📚 Multiplying and Dividing Fractions | 分数的乘法和除法
Multiplying and dividing fractions is a fundamental skill in KS3 mathematics. It builds on your understanding of fractions and prepares you for more advanced topics such as algebra, ratio, and proportional reasoning. In this article, we will break down the rules, explore step-by-step methods, and tackle common problem types with plenty of examples. The key is to master the concepts of multiplying numerators and denominators, using reciprocals, and simplifying before multiplying. With practice, you will find these operations straightforward and even enjoyable!
分数的乘法和除法是KS3数学中的一项基本技能。它建立在你对分数的理解之上，并为代数、比率和比例推理等更高级的主题做好准备。在本文中，我们将分解规则、探索逐步方法，并通过大量示例解决常见问题类型。关键在于掌握分子与分母相乘、使用倒数以及在乘法前进行约分。通过练习，你会觉得这些运算既简单又有趣！
1. Understanding Multiplication of Fractions | 理解分数的乘法
Multiplication of fractions can be thought of as finding a part of another part. For example, '¹/₂ of ³/₄' means we want half of three-quarters. If you imagine a shape divided into quarters and shade three of them, taking half of that shaded area gives you three-eighths of the whole shape. This visual helps make sense of the rule: multiply the numerators together and multiply the denominators together.
分数的乘法可以理解为求一个部分的又一部分。例如，“¹/₂ 的 ³/₄”意思是我们想要四分之三的一半。想象一个图形被分成四等份，并涂色其中的三份，再取这个涂色部分的一半，你就会得到整个图形的八分之三。这种可视化有助于理解规则：分子与分子相乘，分母与分母相乘。
en you multiply fractions, the pieces become smaller, which is why we multiply denominators to get a larger denominator, often resulting in a smaller fraction.
用符号表示，a/b × c/d = (a × c)/(b × d)。用我们的例子：¹/₂ × ³/₄ = (1×3)/(2×4) = ³/₈。这里有一个清晰的逻辑：当你乘以分数时，每一份变得更小，这就是为什么分母相乘会让分母变大，通常得到更小的分数。
2. Multiplying Fractions by Whole Numbers | 分数乘以整数
When multiplying a fraction by a whole number, remember that any whole number can be written as a fraction with denominator 1. For example, 4 is the same as ⁴/₁. Then you simply follow the rule for multiplying two fractions.
当分数乘以整数时，请记住任何整数都可以写成分母为1的分数。例如，4 等于 ⁴/₁。然后你只需按照两个分数相乘的规则进行计算。
leave it as it is or convert it to a mixed number. Another example: 5 × ³/₈ = ¹⁵/₈ = 1⁷/₈.
示例：3 × ²/₇ = ³/₁ × ²/₇ = ⁶/₇。如果结果是假分数，你可以保留它或将其转换为带分数。另一个示例：5 × ³/₈ = ¹⁵/₈ = 1⁷/₈。
on: ²/₉ × 6 = ²/₉ × ⁶/₁ = ¹²/₉ = ⁴/₃ = 1¹/₃. Always simplify the final answer if possible.
当分数后面跟着整数时，这个方法也适用：²/₉ × 6 = ²/₉ × ⁶/₁ = ¹²/₉ = ⁴/₃ = 1¹/₃。如果可能，始终简化最终答案。
3. Multiplying Two Proper Fractions | 两个真分数相乘
The procedure for multiplying two proper fractions is straightforward: multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator. Then simplify the fraction if common factors exist.
两个真分数相乘的步骤很简单：分子相乘得到新分子，分母相乘得到新分母。然后如果存在公因数就约分。
5. So the answer is ⁸/₁₅. This fraction is already in its simplest form because 8 and 15 share no common factors other than 1.
考虑 ²/₃ × ⁴/₅。分子相乘：2×4=8。分母相乘：3×5=15。所以答案是 ⁸/₁₅。这个分数已经是最简形式，因为8和15除了1之外没有公因数。
= ¹/₄. But it is more efficient to cancel before multiplying, which we will explore next.
另一个例子：³/₈ × ²/₃。这里我们可以在乘法后化简：⁶/₂₄ = ¹/₄。但在乘法前约分更为高效，我们接下来会探讨。
4. Simplifying Before Multiplying (Cross-Cancelling) | 乘法前的约分（交叉约分）
Cross-cancelling, also known as simplifying before multiplying, makes calculations much easier and reduces the risk of handling large numbers. The idea is to look for any common factor between a numerator and a denominator across the multiplication – not necessarily from the same fraction – and divide them out.
交叉约分，也称为乘法前化简，使计算变得更加容易，并降低了处理大数字的风险。其思路是在整个乘法中寻找任何分子与分母之间的公因数——不一定是来自同一个分数——并约去它们。
÷ 7 = 2. Now the problem becomes ³/₁ × ²/₉ = ⁶/₉ = ²/₃. No large numbers were multiplied.
例如，³/₇ × ¹⁴/₉。注意到7和14有公因数7。约分：7÷7=1，14÷7=2。现在问题变成 ³/₁ × ²/₉ = ⁶/₉ = ²/₃。没有涉及大数字的乘法。
11, giving ²/₁ × ²/₃ = ⁴/₃. This step-by-step cross-cancelling prevents messy arithmetic.
你可以多次对角约分：⁶/₁₁ × ²²/₉。将6和9用3约分，得到 ²/₁₁ × ²²/₃。再将11和22用11约分，得到 ²/₁ × ²/₃ = ⁴/₃。这种循序渐进的交叉约分避免了繁琐的算术。
5. Understanding Division of Fractions | 理解分数的除法
Division of fractions asks: how many of the divisor fit into the dividend? For example, 3 ÷ ¹/₂ means 'how many halves are there in 3?' ¹/₂ = 6. This interpretation leads naturally to the rule of multiplying by the reciprocal.
分数的除法是在问：被除数里有多少个除数？例如，3 ÷ ¹/₂ 表示“3里面有多少个一半？”。因为每个1里面有2个一半，所以3里面一定有6个一半。这就是为什么 3 ÷ ¹/₂ = 6。这种解释自然地引出了乘以倒数的规则。
But what about dividing a fraction by another fraction, e.g., ¹/₄ ÷ ¹/₈? r equals two eighths, the answer is 2. Notice that ¹/₄ × ⁸/₁ = ²/₁ = 2. The pattern holds.
那如果是分数除以另一个分数呢，比如 ¹/₄ ÷ ¹/₈？这是在问一个四分之一里有多少个八分之一。因为一个四分之一等于两个八分之一，答案是2。注意 ¹/₄ × ⁸/₁ = ²/₁ = 2。这一规律成立。
6. Reciprocal of a Fraction | 分数的倒数
e, the reciprocal of ²/₅ is ⁵/₂. The reciprocal of a whole number n is ¹/ₙ, since n = n/1.
分数的倒数是通过交换其分子和分母得到的。对于分数 a/b，它的倒数是 b/a。例如，²/₅ 的倒数是 ⁵/₂。整数 n 的倒数是 ¹/ₙ，因为 n = n/1。
For instance, ²/₃ × ³/₂ = ⁶/₆ = 1. This property is the basis for the division algorithm.
需要注意的是，一个分数乘以它的倒数总是得到1。例如，²/₃ × ³/₂ = ⁶/₆ = 1。这一性质是除法算法的基础。
Also, the reciprocal of a mixed number requires converting it to an improper fraction first. The reciprocal of 1³/₄ is the reciprocal of ⁷/₄, which is ⁴/₇.
此外，求带分数的倒数需要先将其转换为假分数。1³/₄ 的倒数是 ⁷/₄ 的倒数，即 ⁴/₇。
7. Dividing Fractions by Whole Numbers | 分数除以整数
To divide a fraction by a whole number, rewrite the whole number as a fraction over 1, then multiply by its reciprocal. This turns the division into a multiplication problem.
要将分数除以整数，先将整数写成分母为1的分数，然后乘以它的倒数。这便将除法转化为乘法问题。
of ⁴/₁ is ¹/₄. So ³/₅ ÷ 4 = ³/₅ × ¹/₄ = ³/₂₀. Another example: ⁷/₈ ÷ 2 = ⁷/₈ × ¹/₂ = ⁷/₁₆.
示例：³/₅ ÷ 4。将4写为 ⁴/₁。⁴/₁ 的倒数是 ¹/₄。所以 ³/₅ ÷ 4 = ³/₅ × ¹/₄ = ³/₂₀。另一个示例：⁷/₈ ÷ 2 = ⁷/₈ × ¹/₂ = ⁷/₁₆。
This method is reliable and works for any whole number divisor. It also sets the stage for the general division of fractions, where the divisor is also a fraction.
这种方法可靠，适用于任何整数除数。它也为一般的分数除法奠定了基础，即除数也是分数的情况。
8. Dividing by a Fraction (Keep-Change-Flip) | 除以一个分数（保持-变号-翻转）
The most popular method for dividing by a fraction is called Keep-Change-Flip, or KCF. It stands for: Keep the first fraction exactly as it is; Change the division sign (÷) to a multiplication sign (×); Flip the second fraction (take its reciprocal).
除以一个分数最流行的方法叫做保持-变号-翻转（KCF）。它代表：保持第一个分数不变；将除号（÷）变为乘号（×）；翻转第二个分数（取其倒数）。
Let' we have ³/₄ × ⁵/₂ = ¹⁵/₈ = 1⁷/₈. This method eliminates confusion and is easy to remember.
我们将KCF应用于 ³/₄ ÷ ²/₅。保持 ³/₄ 不变。将 ÷ 变为 ×。翻转 ²/₅ 得到 ⁵/₂。现在我们有 ³/₄ × ⁵/₂ = ¹⁵/₈ = 1⁷/₈。这种方法消除了混淆，且容易记忆。
| Step | Action | Example: ⁵/₆ ÷ ²/₃ |
| Keep | First fraction unchanged | ⁵/₆ |
| Change | ÷ becomes × | × |
| Flip | Second fraction flipped | ³/₂ |
| Result | Multiply: ⁵/₆ × ³/₂ = ¹⁵/₁₂ = ⁵/₄ |
Using KCF consistently ensures accuracy. Remember to simplify your final answer whenever possible.
坚持使用KCF可以确保准确性。请记住尽可能简化最终答案。
9. Mixed Numbers and Improper Fractions | 带分数与假分数
Mixed numbers (like 1³/₅) combine a whole number and a fraction. Before multiplying or dividing, you must convert every mixed number into an improper fraction. To do this, multiply the whole number by the denominator and add the numerator; place this sum over the original denominator.
带分数（如 1³/₅）由一个整数和一个分数组合而成。在乘除之前，你必须将每个带分数转换为假分数。方法：用整数乘以分母再加上分子；将这个和放在原分母之上。
is crucial; otherwise, you might make mistakes like multiplying only the fractional part.
示例：转换 2²/₃。计算 2×3=6，加上2得到8。所以 2²/₃ = ⁸/₃。另一个：4¹/₆ = (4×6+1)/6 = ²⁵/₆。这一步至关重要；否则，你可能出错，比如只乘以了分数部分。
After performing the operation, you may convert an improper fraction back to a mixed number by dividing the numerator by the denominator. The quotient is the whole number, and the remainder becomes the new numerator.
完成运算后，你可以通过分子除以分母将假分数转换回带分数。商是整数部分，余数成为新的分子。
10. Multiplying and Dividing Mixed Numbers | 带分数的乘法和除法
are converted to improper fractions, the same multiplication and division rules apply. Let's look at a multiplication example: 2¹/₂ × 1³/₄.
一旦带分数被转换为假分数，就可以应用同样的乘法和除法规则。我们看一个乘法例子：2¹/₂ × 1³/₄。
ltiply: ⁵/₂ × ⁷/₄ = ³⁵/₈. Convert back to a mixed number: 35 ÷ 8 = 4 remainder 3, so 4³/₈.
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