📚 Direct and Inverse Proportion | 正比与反比
Proportion is one of the most frequently tested topics in the Edexcel IGCSE Mathematics syllabus. It connects ratios, algebra, and graph plotting into a single coherent skill set. Whether you are aiming for a Grade 9 or simply trying to secure the core marks, mastering direct and inverse proportion is essential.
比例是 Edexcel IGCSE 数学考纲中最高频考查的知识点之一。它将比、代数与图像绘制融合为一套完整的技能体系。无论你是志在冲刺 9 分，还是希望稳稳拿到基础分，掌握正比与反比都至关重要。
1. Understanding Direct Proportion | 理解正比
Two quantities are in direct proportion when they increase or decrease at the same rate. In other words, their ratio remains constant. If one quantity doubles, the other doubles too; if one is halved, the other is halved as well.
当两个量以相同的速率增大或减小时，它们就成正比关系。换句话说，它们的比值保持不变。如果一个量翻倍，另一个量也翻倍；如果一个量减半，另一个量也减半。
x. The constant k is called the constant of proportionality, or the constant of variation.
正式定义是：若存在非零常数 k 使得 y = kx，则称 y 与 x 成正比。常数 k 称为比例常数，或变化常数。
For example, the cost of apples is directly proportional to the weight you buy. If 1 kg costs 3 pounds, then 2 kg costs 6 pounds, and 0.5 kg costs 1.5 pounds. The ratio cost ÷ weight is always 3.
例如，购买苹果的总价与所买重量成正比。如果 1 千克需 3 英镑，那么 2 千克需 6 英镑，0.5 千克需 1.5 英镑。总价 ÷ 重量的比值始终为 3。
2. The Equation y = kx | 方程 y = kx
en substitute the given values to find k. Once k is known, you can find any unknown value.
处理正比问题时，第一步永远是写出通式 y = kx，然后将已知数值代入求出 k。一旦求出 k，就可以计算任意未知量。
Worked example: y is directly proportional to x. When x = 4, y = 20. Find y when x = 7.
例题：y 与 x 成正比。当 x = 4 时，y = 20。求当 x = 7 时 y 的值。
y = kx → 20 = k × 4 → k = 5
y = 5 × 7 = 35
he rate of change. If y represents distance and x represents time, then k represents speed. This is why the graph of direct proportion is always a straight line through the origin.
注意常数 k = 5 同时也是变化率。若 y 表示距离，x 表示时间，那么 k 就代表速度。这就是为什么正比的图像总是一条经过原点的直线。
In the Edexcel exam, you may also see the phrase "y varies directly as x" or "y is proportional to x". Both mean exactly the same thing: y = kx.
在 Edexcel 考试中，你可能会看到 "y varies directly as x" 或 "y is proportional to x" 的表述。二者的含义完全相同：y = kx。
3. Graphs of Direct Proportion | 正比图像
at passes through the origin (0, 0). The gradient of the line is equal to k. A larger value of k produces a steeper line; a smaller value of k produces a shallower line.
y = kx 的图像是一条经过原点 (0, 0) 的直线。直线的斜率等于 k。k 值越大，直线越陡；k 值越小，直线越平缓。
Key features to remember:
需要牢记的关键特征：
- The line always passes through (0, 0).
- The gradient is k, not just "rise over run" but the constant of proportionality itself.
- If k > 0, the line slopes upward from left to right.
- If k < 0, the line slopes downward, but this is rarely tested in simple proportion questions.
- 直线必过 (0, 0)。
- 斜率就是 k，它不仅是"纵增量÷横增量"，更就是比例常数本身。
- 若 k > 0，直线从左向右上升。
- 若 k < 0，直线从左向右下降，但简单比例题中很少考查。
When identifying direct proportion from a graph, check two things: is it a straight line, and does it pass through the origin? If both answers are yes, the relationship is directly proportional.
从图像判断正比时需检查两点：是否为直线，以及是否经过原点。如果两个答案都是"是"，则该关系为正比关系。
4. Real-Life Examples of Direct Proportion | 正比的实际应用
Direct proportion appears frequently in real-world contexts. Recognising it quickly will save you valuable time in the exam.
正比在现实生活情境中出现得非常频繁。快速识别正比关系可以在考试中为你节省宝贵时间。
Distance = Speed × Time (when speed is constant)
距离 = 速度 × 时间（当速度恒定时）
Cost = Price per unit × Number of units
总价 = 单价 × 数量
Area of a circle = π × r² (proportional to r², not r)
圆面积 = π × r²（与 r² 成正比，而非与 r 成正比）
o the radius itself. If the radius doubles, the area is multiplied by 2² = 4. This type of "proportional to the square" question is very popular on Edexcel papers.
最后一个例子需要特别注意：面积与半径的平方成正比，而不是与半径本身成正比。如果半径翻倍，面积变为原来的 2² = 4 倍。这种"与平方成正比"的题型在 Edexcel 试卷中非常流行。
en the amount in EUR is directly proportional to the amount in GBP. The constant k = 1.15.
另一个常见例子是货币兑换：若 1 英镑 = 1.15 欧元，那么欧元的金额与英镑的金额成正比。此时常数 k = 1.15。
5. Understanding Inverse Proportion | 理解反比
Two quantities are in inverse proportion when one increases while the other decreases, and their product remains constant. If one quantity doubles, the other halves; if one triples, the other becomes one third.
当一个量增大而另一个量减小，且两者的乘积保持不变时，这两个量成反比。如果一个量翻倍，另一个量减半；如果一个量变为三倍，另一个量变为原来的三分之一。
onal to x if there exists a non-zero constant k such that xy = k, or equivalently y = k/x.
正式定义：若存在非零常数 k 使得 xy = k，即 y = k/x，则称 y 与 x 成反比。
4 hours. Speed × time = 120 in every case. The product of the two quantities is constant.
最经典的例子是速度与行程时间。若一段路程为 120 公里，以 60 km/h 行驶需 2 小时，以 30 km/h 行驶则需 4 小时。每种情况下 速度 × 时间 = 120 都是常数。两个量的乘积保持恒定。
6. The Equation y = k/x | 方程 y = k/x
ation y = k/x, substitute the known pair of values to find k, and then answer the unknown.
求解反比问题，应先写出方程 y = k/x，代入已知的一对数值求出 k，再解出未知量。
Worked example: y is inversely proportional to x. When x = 3, y = 8. Find y when x = 6.
例题：y 与 x 成反比。当 x = 3 时，y = 8。求当 x = 6 时 y 的值。
y = k/x → 8 = k/3 → k = 24
y = 24/6 = 4
Notice how the answer makes sense: x doubled from 3 to 6, so y must halve from 8 to 4. Always check whether your final answer is sensible. If x increases, y must decrease; otherwise you have made a sign or arithmetic error.
注意此结果的合理性：x 从 3 变为 6 翻了一倍，因此 y 应从 8 减半为 4。务必检查最终答案是否符合逻辑。如果 x 增大，y 必须减小；否则说明你在符号或算术上出了错。
Sometimes the question states "y is inversely proportional to the square of x", giving y = k/x². The same method applies: substitute, find k, then evaluate.
有时题目会说 "y 与 x 的平方成反比"，即 y = k/x²。方法完全相同：代入求 k，再求值。
7. Graphs of Inverse Proportion | 反比图像
ectangular hyperbola. It has two branches: one in the first quadrant (where x > 0, y > 0) and one in the third quadrant (where x < 0, y < 0), assuming k is positive.
y = k/x 的图像是一条称为等轴双曲线的曲线。它有两支：一支在第一象限（x > 0, y > 0），另一支在第三象限（x < 0, y < 0），此处假设 k 为正。
Key features to remember:
需要牢记的关键特征：
- The curve never touches the x-axis (y = 0) or the y-axis (x = 0).
- Both axes are asymptotes of the curve.
- As x increases, y decreases, but never reaches zero.
- The curve is symmetric about the line y = x.
- 曲线永远不会触及 x 轴 (y = 0) 或 y 轴 (x = 0)。
- 两条坐标轴都是曲线的渐近线。
- 随着 x 增大，y 减小，但永远不会到零。
- 曲线关于直线 y = x 对称。
In the exam, you will often be asked to identify which graph represents inverse proportion. The shape of the curve — curving down towards both axes without touching them — is your visual clue.
考试中常见题型是识别哪条图表示反比关系。曲线的形状——同时向下弯向两条坐标轴且不相交——就是你的视觉线索。
8. Real-Life Examples of Inverse Proportion | 反比的实际应用
Inverse proportion models many real situations where a fixed total is shared out over a varying number of parts.
反比可以模拟许多现实情境，在这些情境中，一个固定的总量被分配到数量变化的若干份中。
Speed = Distance ÷ Time (when distance is fixed)
速度 = 距离 ÷ 时间（当距离固定时）
Time to complete a task = Work ÷ Number of workers (each worker works at the same rate)
完成任务时间 = 总工作量 ÷ 工人数（设每个工人效率相同）
The classic exam question: "20 workers can build a wall in 9 days. How long would it take 30 workers?" Here, the number of workers and the time are inversely proportional, because the total number of worker-days is fixed.
经典考题："20 名工人用 9 天能砌完一堵墙。30 名工人需要多少天？"此处工人数与天数成反比，因为总"人·天"数是固定的。
Total worker-days = 20 × 9 = 180
总人·天数 = 20 × 9 = 180
Time for 30 workers = 180 ÷ 30 = 6 days
30 名工人所需天数 = 180 ÷ 30 = 6 天
Another common example is the "taps filling a tank" question: more taps mean less time, and the product of taps and time is constant.
另一个常见例子是"水管注满水池"问题：水管越多，所需时间越短，水管数与时间的乘积为常数。
9. Problem-Solving Strategy | 解题策略
Follow this four-step strategy for every proportion question, and you will not lose marks.
解所有比例题时遵循以下四步策略，就不容易失分。
- rite the general equation. For direct proportion: y = kx. For inverse proportion: y = k/x.
- Step 2: Substitute the known pair of values to find k.
- Step 3: Write the complete equation with the value of k.
- Step 4: Substitute the new value and solve.
- 第一步：写出通式。正比：y = kx。反比：y = k/x。
- 第二步：代入已知的一对数值求出 k。
- 第三步：写出含有 k 值的完整方程。
- 第四步：代入新值并求解。
Edexcel papers, method marks are often awarded even if the final answer is wrong. Writing "y = kx" and showing the substitution of values earns you at least one or two marks.
一定要清晰展示解题过程。在 Edexcel 试卷中，即使最终答案错误，方法分通常仍然有效。写出 "y = kx" 并展示代入过程，至少能帮你拿到一两分。
For word problems, first identify the two quantities involved and assign variables. Then decide from the phrasing whether the relationship is direct or inverse. Look for cue phrases: "increases at the same rate" means direct; "as one doubles the other halves" means inverse.
对于文字题，先明确涉及的两个量并赋予变量。然后根据措辞判断关系是正比还是反比。注意提示短语："以相同速率增大"指向正比；"一个翻倍另一个减半"指向反比。
10. Exam-Style Questions | 考试题型
Let us work through a selection of genuine exam-style questions to consolidate the method.
下面通过一组仿真考试题目来巩固方法。
Question 1: p is directly proportional to q². When q = 5, p = 50. Find p when q = 10.
第 1 题：p 与 q² 成正比。当 q = 5 时，p = 50。求当 q = 10 时 p 的值。
p = kq² → 50 = k × 25 → k = 2
p = 2 × 100 = 200
Question 2: y is inversely proportional to √x. When x = 4, y = 6. Find y when x = 16.
第 2 题：y 与 √x 成反比。当 x = 4 时，y = 6。求当 x = 16 时 y 的值。
y = k ÷ √x → 6 = k ÷ 2 → k = 12
y = 12 ÷ 4 = 3
Question 3: 8 machines produce 1200 items in 5 days. How many days would 10 machines take to produce 1500 items?
第 3 题：8 台机器 5 天生产 1200 件产品。10 台机器生产 1500 件需要多少天？
This question combines two proportions. First find the production rate per machine per day:
此题结合了两种比例关系。首先求每台机器每天的生产率：
1200 ÷ (8 × 5) = 30 items per machine per day
1200 ÷ (8 × 5) = 每台机器每天 30 件
10 machines → 300 items per day → 1500 ÷ 300 = 5 days
10 台机器 → 每天 300 件 → 1500 ÷ 300 = 5 天
Question 4 (tables): Determine whether each table shows direct proportion, inverse proportion, or neither.
第 4 题（表格）：判断下列各表所示关系是正比、反比还是均非。
| Table A | x: 2, 4, 6 | y: 5, 10, 15 |
| Table B | x: 2, 4, 8 | y: 8, 4, 2 |
= 2.5, 15/6 = 2.5. The ratio is constant, so Table A shows direct proportion with k = 2.5.
对于表 A，计算 y/x：5/2 = 2.5，10/4 = 2.5，15/6 = 2.5。比值恒定，所以表 A 为正比关系，k = 2.5。
16, 8 × 2 = 16. The product is constant, so Table B shows inverse proportion with k = 16.
对于表 B，计算 xy：2 × 8 = 16，4 × 4 = 16，8 × 2 = 16。乘积恒定，所以表 B 为反比关系，k = 16。
11. Common Mistakes | 常见错误
Below are the most common errors students make in proportion questions, along with advice on how to avoid them.
以下是学生在比例题中最常犯的错误，以及避免这些错误的建议。
Mistake 1: Confusing direct and inverse proportion. Always check the wording carefully. "Doubling x halves y" means inverse. "Doubling x doubles y" means direct.
错误一：混淆正比与反比。务必仔细读题。"x 翻倍导致 y 减半"是反比。"x 翻倍导致 y 翻倍"是正比。
Mistake 2: Forgetting units. If the question involves real measurements, always include the correct units in your final answer. A number without units may lose a mark.
错误二：忘记单位。若题目涉及实际测量，最终答案务必包含正确单位。没有单位的数值可能被扣一分。
common error is writing y = kx instead of y = k/x. Match the equation to the relationship.
错误三：反比时误写为 y = kx 而不是 y = k/x。反比关系最常见错误就是把方程写反。让方程匹配关系类型。
Mistake 4: Taking square roots instead of squares. If y is proportional to x², remember that doubling x multiplies y by 4, not by 2. Conversely, if x is proportional to √y, then doubling y multiplies x by √2, not by 2.
错误四：该平方时取了平方根。若 y 与 x² 成正比，x 翻倍时 y 变为原来的 4 倍而非 2 倍。反之，若 x 与 √y 成正比，y 翻倍时 x 变为原来的 √2 倍而非 2 倍。
Mistake 5: Not checking if the answer makes sense. For inverse proportion, if x gets bigger, y must get smaller. For direct proportion, if x gets bigger, y must get bigger. A quick sense-check catches most arithmetic slips.
错误五：不检查答案是否合理。反比关系中，x 增大则 y 必须减小。正比关系中，x 增大则 y 必须增大。快速检查逻辑能发现大多数计算失误。
12. Summary | 总结
portion means y = k/x: a curved graph never touching the axes, with a constant product xy.
正比即 y = kx：图像是经过原点的直线，比值 y/x 恒定。反比即 y = k/x：图像是永不触及坐标轴的曲线，乘积 xy 恒定。
The golden rule for every proportion question is to find the constant k first. Write the general equation, substitute the known values, solve for k, then answer the emerging question.
解所有比例题的金科玉律是：先求出常数 k。写出通式，代入已知值，解出 k，再回答题目所求。
With regular practice, proportion questions become some of the simplest marks on the paper. They follow predictable patterns and reward clear, methodical working. Review the worked examples in this article, attempt as many past paper questions as you can, and you will be fully prepared for this topic in your Edexcel IGCSE Mathematics examination.
通过定期练习，比例题会成为试卷上最容易得分的题目之一。它们模式固定，且奖励清晰有条理的解题过程。复习本文中的例题，尽可能多做历年真题，你就能在 Edexcel IGCSE 数学考试中从容应对这一话题。
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