比例（爱德思IGCSE数学A 高阶层级）

Q: 我是否必须在解题过程中写出常数？

是的，爱德思的评卷官会为引入并写出完整比例方程的步骤明确给出1分，即使你可以心算得出最终答案。

爱德思国际GCSE数学A· 2.5· 25 分钟阅读

1. 正比例关系★★☆☆☆Higher 专属⏱ 8 min

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📘 定义

正比例

$y \propto f (x)$

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： $y = k f (x)$ ，其中 $k$ 为非零的比例常数。

例:

If $y$ is directly proportional to the square of $x$ , then $y = k x^{2}$

Allowed direct proportion relationships: $y \propto x$ , $y \propto x^{2}$ , $y \propto x^{3}$ , $y \propto x$
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， $C = £ 2.40$ 。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 $k$ ： $2.40 = k (4)^{2} = 16 k$ ，因此 $k = 2.40/16 = 0.15$
4
写出完整等式： $C = 0.15 r^{2}$
5
代入 $r = 7$ ： $C = 0.15 (49) = 7.35$
6
最终答案： $£ 7.35$

Exam tip:

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。

2. 反比例关系★★★☆☆Higher 专属⏱ 8 min

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📘 定义

反比例

$y \propto \frac{1}{f ( x )}$

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。

例:

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

Allowed inverse proportion relationships: $y \propto 1/ x$ , $y \propto 1/ x^{2}$ , $y \propto 1/ x^{3}$ , $y \propto 1/ x$
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： $t \propto \frac{1}{p}$
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 $k$ ： $90 = \frac{k}{16} = \frac{k}{4}$ ，因此 $k = 360$
4
写出完整等式： $t = \frac{360}{p}$
5
代入 $t = 60$ ： $60 = \frac{360}{p}$ → $p = 6$ → $p = 36$
6
最终答案：36台水泵

Exam tip:

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。

3. 比例方程与图像的关联★★★☆☆Higher 专属⏱ 7 min

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📐 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。

✓ 快速检测

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
显示答案
1 —
Correct: All direct power relationships other than linear give upward curving lines through the origin.

Exam tip:

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。

4. 常见陷阱

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 $y = k f (x)$ (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 $y = k x$ 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; $y \propto x^{2}, x^{3}, x$ 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.

5. 速查表

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

6. 常见问题

我是否必须在解题过程中写出常数 $k$ ？

是的，爱德思的评卷官会为引入 $k$ 并写出完整比例方程的步骤明确给出1分，即使你可以心算得出最终答案。

我如何区分正比例图像和反比例图像？

所有正比例图像都经过原点(0,0)。所有反比例图像都是递减曲线，且永远不会与x轴或y轴相交，因为除以零是无定义的。

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.