📚 Direct and Inverse Proportion | 正比例与反比例
Direct and inverse proportion describe how two quantities relate to each other. When two values are in direct proportion, as one increases, the other increases at the same rate. Inverse proportion means that as one quantity increases, the other decreases at a consistent rate. These concepts appear regularly in IGCSE Edexcel exams, from algebraic manipulation to graph interpretation and real‑world modelling.
正比例和反比例描述了两个量之间的变化关系。当两个量成正比例时，一个量增加，另一个量以相同的速率增加。反比例则表示一个量增加，另一个量以一定的速率减少。这些概念在IGCSE Edexcel考试中经常出现，涉及代数运算、图像解释以及实际问题的建模。
Understanding proportionality helps you set up equations, sketch graphs, and solve word problems efficiently. This article covers direct and inverse proportion in detail, including higher‑tier topics involving squares, cubes, and their reciprocals.
理解比例关系能帮助你快速建立方程、画出图像并解决应用题。本文详细讲解正比例与反比例，还包括涉及平方、立方及它们倒数的进阶内容。
1. What is Direct Proportion? | 什么是正比例？
Two quantities, x and y, are in direct proportion if their ratio remains constant. This means y divided by x is always the same number. We write y ∝ x, where the symbol ∝ means 'is proportional to'.
如果两个量 x 和 y 的比值保持不变，它们就成正比例。也就是说 y 除以 x 始终是同一个数。我们写作 y ∝ x，其中 ∝ 表示“正比于”。
For example, if you buy 3 pens for £1.50, 6 pens will cost £3.00. The cost is directly proportional to the number of pens. Doubling the number of pens doubles the cost.
例如，买 3 支笔花费 1.50 英镑，买 6 支笔就花费 3.00 英镑。总价与笔的数量成正比例。笔的数量翻倍，总价也翻倍。
2. Equation and Constant of Proportionality | 方程与比例常数
The relationship y ∝ x can be turned into an equation by introducing a constant k: y = kx. Here k is called the constant of proportionality. To find k, simply divide a known y by its corresponding x: k = y / x.
关系式 y ∝ x 可以通过引入常数 k 转化为方程：y = kx。这里的 k 叫做比例常数。要求出 k，只需用已知的 y 值除以对应的 x 值：k = y / x。
Once k is known, you can calculate y for any x, or x for any y. This forms the basis of most direct proportion questions in the exam.
一旦知道 k，就可以计算任意 x 对应的 y 值，或任意 y 对应的 x 值。这是考试中大多数正比例问题的基础。
| Statement | Equation |
| y is directly proportional to x | y = kx |
| y ∝ x | y = kx |
k = y / x
3. Graphs of Direct Proportion | 正比例图像
The graph of y = kx is a straight line that passes through the origin (0,0). The gradient of the line is exactly the constant k. A larger k means a steeper slope.
y = kx 的图像是一条经过原点 (0,0) 的直线。直线的斜率正好就是常数 k。k 越大，直线越陡。
When you plot two directly proportional quantities, all points lie on a line through the origin. If a straight line does NOT go through the origin, the relationship is linear but not directly proportional.
绘制两个成正比例的量时，所有点都落在一条过原点的直线上。如果一条直线不经过原点，那么它是线性关系，但不是正比例关系。
In exams, you may be asked to determine whether a table of values shows direct proportion by checking if y / x is constant. Alternatively, you can plot points and see if they form a straight line through the origin.
考试中可能要求你通过检查 y / x 是否为常数，或者绘制点看它们是否形成一条穿过原点的直线，来判断一组值是否成正比例。
4. What is Inverse Proportion? | 什么是反比例？
Two quantities are inversely proportional if their product is constant. As one increases, the other decreases so that xy always equals the same value. We write y ∝ 1/x.
如果两个量的乘积保持不变，它们就成反比例。一个量增加，另一个量就相应减少，使得 xy 始终等于同一个值。我们写作 y ∝ 1/x。
For example, if a journey of 120 km takes 2 hours at a speed of 60 km/h, it will take 4 hours at 30 km/h. Time taken is inversely proportional to speed.
例如，120 公里的路程以 60 公里/小时的速度行驶需要 2 小时，以 30 公里/小时则需要 4 小时。所花时间与速度成反比例。
5. Equation and Constant for Inverse Proportion | 反比例方程与常数
The inverse proportion statement y ∝ 1/x becomes an equation with constant k: y = k / x, or equivalently xy = k. To find k, multiply a known pair x and y: k = xy.
反比例关系 y ∝ 1/x 引入常数 k 后变为方程：y = k / x，或者写成 xy = k。要求出 k，只需将一组已知的 x 与 y 相乘：k = xy。
Once k is found, the equation can be used to find unknown values. This method is extremely common in IGCSE problems, especially those involving speed, time, workers, or concentration.
求出 k 后，就可以用方程求未知量。这种方法在 IGCSE 题目中极为常见，特别是在速度、时间、工人数量或浓度等问题中。
| Statement | Equation |
| y is inversely proportional to x | y = k / x |
| y ∝ 1/x | y = k / x |
k = xy
6. Graphs of Inverse Proportion | 反比例图像
The graph of y = k / x (for positive k) is a hyperbola in the first and third quadrants. For positive x and y values, the curve slopes downwards, never touching the axes.
y = k / x（k 取正值）的图像是一条位于第一、第三象限的双曲线。对于正值的 x 和 y，曲线向下倾斜，永远不会碰到坐标轴。
As x gets very large, y approaches zero. As x gets very close to zero, y becomes extremely large. This shape helps you recognise inverse proportion in sketch questions.
当 x 非常大时，y 趋近于零；当 x 非常接近零时，y 变得非常大。这个形状能帮助你在草图题中识别反比例关系。
Remember: an inverse proportion graph is NOT a straight line. Checking whether xy is the same for each data point is the easiest way to confirm inverse proportion in a table.
记住：反比例的图像不是一条直线。检查数据表中每一组 xy 是否相同，是判断反比例最简单的方法。
7. Direct Proportion with Powers | 带幂的正比例
IGCSE higher tier also tests direct proportion to squares, cubes, or square roots. If y ∝ x², then y = kx². The graph is a parabola through the origin. For y ∝ x³, the equation is y = kx³, giving a cubic curve.
IGCSE 进阶部分还会考查与平方、立方或平方根成正比例的情况。如果 y ∝ x²，则 y = kx²，图像是经过原点的抛物线。对于 y ∝ x³，方程为 y = kx³，图像是三次曲线。
To find the constant k, rearrange the equation: k = y / x² (or k = y / x³). Graphically, these relationships are not linear, but you can still test proportionality by checking if y / x² is constant.
要求常数 k，只需将方程变形：k = y / x²（或 k = y / x³）。这些关系的图像不是线性的，但你仍然可以通过检查 y / x² 是否保持不变来验证比例关系。
| Proportionality | Equation | Finding k |
| y ∝ x² | y = kx² | k = y / x² |
| y ∝ x³ | y = kx³ | k = y / x³ |
| y ∝ √x | y = k √x | k = y / √x |
8. Inverse Proportion with Powers | 带幂的反比例
Similarly, inverse proportion can involve powers: y ∝ 1/x² gives y = k / x², or y ∝ 1/√x gives y = k / √x. The constant is found by multiplying: k = x² y or k = √x y.
类似地，反比例也可以带有幂次：y ∝ 1/x² 给出 y = k / x²，或 y ∝ 1/√x 给出 y = k / √x。常数通过乘法求得：k = x² y 或 k = √x y。
In physics-style questions, you might see statements like 'intensity is inversely proportional to the square of the distance'. This translates to I = k / d².
在物理类问题中，你可能会看到“强度与距离的平方成反比”这样的表述，这可以转化为 I = k / d²。
Always write down the proportionality statement first, then replace ∝ with = k to form an equation. Substitute known values to find k, and use the full equation for the remaining calculations.
解答时一定要先写下比例关系式，然后用 = k 替换 ∝ 得到方程。代入已知值求出 k，再用完整的方程进行后续计算。
9. Solving Real-world Proportion Problems | 解决实际比例问题
Typical IGCSE questions provide a scenario: 'The cost of apples varies directly with mass. 2 kg costs £3. Find the cost of 5 kg.' Set up C ∝ m, so C = km. Using 2 kg → £3 gives 3 = k × 2, so k = 1.5. Then for 5 kg: C = 1.5 × 5 = £7.50.
典型的 IGCSE 题目会给出情景：“苹果的价钱与质量成正比。2 公斤售价 3 英镑，求 5 公斤的价格。”建立 C ∝ m，即 C = km。代入 2 公斤→£3 得 3 = k × 2，所以 k = 1.5。然后 5 公斤：C = 1.5 × 5 = £7.50。
For inverse proportion: '4 workers build a wall in 6 days. How long will 3 workers take?' Assume time t ∝ 1/n (number of workers). So t = k / n. With 4 workers, 6 = k / 4 → k = 24. For 3 workers: t = 24 / 3 = 8 days.
对于反比例：“4 个工人砌一堵墙需要 6 天。3 个工人需要多少天？”假设时间 t ∝ 1/n（工人数），即 t = k / n。4 个工人时 6 = k / 4 → k = 24。3 个工人时 t = 24 / 3 = 8 天。
Always check whether the relationship is direct or inverse by thinking logically: would the unknown increase or decrease when the other variable increases?
一定要通过逻辑判断关系是正比还是反比：当一个变量增加时，另一个是增加还是减少？
10. Tips for Exams and Common Mistakes | 考试技巧与常见错误
students confuse direct and inverse proportion formulas. Remember: direct → divide to find k (y / x); inverse → multiply to find k (xy).
许多学生容易混淆正比和反比的公式。记住：正比用除法求 k（y / x）；反比用乘法求 k（xy）。
When plotting graphs, do not force a line of best fit through the origin if it does not pass through the data points approximately. For inverse proportion, do NOT draw a straight line through points that follow a curve.
画图像时，如果数据点并不大致经过原点，不要强行画一条穿过原点的拟合直线。对于反比例，也不要把符合曲线的点画成直线。
Finally, when a question says 'state the relationship', your answer should be in the form 'y is directly proportional to x²' or similar, not just an equation.
最后，当题目要求“描述关系”时，你的回答应该写成“y 与 x² 成正比”等形式，而不只是给出一个方程。
By practising a variety of word problems and past paper questions, you will become confident in handling any proportionality task in your IGCSE exam.
通过练习各种文字题和历年试题，你能自信地应对 IGCSE 考试中任何与比例相关的题目。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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