Inverse Proportion | 反比例

📚 Inverse Proportion | 反比例

In IGCSE Edexcel Mathematics, understanding the relationship between two variables is a core skill. While direct proportion implies that variables move in the same direction, inverse proportion describes a situation where one quantity increases while the other decreases at the same rate. This article will guide you through the principles, equations, and graphs of inverse proportion, ensuring you are fully prepared for your exams.

在 IGCSE Edexcel 数学中，理解两个变量之间的关系是一项核心技能。正比例意味着变量朝同一方向变化，而反比例则描述了一个量增加而另一个量以相同速率减少的情况。本文将指导你理解反比例的原理、方程和图像，确保你为考试做好充分准备。

1. What is Inverse Proportion? | 什么是反比例？

Inverse proportion describes the relationship between two variables whose product is constant. If one variable doubles, the other halves, meaning that as one quantity increases, the other decreases proportionally.

反比例描述的是两个变量乘积为常数的关系。如果一个变量加倍，另一个变量减半，这意味着当一个量增大时，另一个量按比例减小。

Formally, we write y ∝ 1 / x. This is read as “y is inversely proportional to x”.

形式上，我们写作 y ∝ 1 / x，读作 “y 与 x 成反比”。

This means that as x increases, y decreases such that the product x × y remains the same.

这意味着当 x 增大时，y 减小，但 x 与 y 的乘积保持不变。

2. The Equation and Constant k | 方程与常数 k

The general equation for inverse proportion is y = k / x, where k is the constant of proportionality. This constant k remains the same for every pair of values (x, y) that satisfy the relationship.

反比例的一般方程是 y = k / x，其中 k 是比例常数。对于满足该关系的每一对值 (x, y)，这个常数 k 都保持不变。

Multiplying both sides by x, we get xy = k. This form is often the most useful for solving exam problems because it allows you to find k with a single multiplication.

将等式两边乘以 x，我们得到 xy = k。这种形式在解决考试问题时通常最有用，因为它允许你通过一次乘法就找到 k。

y = k / x ⇔ xy = k ⇔ x = k / y

3. Finding k in Exam Questions | 在考试题中求 k

Typically, Edexcel questions give you a pair of values (x, y) that satisfy the relationship. Substitute them into xy = k to find the numerical value of k.

通常，Edexcel 题目会给你一组满足关系的值 (x, y)。将它们代入 xy = k 即可求出 k 的数值。

For example, if y = 4 when x = 3, then k = 3 × 4 = 12. This means the complete equation of the relationship is y = 12 / x.

例如，当 x = 3 时 y = 4，那么 k = 3 × 4 = 12。这意味着该关系的完整方程为 y = 12 / x。

Then, if the question asks for the value of y when x = 2, simply substitute: y = 12 / 2 = 6. Always calculate k before attempting to find any missing values.

然后，如果题目要求当 x = 2 时 y 的值，直接代入即可：y = 12 / 2 = 6。在尝试求任何缺失值之前，务必先计算出 k。

4. The Graph of y = k/x | y = k/x 的图像

The graph of an inverse proportion is a curve called a hyperbola (or reciprocal graph). Unlike a straight line, it has two distinct branches.

反比例的图像是一条称为双曲线（或倒数曲线）的曲线，它有两条不同的分支。与直线不同，它有两条明显的分支。

Crucially, the curve never touches the x-axis or the y-axis. As x approaches infinity, y approaches zero. As x approaches zero, y approaches infinity. These axes act as asymptotes.

关键在于，曲线永远不与 x 轴或 y 轴相交。当 x 趋近于无穷大时，y 趋近于零；当 x 趋近于零时，y 趋近于无穷大。坐标轴充当渐近线。

For k > 0, the branches are in the first (top right) and third (bottom left) quadrants. For k < 0, the branches are in the second (top left) and fourth (bottom right) quadrants.

当 k > 0 时，两条分支位于第一（右上）和第三（左下）象限。当 k < 0 时，两条分支位于第二（左上）和第四（右下）象限。

5. Identifying Inverse Proportion from Tables | 从表格中识别反比例

Edexcel often provides a table of values and asks whether they represent inverse proportion. To check, calculate the product of each corresponding pair (x × y). If all the products are equal, the relationship is inverse proportion.

Edexcel 通常会提供一个数值表，询问它们是否表示反比例关系。要进行检查，计算每对对应值（x × y）的乘积。如果所有乘积都相等，则该关系为反比例。

x	1	2	4	5
y	20	10	5	4

Here, 1 × 20 = 20, 2 × 10 = 20, 4 × 5 = 20, 5 × 4 = 20. Since the products are all equal (constant), y is inversely proportional to x.

这里，1 × 20 = 20, 2 × 10 = 20, 4 × 5 = 20, 5 × 4 = 20。由于乘积都相等（为常数），y 与 x 成反比。

6. Solving Word Problems | 解决应用题

Many Edexcel IGCSE questions contextualize inverse proportion. For instance, “It takes 5 workers 12 days to build a wall. How long would it take 10 workers?”

许多 Edexcel IGCSE 问题将反比例置于实际情境中。例如，”5 名工人需要 12 天才能砌好一堵墙。10 名工人需要多少天？”

Let the number of workers be w and the time be t. These are inversely proportional: w × t = k. Substituting the initial conditions: 5 × 12 = 60. Thus, the full equation is w × t = 60. When w = 10, we have 10 × t = 60, so t = 60 / 10 = 6 days.

设工人数为 w，时间为 t。它们成反比：w × t = k。代入初始条件：5 × 12 = 60。因此完整方程为 w × t = 60。当 w = 10 时，我们得到 10 × t = 60，所以 t = 60 / 10 = 6 天。

Always state the quantities you are defining clearly. This will help you set up the correct equation and earn method marks in the exam.

在解题时，始终清晰地定义你所设的量。这将帮助你建立正确的方程，并在考试中获得方法分。

7. Inverse Proportion in Formulas | 公式中的反比例

You might be given a formula, such as v = k / (d²). Here, v is inversely proportional to the square of d. This is known as the inverse square law, which frequently appears in higher-tier IGCSE Maths and Physics.

你可能会看到一个公式，例如 v = k / (d²)。这里，v 与 d 的平方成反比。这被称为平方反比定律，在 IGCSE 高等数学和物理中经常出现。

The method is the same: substitute the given pair of values to find k first. If v = 2 when d = 3, then k = v × d² = 2 × 9 = 18. The full formula is v = 18 / d².

解题方法相同：先代入给定的一对值求出 k。如果当 d = 3 时 v = 2，那么 k = v × d² = 2 × 9 = 18。完整公式为 v = 18 / d²。

Watch out for the square or cube on the denominator. You must square or cube x before multiplying by y to find k.

注意分母上的平方或立方。在乘以 y 求 k 之前，你必须先对 x 进行平方或立方。

8. Worked Exam Question | 考试例题详解

Question / 问题: y is inversely proportional to x. When x = 5, y = 8. Find the value of y when x = 2.

y 与 x 成反比。当 x = 5 时，y = 8。求当 x = 2 时 y 的值。

Solution / 解题过程:

1. Write the general formula: y = k / x. 2. Substitute x = 5 and y = 8 into xy = k: k = 5 × 8 = 40. 3. Therefore, the specific equation is y = 40 / x. 4. Substitute x = 2: y = 40 / 2 = 20.

1. 写出一般公式：y = k / x。 2. 将 x = 5 和 y = 8 代入 xy = k：k = 5 × 8 = 40。 3. 因此，具体方程为 y = 40 / x。 4. 代入 x = 2：y = 40 / 2 = 20。

Final Answer / 最终答案: y = 20

9. Common Mistakes to Avoid | 避免常见错误

Mistake 1: Confusing direct and inverse proportion. In direct proportion, y / x = k. In inverse proportion, xy = k. Always analyze the relationship before writing the equation.
错误 1：混淆正比例和反比例。正比例中 y / x = k，反比例中 xy = k。在写出方程之前，务必先分析关系。
Mistake 2: Trying to solve without finding k first. Always find the constant of proportionality before substituting other values. Skipping this step will lead to an incorrect answer.
错误 2：不先求 k 就尝试求解。在代入其他值之前，务必先求出比例常数。跳过此步骤将导致错误答案。
Mistake 3: Forgetting to check if the relationship is indeed inverse (the product xy is constant) when given a table of values.
错误 3：当给定数值表时，忘记检查关系是否确实是反比例（乘积 xy 是否为常数）。
Mistake 4: When dealing with square or cube relationships, forgetting to square or cube x. For example, if y ∝ 1/x², then k must be calculated as y × x².
错误 4：在处理平方或立方关系时，忘记对 x 进行平方或立方。例如，如果 y ∝ 1/x²，那么 k 必须通过 y × x² 来计算。

10. Key Points for Edexcel IGCSE | Edexcel IGCSE 关键要点

Remember that Edexcel IGCSE often uses inverse proportionality in multi-step problems, such as those involving density, speed, and rates of pay. You may need to combine it with direct proportion or other algebraic skills.

请记住，Edexcel IGCSE 经常在多步骤问题中使用反比例，例如涉及密度、速度和工资率的问题。你可能需要将其与正比例或其他代数技能结合使用。

Always show your working clearly. If you define k explicitly, you will likely earn method marks even if your final arithmetic is incorrect.

始终清晰地展示你的解题步骤。如果你明确写出了 k，即使最终计算错误，你也很可能获得方法分。

Pay attention to the wording: “inversely proportional to the square of x” means y ∝ 1/x², while “inversely proportional to x” means y ∝ 1/x.

注意措辞：”与 x 的平方成反比” 意味着 y ∝ 1/x²，而 “与 x 成反比” 意味着 y ∝ 1/x。

11. Quick Practice Questions | 快速练习题

Q1: y is inversely proportional to x². When x = 2, y = 5. Find y when x = 5.

练习 1：y 与 x² 成反比。当 x = 2 时，y = 5。求当 x = 5 时 y 的值。

Solution 1: k = 2² × 5 = 20. Therefore, y = 20 / x². When x = 5, y = 20 / 5² = 20 / 25 = 0.8.

解答 1：k = 2² × 5 = 20。因此 y = 20 / x²。当 x = 5 时，y = 20 / 5² = 20 / 25 = 0.8。

Q2: It takes 3 machines 8 hours to fill a warehouse. How long will it take 6 machines?

练习 2：3 台机器需要 8 小时才能装满一个仓库。6 台机器需要多长时间？

Solution 2: k = 3 × 8 = 24. Time = 24 / 6 = 4 hours.

解答 2：k = 3 × 8 = 24。时间 = 24 / 6 = 4 小时。

12. Conclusion & Revision Checklist | 结论与复习清单

Master the algebraic form, the table method, and the graph. Understanding how these three representations connect is the key to succeeding with inverse proportion in your Edexcel IGCSE Mathematics exam.

掌握代数形式、表格方法和图像。理解这三种表示方式之间的联系，是在 Edexcel IGCSE 数学考试中解决反比例问题的关键。

I know that inverse proportion means xy = k. (我知道反比例意味着 xy = k。)
I can find the constant k given a pair of values. (给定一对值，我能求出常数 k。)
I can recognize the reciprocal graph and its asymptotes. (我能识别反比例图像及其渐近线。)
I can solve word problems involving inverse proportion. (我能解决涉及反比例的应用题。)
I can distinguish between inverse proportion and inverse square proportion. (我能区分反比例和平方反比。)

Review these concepts with TutorHao’s revision notes and past papers to consolidate your understanding.

使用 TutorHao 的复习笔记和历年试卷复习这些概念，以巩固你的理解。

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Inverse Proportion | 反比例