📚 Direct and Inverse Proportion | 正比例与反比例
Direct and inverse proportion describe how two quantities are linked. In direct proportion, as one quantity increases, the other increases at the same rate, while in inverse proportion, an increase in one leads to a decrease in the other so that their product remains constant. These ideas are fundamental in IGCSE Edexcel mathematics and appear in both algebraic and graphical forms.
正比例和反比例描述了两个量之间的关联方式。在正比例中，一个量增加时，另一个量以相同的速率增加；而在反比例中，一个量增加会导致另一个量减少，从而使它们的乘积保持不变。这些概念是 IGCSE Edexcel 数学的基础，常以代数和图形的形式出现。
1. Understanding Direct Proportion | 理解正比例
Two quantities are in direct proportion if they change in such a way that their ratio is constant. In other words, if one quantity is multiplied by a factor, the other is multiplied by the same factor. The symbol for direct proportion is ∝, so we write y ∝ x, which means y is directly proportional to x.
如果两个量的比值保持不变，则它们成正比例。换言之，如果一个量乘以某个倍数，另一个量也会乘以相同的倍数。正比例的符号是 ∝，因此我们写作 y ∝ x，表示 y 与 x 成正比。
ough the origin (0,0) because when x = 0, y must also be 0 for the ratio to stay constant.
对于正比例，x 加倍会使 y 加倍；x 变三倍则 y 变三倍。如果 x 减半，y 也会减半。这种关系总是通过原点 (0,0)，因为当 x = 0 时，y 也必须为 0 才能保持比值不变。
2. The Proportionality Constant k | 比例常数 k
/x. Once k is found from a given pair of values, the equation can be used to find unknown values.
为了将正比例写成方程，我们引入比例常数，通常用 k 表示。方程变为 y = kx，其中 k = y/x。一旦从给定的一对数值中求出 k，便可根据方程求出未知值。
y = kx
15 when x = 3, then k = 15/3 = 5. The relationship is y = 5x. When x = 8, y = 5 × 8 = 40.
例如，如果当 x = 3 时 y = 15，那么 k = 15/3 = 5。关系式为 y = 5x。当 x = 8 时，y = 5 × 8 = 40。
3. Graphical Representation of Direct Proportion | 正比例的图形表示
ssing through the origin. The gradient of the line equals the constant k. If k is positive, the line slopes upward from left to right; if k is negative, it slopes downward, but the intercept on both axes is zero.
y = kx 的图像是一条通过原点的直线。该直线的斜率等于常数 k。如果 k 为正，直线从左到右向上倾斜；如果 k 为负，则向下倾斜，但与两轴的交点均为零。
The key feature that distinguishes a direct proportion graph from any other straight line is that it must go through (0,0). In exams, always check that the line passes through the origin when identifying a direct proportion from a graph.
区分正比例图像与其他直线的关键特征是它必须通过 (0,0)。在考试中，当从图形识别正比例时，务必检查直线是否通过原点。
4. Solving Direct Proportion Problems | 解决正比例问题
kx to find the missing value. Always pay attention to units and state the answer clearly.
典型的考题会提供两个对应的值，并要求求出第三个值。首先，用给定的一对值计算 k = y/x。然后，将新的 x 或 y 代入 y = kx 中求出缺失的值。始终注意单位，并清晰地写出答案。
Example: y is directly proportional to x. When x = 4, y = 10. Find y when x = 14.
k = 10/4 = 2.5, so y = 2.5 × 14 = 35.
示例：y 与 x 成正比。当 x = 4 时，y = 10。求当 x = 14 时的 y 值。
k = 10/4 = 2.5，因此 y = 2.5 × 14 = 35。
5. Understanding Inverse Proportion | 理解反比例
Two quantities are inversely proportional if their product is constant. This means as one quantity increases, the other decreases in such a way that multiplying them always gives the same number. The symbol is y ∝ 1/x, read as 'y is inversely proportional to x'.
如果两个量的乘积是常数，则它们成反比例。这意味着当一个量增加时，另一个量相应减少，使得两者相乘的结果始终相同。符号为 y ∝ 1/x，读作“y 与 x 成反比”。
For inverse proportion, doubling x halves y; tripling x makes y one third of its original value. It is important to note that neither variable can be zero, because division by zero is undefined.
对于反比例，x 加倍会使 y 减半；x 变为三倍会使 y 变为原来的三分之一。注意，两个变量都不能为零，因为除以零没有定义。
6. Inverse Proportion Equation y = k/x | 反比例方程 y = k/x
o find k, simply multiply a given pair of x and y values. Once k is known, you can determine either variable when the other is given.
反比例关系表示为 y = k/x，其中 k 是比例常数。也可以写作 xy = k。要计算 k，只需将给定的一对 x 和 y 值相乘。一旦知道 k，就可以在已知一个变量时求出另一个。
y = k / x or xy = k
y = 6 when x = 2, then k = 6 × 2 = 12. The equation is y = 12/x. When x = 4, y = 12/4 = 3.
例如，如果当 x = 2 时 y = 6，那么 k = 6 × 2 = 12。方程为 y = 12/x。当 x = 4 时，y = 12/4 = 3。
7. Graph of Inverse Proportion | 反比例图形
angular hyperbola. It consists of two smooth curves, one in the first quadrant and one in the third quadrant. The curves approach the x-axis and y-axis but never touch them; these lines are called asymptotes.
y = k/x（k > 0）的图像是等轴双曲线。它由两条平滑曲线组成，一条在第一象限，另一条在第三象限。曲线无限接近 x 轴和 y 轴，但永远不会接触；这些直线称为渐近线。
If k is negative, the branches lie in the second and fourth quadrants. In IGCSE exams, you will typically work with positive constants only. Recognising the shape helps you check your algebraic answers.
如果 k 为负，分支将位于第二象限和第四象限。在 IGCSE 考试中，通常只涉及正常数。认识图形形状有助于检查代数答案。
8. Solving Inverse Proportion Problems | 解决反比例问题
the known pair to find k, then use the equation to find the unknown. Always verify that the product xy remains the same for any pair of values given by the relationship.
要解决反比例问题，先写出 xy = k。代入已知一对值求出 k，然后利用方程求出未知量。务必验证对于关系式给出的任意一对值，乘积 xy 始终保持不变。
Example: y is inversely proportional to x. When x = 8, y = 3. Find y when x = 12.
k = 8 × 3 = 24, so y = 24/12 = 2.
示例：y 与 x 成反比。当 x = 8 时，y = 3。求当 x = 12 时的 y。
k = 8 × 3 = 24，因此 y = 24/12 = 2。
9. Recognising Proportional Relationships from Tables | 从表格中识别比例关系
Tables of values can reveal whether two quantities are in direct or inverse proportion. For direct proportion, calculate the ratio y/x for each pair. If the ratio is constant, the relationship is direct. For inverse proportion, calculate the product xy; if it is constant, the relationship is inverse.
数值表格可以揭示两个量之间是正比例还是反比例。对于正比例，计算每一对的比值 y/x。如果比值恒定，则关系为正比例。对于反比例，计算乘积 xy；如果乘积恒定，则关系为反比例。
= 2 every time. The second row shows an inverse proportion because xy = 12 for every pair.
请看下表。第一行显示的是正比例，因为每次 y/x = 2。第二行显示的是反比例，因为每一对的 xy = 12。
| x | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| y (direct) | 4 | 8 | 12 | 16 |
| y (inverse) | 6 | 3 | 2 | 1.5 |
10. Real-life Applications | 实际应用
speed. If 1 kg of apples costs 2 pounds, then the cost C = 2m, where m is the mass in kg.
正比例出现在许多情景中，如货币兑换、计算相同单价的多件商品的费用，或者以恒定速度行驶的距离。如果 1 公斤苹果售价 2 英镑，那么费用 C = 2m，其中 m 是以千克计的质量。
Inverse proportion is common in speed– umber of workers and time taken to complete a job, or pressure and volume of a gas at constant temperature (Boyle's Law). For a fixed journey, the faster you go, the less time it takes.
反比例常见于距离固定时的速度与时间关系（时间 = 距离/速度）、完成一项工作的工人数量与所需时间，或恒温下气体的压强与体积（波义耳定律）。对于固定的路程，速度越快，所用时间越短。
11. Common Exam Pitfalls | 常见考试误区
Many students confuse direct and inverse proportion. Always check whether the question expects the product xy to be constant (inverse) or the ratio y/x to be constant (direct). Another mistake is assuming that a straight-line graph always represents direct proportion; it must go through the origin.
许多学生会混淆正比例和反比例。务必检查题目是否要求乘积 xy 为常数（反比）还是比值 y/x 为常数（正比）。另一个常见错误是认为直线图形总是表示正比例；它必须通过原点才行。
Forgetting to find k before answering the final part of a problem is also a common error. Always calculate k using the given complete pair, and then use the equation. Also, ensure you do not mix up x and y when substituting values.
忘记在回答问题最后部分之前求出 k 也是常见错误。务必先用给定的完整一对值计算出 k，再使用方程。此外，代入数值时确保不要混淆 x 和 y。
12. Summary and Key Points | 总结要点
stant k is found from a known pair of values, and it links the two quantities in a predictable way.
总结起来，正比例由 y = kx 描述，图像是通过原点的直线。反比例由 y = k/x 或 xy = k 描述，图像是双曲线。常数 k 从已知的一对值中求出，它以可预测的方式将两个量联系起来。
Always double-check whether the relationship involves direct or inverse proportion by examining the product xy or the ratio y/x. Mastering these two types of proportionality will help you tackle a wide variety of IGCSE questions confidently.
务必通过检查乘积 xy 或比值 y/x，再次确认关系是正比例还是反比例。掌握这两类比例关系将帮助你自信地应对各种 IGCSE 考题。
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