📚 Direct and Inverse Proportion | 正比与反比
In mathematics, understanding how quantities relate to one another is essential. Proportion describes a special relationship where a change in one quantity leads to a predictable change in another. This article covers direct and inverse proportion as required by the Edexcel IGCSE Mathematics syllabus, including how to set up equations, find the constant of proportionality, interpret graphs, and solve real-life problems.
在数学中，理解数量之间的关系至关重要。比例描述了一种特殊关系，其中一个量的变化会引起另一个量可预测的变化。本文涵盖Edexcel IGCSE数学大纲要求的正比与反比内容，包括如何建立方程、求比例常数、解读图形以及解决现实生活中的问题。
1. What is Proportion? | 什么是比例？
Proportion tells us how two variables are related. If two quantities are in proportion, there is a constant multiplier linking them. We use the symbol ∝ to express 'is proportional to'. For instance, if y is proportional to x, we write y ∝ x. This becomes a powerful tool to model real situations, from currency conversion to speed and time calculations.
比例告诉我们两个变量如何关联。如果两个量成比例，就存在一个常数乘子连接它们。我们使用符号 ∝ 表示“与……成正比”。例如，如果 y 与 x 成正比，写作 y ∝ x。这成为模拟真实情况的强大工具，从货币兑换到速度与时间的计算。
2. Direct Proportion: y ∝ x | 正比例：y ∝ x
Direct proportion means that as one quantity increases, the other increases at the same rate. If y is directly proportional to x, then doubling x will double y, tripling x will triple y, and so on. Mathematically, we write y ∝ x. This relationship implies that the ratio y/x is constant. That constant ratio is the key to moving from a proportional statement to an equation.
正比例意味着当一个量增加时，另一个量以相同的速率增加。如果 y 与 x 成正比，那么把 x 加倍，y 也会加倍；扩大三倍，y 也扩大三倍，以此类推。数学上，记作 y ∝ x。这种关系意味着比值 y/x 是常数。这个常数比是从比例关系走向方程的关键。
ind a missing value without explicitly finding the constant k, though finding k is usually more straightforward.
对于正比中的任意两对 (x₁, y₁) 和 (x₂, y₂)，都有 y₁/x₁ = y₂/x₂。这个等式可用于在不显式求出常数 k 的情况下求缺失值，尽管求出 k 通常更直接。
3. The Direct Proportion Equation y = kx | 正比例方程 y = kx
The statement y ∝ tionality. This constant is a fixed number that tells us the rate at which y changes with x. To find k, we need one pair of corresponding values (x, y). Once k is known, we can find y for any x, or x for any y.
关系式 y ∝ x 可以转化为方程 y = kx，其中 k 是比例常数。这个常数是一个固定数值，表示 y 随 x 变化的速率。要求出 k，我们需要一对对应的值 (x, y)。一旦知道 k，就可以求出任意 x 对应的 y，或任意 y 对应的 x。
y = kx
ion becomes y = 2.5x. If we then need y when x = 12, substitution gives y = 2.5 × 12 = 30.
例如，若 x = 4 时 y = 10，则 k = y/x = 10 ÷ 4 = 2.5。因此方程为 y = 2.5x。若我们需要 x = 12 时的 y，代入得 y = 2.5 × 12 = 30。
4. Finding the Constant and Solving Problems | 求常数与解题
To solve direct proportion problems, follow three steps: a) Write the proportion statement: y ∝ x. b) Replace ∝ with = k to get y = kx. c) Substitute the given pair to find k, then use the equation to find the unknown value.
要解正比例问题，请遵循三步： a) 写出比例关系：y ∝ x。 b) 将 ∝ 替换为 = k 得到 y = kx。 c) 代入已知数对求出 k，然后利用方程求未知值。
Example: y is directly proportional to x. When x = 3, y = 21. Find y when x = 8.
示例：y 与 x 成正比。当 x = 3 时 y = 21。求当 x = 8 时 y 的值。
First, find k: k = y/x = 21 ÷ 3 = 7. Equation: y = 7x. Then y = 7 × 8 = 56.
首先求 k：k = y/x = 21 ÷ 3 = 7。方程：y = 7x。那么 y = 7 × 8 = 56。
An alternative method is to set up the proportion y₁/x₁ = y₂/x₂. Here 21/3 = y₂/8 ⇒ y₂ = (21/3) × 8 = 56. This works because k is constant.
另一种方法是建立比例式 y₁/x₁ = y₂/x₂。这里 21/3 = y₂/8 ⇒ y₂ = (21/3) × 8 = 56。这样可行是因为 k 不变。
5. Graphs of Direct Proportion | 正比例的图形
ssing through the origin (0,0). The gradient of the line equals k. If k > 0, the line slopes upwards, showing a positive correlation. If k < 0, the line slopes downwards. This visual representation makes it easy to check whether two quantities are directly proportional: a straight line through the origin confirms it
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