percentages. For instance, finding 25% × 80 = 20. Dividing by a fraction relates to decimal division: 3 ÷ 0.2 is the same as 3 ÷ ⅕ ing these connections can help you check your answers and see the bigger picture in maths.
理解分数的乘除也能强化你处理小数和百分数的能力。例如，求 80 的 25% 等同于 ¼ × 80 = 20。除以一个分数则与小数除法相关：3 ÷ 0.2 等同于 3 ÷ ⅕ = 15。意识到这些联系可以帮助你检查答案，并构建更宏大的数学图景。
When multiplying decimals, you might sometimes find it easier to convert them to fractions, perform the fraction multiplication, and convert back. The same applies to percentages. This flexibility is a key problem-solving skill.
在乘法小数时，有时将它们转换成分数、进行分数乘法，然后再转换回去会更容易。百分数也是如此。这种灵活性是一项关键的解题技能。
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