IGCSE Mathematics Workbook 108: Direct and Inverse Proportion | IGCSE 数学练习册 108：正比例与反比例

📚 IGCSE Mathematics Workbook 108: Direct and Inverse Proportion | IGCSE 数学练习册 108：正比例与反比例

In IGCSE Mathematics, direct and inverse proportion are essential skills for both Core and Extended papers. They appear in algebra, graphs, geometry, and real-life word problems. This workbook section covers the key definitions, algebraic forms, graphical interpretations, common exam techniques, and a set of practice questions with full solutions.

在 IGCSE 数学中，正比例和反比例是基础试卷和扩展试卷的核心技能。它们出现在代数、图像、几何以及实际应用题中。本练习册内容涵盖关键定义、代数形式、图像解释、常见考试技巧，以及一组带有完整解答的练习题。

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

ematically, y is directly proportional to x if y = kx for a constant k. We write this as y ∝ x.

两个量成正比例，是指它们以相同比率增加或减少。一个量翻倍，另一个量也翻倍；一个量除以三，另一个量也除以三。数学上，若 y = kx（k 为常数），则 y 与 x 成正比例，记作 y ∝ x。

y ∝ x → y = kx

The symbol ∝ means "is proportional to". It is only used to describe a relationship, so it must be replaced by an equation with a constant k before calculations can be made.

符号 ∝ 表示“与…成比例”。它只用来描述关系，所以在进行计算之前，必须用含有常数 k 的方程替换它。

2. The Constant of Proportionality | 比例常数

= y/x. For direct proportion, the value of k is the same for every pair of x and y values.

y = kx 中的常数 k 称为比例常数。它可以通过变形 k = y/x 求出。对于正比例，每一对 x 和 y 所对应的 k 值都相同。

k = y/x

If k is positive, the two quantities increase together. If k is negative, one quantity increases while the other decreases, but the graph is still a straight line passing through the origin.

若 k 为正，两个量同增同减；若 k 为负，一个量增加而另一个量减少，但图像仍然是一条经过原点的直线。

3. Direct Proportion in Tables and Graphs | 表格与图像中的正比例

mple, the table below shows y = 3x, where each y value is exactly three times its x value.

在表格中，如果每一对数据的 y/x 比值相同，那么关系就是正比例。例如，下表显示 y = 3x，其中每个 y 值都正好是对应 x 值的三倍。

x	1	2	3	4
y	3	6	9	12

On a graph, direct proportion always produces a straight line passing through the origin. The gradient of that line is equal to the constant k.

在图像中，正比例总是一条经过原点的直线。该直线的斜率等于常数 k。

Gradient = k

It may be a linear relationship of the form y = mx + c, but it is not a direct proportion.

如果绘制一个比例关系的图像，但直线不经过 (0,0)，那么它就不是正比例。它可能是 y = mx + c 形式的线性关系，但不是正比例。

4. What is Inverse Proportion? | 什么是反比例？

= k, or y = k/x. If x doubles, y becomes half. If x is multiplied by 4, y is divided by 4.

两个量成反比例，是指它们的乘积保持不变。一个量增加，另一个量以 xy = k 或 y = k/x 的方式减少。x 翻倍，y 变为原来的一半；x 乘以 4，y 除以 4。

y ∝ 1/x → xy = k → y = k/x

Inverse proportion is common in situations where a fixed amount is shared or spread over a changing number of items. The key test is that the product of the two quantities remains constant.

反比例常见于固定的量被分配到数量变化的项目中。关键判断标准是两个量的乘积始终保持不变。

5. Inverse Proportion Graphs and Real-Life Contexts | 反比例图像与实际情境

y large when x is very small, but neither value ever reaches zero in the equation y = k/x.

y = k/x 的图像是一条等轴双曲线。它永远不会与 x 轴或 y 轴相交，这两条轴是渐近线。这意味着当 x 非常小时，y 可以变得非常大，但在 y = k/x 中两个变量的值都不会等于零。

Real-life examples include the time taken to travel a fixed distance at different speeds, the number of workers sharing a fixed task, and the pressure of a fixed amount of gas at constant temperature as volume changes.

实际例子包括以不同速度行驶固定距离所需的时间，工人分配固定任务的人数，以及固定量气体在恒温下压力随体积变化的情况。

Speed × Time = Distance

If the distance is constant, then speed and time are inversely proportional. Doubling the speed halves the time, while reducing the speed to one third triples the time.

如果距离一定，那么速度和时间成反比例。速度翻倍，时间减半；速度减少到原来的三分之一，时间变为原来的三倍。

6. Algebraic Forms of Direct and Inverse Proportion | 正比例与反比例的代数形式

re of x, then y = kx². If y is inversely proportional to the square root of x, then y = k/√x. The wording of the question tells you which form to use.

比例关系也可能涉及幂和根。如果 y 与 x 的平方成正比，则 y = kx²。如果 y 与 x 的平方根成反比，则 y = k/√x。题目中的措辞会告诉你要使用哪种形式。

y ∝ x² → y = kx²

y ∝ x³ → y = kx³

y ∝ 1/x² → y = k/x²

y ∝ 1/√x → y = k/√x

You must read phrases such as "the square of x", "the cube of x", or "the square root of x" carefully, because they change the structure of the equation before you substitute values.

你必须仔细阅读“x 的平方”、“x 的立方”或“x 的平方根”等短语，因为它们会改变代入数值之前方程的结构。

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7. Square and Cubic Proportion | 平方与立方比例

ional to r² with constant π. For spheres, volume V is proportional to r³, so V = (4/3)πr³.

x² 和 x³ 的正比例在几何中很常见。圆的面积 A = πr² 表示 A 与 r² 成正比，比例常数为 π。球的体积 V 与 r³ 成正比，因此 V = (4/3)πr³。

A ∝ r²

V ∝ r³

Inverse square proportion appears in physics: gravitational force F between two masses is inversely proportional to the square of the distance d between them, so F ∝ 1/d².

平方反比例在物理学中出现：两个质量之间的万有引力 F 与它们之间距离 d 的平方成反比，因此 F ∝ 1/d²。

F ∝ 1/d²

In exam questions, these power forms are usually tested by giving one pair of values, asking you to find k, and then asking for an unknown value at a different input.

在考试题目中，这些幂形式通常会给出一个数据对，要求你求出 k，然后要求你在不同输入下求未知量。

8. Solving Word Problems: A Step-by-Step Method | 解应用题：分步方法

First write the proportion statement using ∝. Second replace ∝ oot. Third substitute the given pair of values to find k. Fourth rewrite the equation with k. Fifth substitute the new value to find the unknown quantity.

首先用 ∝ 写出比例关系；其次将 ∝ 替换为 = k 并配上适当的幂或根；第三步代入已知数据对求出 k；第四步写出含 k 的方程；第五步代入新数值求未知量。

• Step 1: Write the relationship. | 步骤 1：写出关系式。
• Step 2: Introduce the constant k. | 步骤 2：引入常数 k。
• Step 3: Substitute the known pair. | 步骤 3：代入已知数据对。
• Step 4: Find k. | 步骤 4：求出 k。
• Step 5: Use the complete formula. | 步骤 5：使用完整公式。

This method works for direct, inverse, square, cubic, and square-root proportion. Keeping the steps separate helps avoid calculation errors and shows clear working for method marks.

这种方法适用于正比例、反比例、平方、立方和平方根比例。分步进行有助于避免计算错误，也能展示清晰的解题过程以获得方法分。

9. Common Mistakes and Exam Tips | 常见错误与应试技巧

Do not confuse direct and inverse proportion. For direct proportion, y/x is constant. For inverse proportion, xy is constant. Always decide which quantity must stay fixed before choosing the form.

不要混淆正比例和反比例。正比例中 y/x 恒定；反比例中 xy 恒定。在选择形式之前，一定要先确定哪个量保持不变。

Always find k before solving for an unknown. Many students forget to calculate k and try to use ratios directly, which can work for simple direct proportion but often fails for inverse and power proportion.

解未知量之前一定要先求 k。许多学生忘记计算 k，直接使用比值，这在简单的正比例中可能有效，但在反比例和幂比例中往往会出错。

Check that your answer makes sense. In inverse proportion, increasing one quantity should reduce the other. If your calculation gives a larger value when it should be smaller, you have likely used the wrong form.

检查答案是否合理。在反比例中，一个量增加另一个量应减少。如果计算得到的结果本应更小却更大，那么你可能使用了错误的形式。

• Direct proportion: y/x constant | 正比例：y/x 恒定
• Inverse proportion: xy constant | 反比例：xy 恒定
• Power proportion: substitute carefully | 幂比例：仔细代入
• Show all working | 展示所有解题步骤

10. Practice Questions from Workbook 108 | 练习册 108 练习题

Attempt these questions without looking at the solutions first. They cover direct proportion, inverse proportion, square proportion, and a real-life inverse proportion context.

先尝试回答这些问题，不要查看答案。它们涵盖正比例、反比例、平方比例以及实际反比例情境。

• Question 1: y is directly proportional to x. When x = 4, y = 12. Find y when x = 10. | 第 1 题：y 与 x 成正比。当 x = 4 时，y = 12。求 x = 10 时的 y。
• Question 2: p is inversely proportional to q. When q = 5, p = 8. Find p when q = 2. | 第 2 题：p 与 q 成反比。当 q = 5 时，p = 8。求 q = 2 时的 p。
• Question 3: y is directly proportional to x². When x = 3, y = 18. Find y when x = 5. | 第 3 题：y 与 x² 成正比。当 x = 3 时，y = 18。求 x = 5 时的 y。
• Question 4: The time T taken to paint a wall is inversely proportional to the number n of painters. If 4 painters take 6 hours, how long will 3 painters take? | 第 4 题：粉刷一面墙所需的时间 T 与油漆工数量 n 成反比。如果 4 个油漆工需要 6 小时，那么 3 个油漆工需要多长时间？
• Question 5: A spring stretches by length L cm under a mass m kg. L is directly proportional to m. If a 2 kg mass stretches it by 5 cm, what mass stretches it by 12.5 cm? | 第 5 题：弹簧在质量 m kg 下拉长 L cm。L 与 m 成正比。如果 2 kg 的质量使弹簧拉长 5 cm，那么什么质量会使弹簧拉长 12.5 cm？

11. Answers and Worked Solutions | 答案与详细解答

Question 1: y ∝ x = 4, y = 12: 12 = k × 4, so k = 3. The formula is y = 3x. When x = 10, y = 3 × 10 = 30.

第 1 题：y ∝ x → y = kx。代入 x = 4, y = 12：12 = k × 4，因此 k = 3。公式为 y = 3x。当 x = 10 时，y = 3 × 10 = 30。

Question 2: p ∝ tute q = 5, p = 8: 8 = k/5, so k = 40. The formula is p = 40/q. When q = 2, p = 40/2 = 20.

第 2 题：p ∝ 1/q → p = k/q。代入 q = 5, p = 8：8 = k/5，因此 k = 40。公式为 p = 40/q。当 q = 2 时，p = 40/2 = 20。

Question 3: y ∝ 18 = k × 3² = 9k, so k = 2. The formula is y = 2x². When x = 5, y = 2 × 5² = 2 × 25 = 50.

第 3 题：y ∝ x² → y = kx²。代入 x = 3, y = 18：18 = k × 3² = 9k，因此 k = 2。公式为 y = 2x²。当 x = 5 时，y = 2 × 5² = 2 × 25 = 50。

Question 4: T ∝ n = 4, T = 6: 6 = k/4, so k = 24. The formula is T = 24/n. When n = 3, T = 24/3 = 8 hours.

第 4 题：T ∝ 1/n → T =

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