Direct and Inverse Proportion | 正比例和反比例

📚 Direct and Inverse Proportion | 正比例和反比例

In Edexcel IGCSE Mathematics, direct and inverse proportion form a key part of the Number and Algebra syllabus. You must be able to write and use equations of the form y = kx and y = k/x, interpret real-life contexts, and solve problems involving proportionality.

在 Edexcel IGCSE 数学中，正比例与反比例是数与代数部分的核心考点。你必须能够写出并运用形如 y = kx 和 y = k/x 的方程，理解实际生活情境，并解决与比例相关的问题。

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1. What is Direct Proportion? | 什么是正比例？

Two quantities are directly proportional if they increase or decrease in the same ratio. If one doubles, the other doubles; if one is multiplied by 3, the other is multiplied by 3. In symbols, y is directly proportional to x means y = kx, where k is a non-zero constant called the constant of proportionality.

若两个量以相同的比率增大或减小，则它们成正比例。一个量翻倍，另一个量也翻倍；一个量乘以3，另一个量也乘以3。用符号表示，y 与 x 成正比例意味着 y = kx，其中 k 是非零常数，称为比例常数。

y = kx ⇔ y ∝ x

The notation y ∝ x is read as “y is directly proportional to x”. To turn this into an equation, you replace ∝ with “= k ×”.

记号 y ∝ x 读作“y 与 x 成正比例”。要将其转化为方程，只需把 ∝ 替换为“= k ×”。

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2. Finding the Constant of Proportionality | 求比例常数

To find k, you need one pair of corresponding values. Substitute the known x and y into y = kx, then solve for k. Once k is found, you can use the equation to find any other value.

要求 k，你需要一对对应的数值。将已知的 x 和 y 代入 y = kx，然后解出 k。一旦求得 k，就可以利用方程求出任意其他值。

Example: If y is directly proportional to x, and y = 20 when x = 5, find y when x = 8.

示例：已知 y 与 x 成正比例，当 x = 5 时 y = 20。求当 x = 8 时 y 的值。

y = kx → 20 = k × 5 → k = 4 → y = 4x → y = 4 × 8 = 32

Always write the proportionality equation first, then substitute. This avoids arithmetic errors and shows clear working.

一定要先写出比例方程，再代入数值。这样可以避免运算错误，并展示清晰的解题过程。

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3. Direct Proportion and Graphs | 正比例与图象

The graph of y = kx is a straight line that passes through the origin (0, 0). The gradient of the line is exactly k. The greater the value of k, the steeper the line.

y = kx 的图象是一条经过原点 (0, 0) 的直线。直线的斜率恰好等于 k。k 的值越大，直线越陡。

In an exam, you may be given a graph and asked whether two quantities are in direct proportion. A straight line through the origin is strong evidence, but be careful: a line not passing through the origin shows a linear relationship, not direct proportion.

考试中，你可能会看到图象并需要判断两个量是否成正比例。过原点的直线是重要证据，但注意：不经过原点的直线只表示线性关系，并不表示正比例。

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4. Inverse Proportion | 反比例

Two quantities are inversely proportional if one increases in the same ratio as the other decreases. For example, if one doubles, the other halves. Algebraically, y is inversely proportional to x means y = k/x, or equivalently xy = k.

若一个量以相同比率增大而另一个量减小，则它们成反比例。例如，一个量翻倍，另一个量减半。代数上，y 与 x 成反比例意味着 y = k/x，等价地 xy = k。

y = k/x ⇔ y ∝ 1/x ⇔ xy = k

The constant k is still called the constant of proportionality. In inverse proportion, the product of the two quantities is always constant.

常数 k 仍然称为比例常数。在反比例中，两个量的乘积始终保持不变。

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5. Solving Inverse Proportion Problems | 解反比例问题

Use the general form y = k/x. Substitute the known pair to find k, then use the equation to find unknown values. Alternatively, use the fact that x₁y₁ = x₂y₂ for two corresponding pairs.

使用一般形式 y = k/x。代入已知的一对值求 k，然后利用方程求未知值。或者使用两个对应量满足 x₁y₁ = x₂y₂ 这一性质。

Example: If p is inversely proportional to q, and p = 6 when q = 4, find p when q = 8.

示例：已知 p 与 q 成反比例，当 q = 4 时 p = 6。求当 q = 8 时 p 的值。

p = k/q → 6 = k/4 → k = 24 → p = 24/q → p = 24/8 = 3

Notice that as q doubled from 4 to 8, p halved from 6 to 3. This is the expected behaviour for inverse proportion.

注意：q 从 4 变成 8 翻倍时，p 从 6 变成 3 减半。这正是反比例的预期行为。

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6. The Graph of Inverse Proportion | 反比例的图象

The graph of y = k/x is a rectangular hyperbola. It has two branches: one in the first quadrant (positive x and y) and one in the third quadrant (negative x and y) if k is positive. The axes are asymptotes — the curve approaches them but never touches them.

y = k/x 的图象是等轴双曲线。它有两条分支：若 k 为正，一条在第一象限（x 和 y 均为正），一条在第三象限（x 和 y 均为负）。坐标轴是渐近线——曲线无限接近但永远不相交。

In IGCSE problems, you usually consider positive values only. The curve slopes downwards from left to right, showing that as x increases, y decreases.

在 IGCSE 题目中，通常只考虑正值。曲线从左到右逐渐下降，表明 x 增大时 y 减小。

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7. Recognising Proportionality from Tables | 从表格判断比例关系

Exam questions often provide a table of values. To check direct proportion, calculate y/x for each pair. If all ratios are equal, the relationship is direct and that common ratio is k. To check inverse proportion, calculate x×y. If all products are equal, the relationship is inverse and that common product is k.

考试题常给出一组数值表格。判断正比例时，计算每一对 y/x。若所有比值相等，则为正比例，该公共比值就是 k。判断反比例时，计算 x×y。若所有乘积相等，则为反比例，该公共乘积就是 k。

Direct: y/x = k constant | Inverse: xy = k constant

This method is quick and reliable. Always show two calculations in your working to demonstrate the pattern.

这种方法快速可靠。在作答中至少展示两组计算，以证明规律存在。

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8. Real-Life Applications | 实际应用

Direct proportion appears in speed–distance–time problems when speed is constant: distance = speed × time. It also appears in currency exchange, ingredient scaling in recipes, and energy bills with a fixed unit price.

正比例出现在速度恒定时的行程问题中：路程 = 速度 × 时间。它也出现在货币兑换、食谱配料按比例调整以及按固定单价计算的电费中。

Inverse proportion appears when work is shared among people: if more people work on a task, each person does less work for a fixed total workload. It also appears in the relationship between speed and journey time at a fixed distance.

反比例出现在多人分担固定工作量时：人数越多，每人分担的工作越少。它也出现在路程固定时速度与行程时间的关系中。

Fixed distance D: D = speed × time ⇒ time = D/speed

When distance is fixed, time is inversely proportional to speed. If you double the speed, the time is halved.

当路程固定时，时间与速度成反比例。速度加倍，时间减半。

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9. Word Problems and Exam Strategy | 文字题与考试策略

Read the problem carefully and decide which type of proportion is involved. Look for phrases such as “y varies directly as x”, “y is proportional to the square of x” (y = kx²), or “inversely proportional to the cube of x” (y = k/x³).

仔细阅读题目，判断属于哪种比例。注意短语如“y 随 x 正变”、“y 与 x 的平方成正比”（y = kx²），或“与 x 的立方成反比”（y = k/x³）。

For square relationships, the same principles apply. If y ∝ x², then y = kx² and the graph of y against x² is a straight line through the origin. Similarly, if y ∝ 1/x², then y = k/x².

对于平方关系，原理相同。若 y ∝ x²，则 y = kx²，y 对 x² 的图象是过原点的直线。类似地，若 y ∝ 1/x²，则 y = k/x²。

Always define the variable k first, write the equation, and then substitute the given pair to find k. Do not attempt to jump directly to the final answer.

始终先定义变量 k，写出方程，再代入已知的一对数据求 k。不要试图直接跳到最终答案。

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10. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Many students confuse direct and inverse proportion. Remember: in direct proportion, both quantities move in the same direction; in inverse proportion, they move in opposite directions. Another common error is forgetting to square or cube a quantity when using y ∝ x².

许多学生混淆正比例和反比例。记住：正比例中两个量同向变化；反比例中两个量反向变化。另一个常见错误是在 using y ∝ x² 时忘记对 x 进行平方或立方。

A third mistake is assuming that a straight line indicates direct proportion. To be directly proportional, the line must pass through the origin. A line with a non-zero y-intercept is linear but not proportional.

第三个错误是认为直线就一定表示正比例。要成为正比例，直线必须经过原点。y 截距非零的直线是线性关系，但不是正比例。

Finally, when finding k from a table, use unrounded values and keep k exact. Only round final answers to a sensible degree of accuracy, usually 3 significant figures unless the question says otherwise.

最后，从表格求 k 时，使用未四舍五入的值并保持 k 精确。只在最终答案时四舍五入到合理精度，通常保留 3 位有效数字，除非题目另有要求。

Practise writing proportional statements both in words and with symbols. This helps you understand the relationship faster in an exam.

练习用文字和符号两种方式书写比例关系。这能帮助你在考试中更快理解关系。

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