📚 Direct and Inverse Proportion | 正比例与反比例
In IGCSE Mathematics, understanding how quantities relate to one another is a fundamental skill. Direct proportion and inverse proportion describe two of the most important types of relationships between variables. These concepts appear not only in pure algebra but also in real-world contexts such as currency conversion, speed–time calculations, and scientific laws. Mastering both direct and inverse proportion will enable you to construct accurate equations, interpret graphs, and solve multi-step problems with confidence.
在IGCSE数学中，理解量之间的关系是一项基本技能。正比例和反比例描述了变量之间最重要的两种关系类型。这些概念不仅出现在纯代数中，还出现在实际情境中，比如货币兑换、速度–时间计算以及科学定律。掌握正比例和反比例将帮助你构建准确的方程、解读图像，并自信地解决多步骤问题。
1. What is Direct Proportion? | 什么是正比例？
y is directly proportional to x, and we write the relationship as y = kx, where k is a non‑zero constant called the constant of proportionality.
当两个量以相同的速率增加或减少时，它们成正比例。这意味着如果一个量乘以某个倍数，另一个量也乘以相同的倍数。我们说 y 与 x 成正比，并将这种关系写成 y = kx，其中 k 是一个非零常数，称为比例常数。
For example, if 3 pens cost £ directly proportional. The unit cost is the constant k: k = 1.50 ÷ 3 = 0.50, so C = 0.50n.
例如，如果 3 支笔花费 1.50 英镑，那么成本（C）与笔的数量（n）成正比。单位成本就是常数 k：k = 1.50 ÷ 3 = 0.50，因此 C = 0.50n。
2. The Equation of Direct Proportion | 正比例的方程
und if you are given one pair of corresponding values. Once k is known, the equation can be used to find any other value of y or x. In Edexcel IGCSE questions, you often need to calculate the constant k first and then use it to solve a problem.
正比例的一般方程是 y = kx。如果给出了一对对应值，就可以求出 k 的值。一旦知道了 k，就可以用方程求出 y 或 x 的任何其他值。在爱德思 IGCSE 试题中，你通常需要先计算出常数 k，然后再用它来解决问题。
ly proportional to x, and y = 24 when x = 8. Find y when x = 15. First, find k: 24 = k × 8 ⇒ k = 3. Then the equation is y = 3x. Substitute x = 15 to get y = 3 × 15 = 45.
例如：已知 y 与 x 成正比，且当 x = 8 时 y = 24。求当 x = 15 时的 y 值。首先求 k：24 = k × 8 ⇒ k = 3。那么方程就是 y = 3x。代入 x = 15，得 y = 3 × 15 = 45。
3. Graphical Representation of Direct Proportion | 正比例的图像表示
line passing through the origin (0,0). The gradient of this line is equal to the constant k. A greater value of k results in a steeper line, while a smaller positive k gives a shallower line. If k is negative, the line slopes downwards as x increases.
y = kx 的图像总是一条经过原点 (0,0) 的直线。这条直线的斜率等于常数 k。k 值越大，直线越陡；正值较小的 k 则给出较缓的直线。如果 k 是负数，直线会随着 x 的增加而向下倾斜。
In coordinate geometry questions, being able to identify direct proportion from a straight line through the origin is essential. Conversely, if a graph is a straight line but does not pass through the origin, the two variables are not directly proportional, though they may share a linear relationship.
在坐标几何问题中，能够通过一条经过原点的直线识别出正比例关系至关重要。反之，如果图像是一条直线但不经过原点，这两个变量就不成正比，尽管它们可能是线性关系。
4. Solving Direct Proportion Problems | 解决正比例问题
t to predict unknown quantities. Always start by writing the proportion statement: y = kx.
IGCSE 考试中关于正比例的题目常常涉及现实生活情景，比如货币兑换、食谱缩放或成本计算。你可能会被要求建立方程、求出常数 k 并用它来预测未知量。始终从写出比例关系式开始：y = kx。
Another common style asks: “The distance is directly proportional to the square of the time.” ng given data, and then substitute the target value. Reading the wording carefully is vital, because many students mistakenly use linear proportion when the problem involves squares or roots.
另一种常见题型问：“距离与时间的平方成正比。”在这种情况下，应写成 d = k t²，利用已知数据求出 k，然后代入目标值。仔细审题非常重要，因为许多学生在问题涉及平方或平方根时，错误地使用了线性比例。
5. Understanding Inverse Proportion | 理解反比例
, and the relationship is written as y = k / x, or equivalently xy = k, where k is the non‑zero constant of proportionality.
如果一个量增加而另一个量以相关的速率减少，则这两个量成反比例。两个变量的乘积保持不变。我们说 y 与 x 成反比，关系式写作 y = k / x，或等价地 xy = k，其中 k 是非零的比例常数。
20 / v. Doubling the speed halves the travel time, which is a hallmark of inverse proportion.
一个日常例子是以不同平均速度行驶固定距离所需的时间。如果一辆车行驶 120 公里，时间 t（小时）与速度 v（公里/小时）成反比：t = 120 / v。速度翻倍则行驶时间减半，这正是反比例的标志。
6. The Equation of Inverse Proportion | 反比例的方程
termined, you can find y for any given x. Always check that k remains constant for all data pairs in the question, as that confirms inverse proportion is the correct model.
标准方程是 y = k / x。要找到 k，需要一对值。将它们相乘：k = xy。一旦确定了 k，就可以求出任意给定 x 所对应的 y 值。务必检查问题中所有数据对应的 k 是否保持不变，以此确认反比例是正确的模型。
st, k = 5 × 10 = 50. Then the equation is p = 50 / q. Substitute q = 4: p = 50 / 4 = 12.5.
例如，p 与 q 成反比。当 p = 5 时，q = 10。求当 q = 4 时的 p 值。首先，k = 5 × 10 = 50。那么方程为 p = 50 / q。代入 q = 4：p = 50 / 4 = 12.5。
7. Graphs of Inverse Proportion | 反比例的图像
ngular hyperbola in the first and third quadrants. The curve approaches but never touches the x‑axis and y‑ eases, y decreases, giving the curve its characteristic shape. For negative k, the graph appears in the second and fourth quadrants.
y = k / x（当 k > 0 时）的图像是一条位于第一和第三象限的等轴双曲线。曲线趋近但永远不会触及 x 轴和 y 轴，它们是渐近线。随着 x 增大，y 减小，使曲线呈现特有的形状。当 k 为负数时，图像会出现在第二和第四象限。
ching curves in transformation questions. If you are given a graph that shows a reciprocal‑ f y = k / x as a possible equation and test it by checking the product of the coordinates.
识别这种图像对于解释实验数据以及在变换问题中画曲线草图非常重要。如果给出的图像显示一条经过已知点的倒数形曲线，你应立即想到 y = k / x 是一个可能的方程，并通过检验坐标的乘积来验证它。
8. Variation with Powers and Roots | 涉及乘方与方根的比例变化
; y is inversely proportional to x³, written y = k / x³; and y is directly proportional to √x, written y = k √x. The same principles apply: use a given data pair to find k, then solve for the required variable.
正比例和反比例可以涉及乘方与方根。常见形式包括：y 与 x² 成正比，写作 y = k x²；y 与 x³ 成反比，写作 y = k / x³；y 与 √x 成正比，写作 y = k √x。同样的原则适用：利用一对已知数据求出 k，然后解出所需的变量。
0. The equation becomes I = 160 / d², allowing you to calculate intensity at any distance.
例如，光的强度（I）与到光源的距离（d）的平方成反比，因此 I = k / d²。如果当 d = 2 时 I = 40，那么 k = 40 × 2² = 160。方程变为 I = 160 / d²，这样你就可以计算任何距离下的光强度。
| Relationship | Form of equation | How to find k |
|---|---|---|
| y is directly proportional to x | y = kx | k = y / x |
| y is inversely proportional to x | y = k / x | k = xy |
| y is proportional to x² | y = k x² | k = y / x² |
| y is inversely proportional to √x | y = k / √x | k = y √x |
9. Mixed Proportion Problems | 混合比例问题
= k x / y. After finding k with a triple of values, you can determine unknown quantities.
更具挑战性的题目要求你判断两个变量是成正比还是成反比，然后处理混合变化关系。例如，题目可能会陈述 z 与 x 成正比、与 y 成反比，写作 z = k x / y。用一组三个值求出 k 后，你就能求出未知量。
A typical exam command is: “ to the mass m and inversely proportional to the square of the distance d. Write down the equation and find F when given m and d.” om a complete set, and then evaluate F. Careful substitution is key to avoiding arithmetic slips.
一个典型的考题指令是：“力 F 与质量 m 成正比，与距离 d 的平方成反比。写出方程，并在给定 m 和 d 时求 F。”你应设 F = k m / d²，利用完整的一组值计算 k，然后再求 F。小心代入是避免算术错误的关键。
10. Testing for Proportion from Tables | 从表格数据检验比例关系
When given a table of x and y values, you can determine the type of proportion by checking which mathematical operation yields a constant. For direct proportion, calculate y / x; if all such quotients are equal, then y ∝ x. For inverse proportion, calculate xy; if the product is constant, then the variables are inversely proportional.
当给出 x 与 y 的数据表时，你可以通过检查哪种数学运算得出常数来确定比例类型。对于正比例，计算 y / x；如果所有这些商都相等，那么 y ∝ x。对于反比例，计算 xy；如果乘积恒定，那么这两个变量成反比。
If neither y/x nor xy is constant, test for y/x², y/√x, etc., to see whether a power or root relationship exists. This method is frequently used in IGCSE experimental data questions, where you must deduce the law linking the variables before making predictions.
如果 y/x 和 xy 都不是常数，可以检验 y/x²、y/√x 等，看看是否存在乘方或方根关系。这种方法常用在 IGCSE 实验数据题中，你必须先推断出变量间的规律，然后才能进行预测。
11. Proportional Reasoning Without Finding k | 无需求出 k 的比例推理
x₂). Cross-multiply to find the unknown. This is particularly useful when dealing with unitary method problems such as “If 5 batteries cost £3, find the cost of 8 batteries.”
有时你可以通过乘除运算解决比例问题，而无需明确计算出常数 k。对于正比例，可以建立比例式：(y₁ / x₁) = (y₂ / x₂)。交叉相乘即可求出未知数。这在处理单价法问题时特别有用，比如“如果 5 节电池花费 3 英镑，求 8 节电池的花费。”
r example, if 6 workers take 10 days to complete a job, how many days would 4 workers take? Here, x₁=6, y₁=10, x₂=4: 6×10 = 4×y₂ ⇒ y₂ = 60/4 = 15 days. This method is quicker for simple numerical examples.
对于反比例，乘积保持不变，因此你可以设 x₁ y₁ = x₂ y₂。例如，如果 6 个工人需要 10 天完成一项工作，那么 4 个工人需要多少天？这里 x₁=6，y₁=10，x₂=4：6×10 = 4×y₂ ⇒ y₂ = 60/4 = 15 天。对于简单的数值例子，这种方法更快捷。
12. Common Mistakes and Exam Tips | 常见错误与考试技巧
A common error is confusing direct with inverse proportion when setting up the initial equation. Always read the wording carefully: "proportional to" means direct, while "inversely proportional to" means inverse. Another mistake is forgetting to square or square‑root the variable when the problem involves a power relationship, leading to a completely wrong equation.
一个常见错误是在建立初始方程时将正比例与反比例混淆。务必仔细阅读措辞：“与…成比例”意味着正比例，而“与…成反比”意味着反比例。另一个错误是当问题涉及乘方关系时忘记对变量进行平方或开方，从而导致方程完全错误。
In the exam, always show the step where you calculate k, even if you could solve by the ratio method. This demonstrates your understanding and can earn marks even if a later substitution is arithmetically incorrect. Finally, check whether your final answer makes sense in the context: for inverse proportion, increasing one variable should always decrease the other.
在考试中，即使你可以用比例法求解，也一定要展示计算 k 的步骤。这能证明你的理解，并且即使后来的代入计算出现算术错误，也能获得分数。最后，检查你的最终答案在情境中是否合理：对于反比例，一个变量的增加应总是导致另一个变量减少。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图