Mastering Direct and Inverse Variation | 掌握正变分与反变分

📚 Mastering Direct and Inverse Variation | 掌握正变分与反变分

In IGCSE Mathematics, the topic of variation appears again and again in Paper 2 and Paper 4. You will be asked to translate a sentence into an equation, find the value of a constant, or use a graph to describe a relationship. This article will give you a complete and exam-focused guide to direct variation and inverse variation, with simple examples and common pitfalls clearly explained.

在 IGCSE 数学中，变分（variation）这一主题在 Paper 2 和 Paper 4 中反复出现。题目通常要求你将一句文字描述转化为方程、求常数 k 的值，或通过图像判断两个量之间的关系。本文将为你提供一份完整且紧扣考点的正变分与反变分指南，并通过简明例题和常见易错点帮助你稳步提分。

1. What Is Variation? | 什么是变分？

Variation describes how one quantity changes when another quantity changes. In IGCSE Mathematics, the word “variation” is used for relationships where two variables are connected by a constant multiplier or divisor. If the ratio between two variables stays the same, it is direct variation. If the product of two variables stays the same, it is inverse variation.

变分描述的是一个量随另一个量的变化而变化的关系。在 IGCSE 数学中，当两个变量之间通过一个固定的乘数或除数相联系时，我们就称它们之间存在变分关系。如果两个变量的比值保持不变，那就是正变分（直接变分）；如果两个变量的乘积保持不变，那就是反变分（反比变分）。

Direct variation: y increases when x increases.
正变分：x 增大时，y 也随之增大。
Inverse variation: y decreases when x increases.
反变分：x 增大时，y 反而减小。
The Greek letter ∝ means “varies directly as”.
希腊字母 ∝ 表示“与……成正比”。

2. Direct Variation: y ∝ x | 正变分：y ∝ x

If y is directly proportional to x, we write y ∝ x. This is the simplest form of direct variation. The definition is: y = kx, where k is a non-zero constant called the constant of proportionality.

如果 y 与 x 成正比，我们写作 y ∝ x。这是正变分中最基本的形式。其定义是：y = kx，其中 k 是一个非零常数，称为比例常数。

y = kx

For example, if y ∝ x and y = 10 when x = 2, then k = 10 ÷ 2 = 5. The equation is y = 5x. You can then use this equation to find y for any value of x.

例如，若 y ∝ x，且当 x = 2 时 y = 10，那么 k = 10 ÷ 2 = 5，方程为 y = 5x。之后你可以用这个方程求出任意 x 所对应的 y 值。

Another common phrase in exams is “directly proportional to the square of x”. This means y ∝ x² and y = kx².

考试中还常见“与 x 的平方成正比”的说法。这表示 y ∝ x²，即 y = kx²。

y = kx²

3. Finding the Constant k | 求比例常数 k

The constant k is the key to solving every variation problem. Once you know k, you can write the full formula. To find k, substitute one known pair of x and y values into the general form.

比例常数 k 是解决所有变分问题的关键。一旦求出 k，你就能写出完整的公式。求 k 的方法是将已知的一组 x 和 y 值代入一般形式。

Example: Given that y ∝ x³ and y = 24 when x = 2, find y when x = 3.

例题：已知 y ∝ x³，且当 x = 2 时 y = 24，求当 x = 3 时 y 的值。

Step 1: Write y = kx³.

第一步：写出 y = kx³。

Step 2: Substitute y = 24 and x = 2: 24 = k × 2³ = 8k. Therefore k = 3.

第二步：代入 y = 24 和 x = 2：24 = k × 2³ = 8k，所以 k = 3。

Step 3: Use y = 3x³. For x = 3, y = 3 × 3³ = 3 × 27 = 81.

第三步：使用 y = 3x³。当 x = 3 时，y = 3 × 3³ = 3 × 27 = 81。

Final answer: y = 81

4. Inverse Variation: y ∝ 1/x | 反变分：y ∝ 1/x

If y is inversely proportional to x, we write y ∝ 1/x. The equation becomes y = k/x. Notice that as x gets larger, y gets smaller, but the product xy stays constant: xy = k.

如果 y 与 x 成反比，我们写作 y ∝ 1/x，对应方程为 y = k/x。注意：当 x 增大时，y 减小，但两者的乘积保持不变：xy = k。

y = k/x or xy = k

Example: y is inversely proportional to x, and y = 8 when x = 3. Find y when x = 6.

例题：y 与 x 成反比，且当 x = 3 时 y = 8，求当 x = 6 时 y 的值。

Using xy = k: 3 × 8 = 24, so k = 24. When x = 6, y = 24 ÷ 6 = 4.

利用 xy = k：3 × 8 = 24，所以 k = 24。当 x = 6 时，y = 24 ÷ 6 = 4。

For inverse square proportion, we write y ∝ 1/x² and y = k/x². This often appears in physics-style IGCSE questions, such as gravity or light intensity.

对于反平方比例，我们写作 y ∝ 1/x²，即 y = k/x²。这类题目常出现在带有物理背景的 IGCSE 问题中，例如万有引力或光照强度。

5. Graphs of Direct and Inverse Variation | 正变分与反变分的图像

You should recognise a direct variation graph without calculation. The graph of y = kx is a straight line passing through the origin. If y = kx², the graph is a parabola passing through the origin, opening upward for k > 0.

你应该能够不经过计算就识别正变分的图像。y = kx 的图像是一条经过原点的直线。若 y = kx²，则图像是一条经过原点的抛物线，当 k > 0 时开口向上。

The graph of inverse variation y = k/x is a curve called a rectangular hyperbola. It has two branches, one in the first quadrant and one in the third quadrant (for positive k). The curve approaches, but never touches, the x-axis and y-axis.

反变分 y = k/x 的图像是一条称为“直角双曲线”的曲线。当 k > 0 时，它有两条分支，一条位于第一象限，另一条位于第三象限。曲线无限接近 x 轴和 y 轴，但永远不会与它们相交。

Relationship	Equation	Shape of graph
Direct	y = kx	Straight line through origin
Direct square	y = kx²	Parabola through origin
Inverse	y = k/x	Hyperbola with two branches
Inverse square	y = k/x²	Decaying curve, all positive

6. Turning Sentences into Equations | 将文字描述转化为方程

Exam questions rarely give you the equation directly. They give you a sentence such as “y is directly proportional to the cube root of x”. You must translate this into a mathematical equation.

考试题通常不会直接给出方程，而会给出类似“y 与 x 的立方根成正比”这样的句子。你必须将其转化为数学方程。

“y is directly proportional to x” → y = kx
「y 与 x 成正比」 → y = kx
“y varies as the square of x” → y = kx²
「y 随 x 的平方而变化」 → y = kx²
“y is inversely proportional to x” → y = k/x
「y 与 x 成反比」 → y = k/x
“y varies inversely as the square root of x” → y = k/√x
「y 与 x 的平方根成反比」 → y = k/√x

Always define k first in your working. If the problem gives you a pair of values, use them immediately to find k. If no pair is given, the answer will be a symbolic expression.

做题时一定要先设出常数 k。如果题目给你了一组对应值，立刻代入求出 k。如果没有给对应值，那么答案通常是一个含有 k 的表达式。

7. Fractional and Root Powers | 分数次幂与根号

IGCSE questions sometimes use fractional powers. You must be comfortable with square roots, cube roots, and negative powers. For example, y ∝ √x means y = k√x, and y ∝ 1/√x means y = k/√x.

IGCSE 题目有时会使用分数幂。你必须熟悉平方根、立方根和负指数。例如，y ∝ √x 表示 y = k√x，而 y ∝ 1/√x 表示 y = k/√x。

y = k√x or y = kx^(1/2)

Example: y is inversely proportional to the square root of x, and y = 5 when x = 16. Find y when x = 4.

例题：y 与 x 的平方根成反比，且当 x = 16 时 y = 5，求当 x = 4 时 y 的值。

Write y = k/√x. Substitute 5 = k/√16 = k/4, so k = 20. Then y = 20/√4 = 20/2 = 10.

先写 y = k/√x。代入 5 = k/√16 = k/4，得 k = 20。因此 y = 20/√4 = 20/2 = 10。

When dealing with cube roots, remember that √[3]x can be typed as x^(1/3). The same substitution method always works.

处理立方根时，请记得 √[3]x 可写作 x^(1/3)。同样的代入法始终适用。

8. Two-Step Proportion Problems | 两步比例应用题

Some exam questions first ask you to find k, and then ask for a different output. To stay accurate, keep k as a fraction if needed, and do not round too early. Only round the final answer to the required level.

有些考试题会先让你求 k，然后再求另一个输出值。为保证准确，如果需要，k 可以保留为分数，不要过早四舍五入。只有在最后答案时才按题目要求的精度进行舍入。

Example: The time T taken for a journey is inversely proportional to the speed v. T = 3 hours when v = 60 km/h. Find T when v = 80 km/h.

例题：完成一段旅程所需的时间 T 与速度 v 成反比。当 v = 60 km/h 时，T = 3 小时。求当 v = 80 km/h 时 T 的值。

T = k/v. From 3 = k/60 we get k = 180. Now T = 180/80 = 2.25 hours, which is 2 hours 15 minutes.

T = k/v。由 3 = k/60 得 k = 180。于是 T = 180/80 = 2.25 小时，即 2 小时 15 分钟。

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

Many students lose marks in variation questions because of careless mistakes. The most common error is writing y = x/k for inverse proportional. Another common error is forgetting that the graph of direct proportion must pass through the origin.

许多学生在变分题中因为粗心丢分。最常见的错误是把反比关系写成 y = x/k。另一个常见错误是忘记正比关系的图像必须经过原点。

Mistake 1: Confusing direct and inverse. Check: does y increase or decrease as x increases?
错误一：混淆正变分与反变分。检查方法：当 x 增大时，y 是增大还是减小？
Mistake 2: Forgetting the square when the question says “square”. Write x², not x.
错误二：题目说“平方”时漏写平方。应写 x²，而不是 x。
Mistake 3: Using y = kx instead of y = k/x in direct and inverse problems.
错误三：在反变分问题中错误使用 y = kx，而不是 y = k/x。
Mistake 4: Failing to show the substitution step in working. In IGCSE, method marks are important.
错误四：没有在解答过程中写出代入步骤。在 IGCSE 中，方法分非常重要。

10. Exam-Style Questions and Strategies | 考试题型与解题策略

In the exam, variation questions may appear as short-answer questions, structured questions, or as part of a graph interpretation task. You should always begin by writing the general equation, then substitute, then solve.

在考试中，变分题可能以简答题、结构化问题或图像理解题的形式出现。你应当始终先写出一般方程，再代入，再求解。

Example exam question: y is directly proportional to x². When x = 4, y = 24. Calculate y when x = 7.

考试例题：y 与 x² 成正比。当 x = 4 时，y = 24。求当 x = 7 时 y 的值。

Solution: y = kx². 24 = k × 16, so k = 24/16 = 3/2. Then y = (3/2) × 49 = 73.5.

解答：y = kx²。24 = k × 16，所以 k = 24/16 = 3/2。因此 y = (3/2) × 49 = 73.5。

For higher marks, you may also be asked to explain what happens to y when x doubles. For direct proportion, y doubles. For inverse proportion, y halves. For y = kx², y becomes four times larger.

对于更高分值的问题，题目可能要求你解释当 x 变为原来的两倍时 y 如何变化。对于正变分，y 变为原来的两倍；对于反变分，y 变为原来的一半；对于 y = kx²，y 变为原来的四倍。

11. Mixed Practice Problems | 综合练习

Try these questions yourself, then check your answers against the solutions below.

请尝试完成以下练习，然后对照下方答案检查。

Question	Answer
y ∝ x, y = 14 when x = 2. Find y when x = 5.	k = 7, y = 35
y ∝ 1/x, y = 6 when x = 8. Find y when x = 12.	k = 48, y = 4
y ∝ x³, y = 5 when x = 2. Find y when x = 4.	k = 5/8, y = 40
y ∝ √x, y = 10 when x = 25. Find x when y = 4.	k = 2, x = 4

After finishing these problems, check your working carefully. Look at each equation and ask yourself: “Is this the correct form of variation?”

完成这些练习后，请仔细检查你的解题过程。观察每一个方程，并问自己：“这个变分形式写对了吗？”

12. Final Summary and Key Takeaways | 总结与核心要点

Direct and inverse variation is a small but highly testable topic in IGCSE Mathematics. The key steps never change: identify the type of variation, write the equation with k, substitute the known values, and then solve for the required value.

正变分与反变分是 IGCSE 数学中范围不大但考查频率很高的知识点。解题关键步骤始终不变：判断变分类型，写出含 k 的方程，代入已知值，然后求解目标值。

The most important formulas to remember are y = kx, y = kx², y = k/x, and y = k/x². Also remember that y ∝ 1/x is the same as xy = k.

最重要的公式是 y = kx、y = kx²、y = k/x 和 y = k/x²。还要记住 y ∝ 1/x 等价于 xy = k。

With enough practice, variation questions become a guaranteed easy win on your exam paper.

只要足够练习，变分问题完全可以成为你试卷中稳拿分的题目。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Mastering Direct and Inverse Variation | 掌握正变分与反变分