# 比例（爱德思IGCSE数学A 高阶层级）

> 爱德思国际GCSE数学A · 4MA1 2016
> 来源: https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s2-proportion/

本指南覆盖爱德思IGCSE数学A（4MA1）高阶层级全部比例相关内容，包括建立方程、求解常数$k$以及将比例关系与图像形状相关联。

**先修:** [公式变形](https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s2-rearranging-formulae/); [绘制线性与非线性图像](https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s3-graphs-basics/)

## 学习目标

- 区分考纲要求的全部8种正比例与反比例关系
- 使用比例常数$k$建立比例方程
- 根据给定数据计算$k$并求解未知量
- 将比例方程与对应的图像形状进行匹配
- 准确解答可拿满分的考试风格比例题型

## 正比例关系

**正比例** — 若两个正量中一个增大时，另一个以固定常数倍数同步增大，则称二者成正比例。该关系可以写为等式：$y = kf(x)$，其中$k$为非零的比例常数。

*记法:* $y \propto f(x)$

*例:* If $y$ is directly proportional to the square of $x$, then $y = kx^2$

- Allowed direct proportion relationships: $y \propto x$, $y \propto x^2$, $y \propto x^3$, $y \propto \sqrt{x}$
- All direct proportion graphs pass through the origin (0,0)
- Only $y \propto x$ gives a straight line graph; other direct relationships give upward-curving lines through the origin

**例题:** 生产金属圆盘的成本$C$与圆盘半径$r$的平方成正比例。当$r = 4$厘米时，$C = \pounds{2.40}$。求半径为7厘米的圆盘的生产成本。

1. 写出比例关系表述：$C \propto r^2$
2. 引入常数$k$以构造等式：

   $$C = kr^2$$
3. 代入给定数值求解$k$：$2.40 = k(4)^2 = 16k$，因此$k = 2.40 / 16 = 0.15$
4. 写出完整等式：$C = 0.15r^2$
5. 代入$r=7$：$C = 0.15(49) = 7.35$
6. 最终答案：$\pounds{7.35}$

> **考试提示:** 请在题目中划出x的幂次（例如平方、立方、平方根），避免在比例方程中写错对应函数。

*计算器:* allowed

## 反比例关系

**反比例** — 若两个正量中一个增大时，另一个以固定常数倍数减小，则称二者成反比例。该关系可以写为等式：$y = \frac{k}{f(x)}$，其中$k$为非零的比例常数。

*记法:* $y \propto \frac{1}{f(x)}$

*例:* If $y$ is inversely proportional to the cube of $x$, then $y = \frac{k}{x^3}$

- Allowed inverse proportion relationships: $y \propto 1/x$, $y \propto 1/x^2$, $y \propto 1/x^3$, $y \propto 1/\sqrt{x}$
- All inverse proportion graphs are decreasing curves in the first quadrant
- Inverse graphs never touch the x or y axes, as dividing by zero is undefined

**例题:** 注满一个游泳池所需的时间$t$与使用的水泵数量$p$的平方根成反比例。当使用16台水泵时，注满泳池需要90分钟。若要在60分钟内注满泳池，需要多少台水泵？

1. 写出比例关系表述：$t \propto \frac{1}{\sqrt{p}}$
2. 引入常数$k$以构造等式：

   $$t = \frac{k}{\sqrt{p}}$$
3. 代入给定数值求解$k$：$90 = \frac{k}{\sqrt{16}} = \frac{k}{4}$，因此$k = 360$
4. 写出完整等式：$t = \frac{360}{\sqrt{p}}$
5. 代入$t=60$：$60 = \frac{360}{\sqrt{p}}$ → $\sqrt{p} = 6$ → $p = 36$
6. 最终答案：36台水泵

> **考试提示:** 请始终反复确认你在反比例关系表述中加入了倒数项，这是该知识点最常见的失分点。

*计算器:* allowed

## 比例方程与图像的关联

> **info**
>
> 在分值为1-2分的选择题或简答题中，你经常会遇到将比例方程与对应图像形状匹配的题型。可以首先使用原点/截距规则快速排除错误选项。

**例题:** 当$x$和$y$均取正值时，说明$y$与$x$的平方成反比例的图像的核心特征。

1. 首先写出完整等式：$y = \frac{k}{x^2}$，其中$k$为正数。
2. 检查截距：当$x=0$时，$y$无定义，因此图像没有y轴截距。当$y=0$时，$1/x^2 = 0$没有解，因此图像也没有x轴截距。
3. 检查形状：随着$x$增大，$y$减小，因此该图像是位于第一象限的向下倾斜曲线。
4. 最终核心特征：无截距，曲线单调递减，趋近于x轴和y轴但永不与之相交。

**概念自测**

1. Which of the following graphs represents $y \propto x^3$?

   - Straight line through origin
   - Upward curving line through origin
   - Decreasing curve with no intercepts
   - Horizontal line

   *答案:* Upward curving line through origin

   *解析:* Correct: All direct power relationships other than linear give upward curving lines through the origin.

> **考试提示:** 如果题目要求你绘制比例图像的草图，请务必标注坐标轴，并在曲线两端添加箭头以表示曲线无限延伸。

*计算器:* allowed

## 常见错误

- **错误做法:** Forgetting to introduce the constant of proportionality $k$ and jumping straight to calculating values.
  - 原因: Examiners award 1 mark for explicitly stating the proportion equation with $k$, so you lose marks even if your final answer is correct.
  - 正确做法: Always write $y = kf(x)$ (direct) or $y = k/f(x)$ (inverse) immediately after your initial proportion statement.
- **错误做法:** Mixing up direct and inverse proportion, e.g., writing $y = kx$ for "y is inversely proportional to x".
  - 原因: This leads to completely incorrect values for $k$ and all subsequent answers.
  - 正确做法: Underline the phrase "directly proportional" or "inversely proportional" in the question, and add the reciprocal for inverse statements immediately.
- **错误做法:** Using rounded decimal values for $k$ instead of exact fractions or integers.
  - 原因: Rounding $k$ early leads to accumulated errors in final answers, which may fall outside the allowed tolerance range.
  - 正确做法: Keep $k$ exact through all calculations, only rounding the final answer if explicitly asked.
- **错误做法:** Assuming all direct proportion graphs are straight lines.
  - 原因: Only $y \propto x$ gives a straight line; $y \propto x^2, x^3, \sqrt{x}$ give curved lines through the origin.
  - 正确做法: Check the power of $x$ in the relationship to determine if the graph is linear or curved.
- **错误做法:** Drawing inverse proportion graphs that touch the x or y axis.
  - 原因: For inverse relationships, $x$ or $y$ can never be zero (division by zero is undefined), so intercepts are impossible.
  - 正确做法: Draw inverse proportion curves approaching but never touching the axes, with arrows at the ends.

## 速查表

| Relationship Type | Proportion Statement | Equation with k | Key Graph Features |
| --- | --- | --- | --- |
| Direct Linear | $y \propto x$ | $y = kx$ | Straight line through origin, gradient k |
| Direct Square | $y \propto x^2$ | $y = kx^2$ | Upward curving line through origin |
| Direct Cube | $y \propto x^3$ | $y = kx^3$ | Steeper upward curving line through origin |
| Direct Square Root | $y \propto \sqrt{x}$ | $y = k\sqrt{x}$ | Flattening upward curve through origin |
| Inverse Linear | $y \propto 1/x$ | $y = k/x$ | Decreasing curve, no intercepts |
| Inverse Square | $y \propto 1/x^2$ | $y = k/x^2$ | Steeper decreasing curve, no intercepts |
| Inverse Cube | $y \propto 1/x^3$ | $y = k/x^3$ | Very steep decreasing curve, no intercepts |
| Inverse Square Root | $y \propto 1/\sqrt{x}$ | $y = k/\sqrt{x}$ | Flattening decreasing curve, no intercepts |

## 下一步

Now that you have mastered proportion for Edexcel IGCSE Maths A Higher Tier, you can apply these skills to solve real-world context problems involving rates, area, volume, and scaling that often appear alongside proportion questions. Proportion questions typically make up 5-8% of the higher tier paper, so consistent practice of past paper questions will help you avoid common errors and secure full marks on this topic. You should also practice combining proportion with rearranging formulae and graph interpretation to tackle the most challenging 4-5 mark exam questions.

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来自 [OwlsPrep](https://www.owlsprep.com) —— A-Level / IB / AP / IGCSE 免费学习指南，依据官方考纲编写。原页面：https://www.owlsprep.com/zh/study/edexcel-igcse-math-a-s2-proportion/