Direct Proportion | 正比例

📚 Direct Proportion | 正比例

In mathematics, when two quantities are directly proportional, they increase or decrease together at the same rate. If one doubles, the other doubles; if one halves, the other halves. This relationship is one of the most fundamental ideas in algebra and appears constantly in science, economics, and everyday life.

在数学中，当两个量成正比时，它们会以相同的比率同时增加或减少。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。这种关系是代数中最基本的概念之一，在科学、经济学和日常生活中反复出现。

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

Two quantities $y$ and $x$ are directly proportional if the ratio between them stays constant. In simpler terms, $y$ is always the same multiple of $x$. For example, if you buy apples at a fixed price per apple, the total cost is directly proportional to the number of apples.

两个量 $y$ 和 $x$ 成正比例，意味着它们之间的比值保持不变。简单来说，$y$ 始终是 $x$ 的同一个倍数。例如，如果你按固定单价购买苹果，那么总价与苹果数量成正比。

We write this as $y \propto x$, which is read as “$y$ is directly proportional to $x$”. The symbol $\propto$ means “is proportional to”.

我们将其写作 $y \propto x$，读作”$y$ 与 $x$ 成正比”。符号 $\propto$ 表示”成正比”。

When two quantities are directly proportional, the graph of $y$ against $x$ is a straight line that passes through the origin. This is a key feature you must be able to recognise in examinations.

当两个量成正比时，以 $y$ 对 $x$ 作图得到的图像是一条经过原点的直线。这是你必须在考试中能够识别的一个关键特征。

2. The Constant of Proportionality | 比例常数

The statement $y \propto x$ can be turned into an equation using a constant. We write $y = kx$, where $k$ is a non-zero number called the constant of proportionality.

语句 $y \propto x$ 可以借助一个常数转化为方程。我们写成 $y = kx$，其中 $k$ 是一个非零的数，称为比例常数。

To find $k$, rearrange the equation as $k = \frac{y}{x}$. This means that for every pair of corresponding values, the quotient $y \div x$ must be the same.

为了求出 $k$，我们将方程变形为 $k = \frac{y}{x}$。这意味着对于每一对对应的数值，$y \div x$ 的商必须相同。

y = kx ⇔ k = y ÷ x ⇔ y ÷ x = k (constant)

Example: Suppose $y \propto x$ and $y = 8$ when $x = 2$. Then $k = 8 \div 2 = 4$, so the equation is $y = 4x$. If $x = 7$, then $y = 4 \times 7 = 28$.

示例：设 $y \propto x$，且当 $x = 2$ 时 $y = 8$。则 $k = 8 \div 2 = 4$，因此方程为 $y = 4x$。若 $x = 7$，则 $y = 4 \times 7 = 28$。

3. Graphs of Direct Proportion | 正比例的图像

When you plot two directly proportional quantities, you will always get a straight line through the origin $(0, 0)$. The gradient of this line is exactly the constant of proportionality $k$.

当你绘制两个成正比量的图像时，总会得到一条经过原点 $(0, 0)$ 的直线。这条直线的斜率恰好就是比例常数 $k$。

For example, $y = 4x$ has gradient 4, so the line rises 4 units for every 1 unit moved to the right. A larger value of $k$ means a steeper line.

例如，$y = 4x$ 的斜率为 4，即直线每向右移动 1 个单位就上升 4 个单位。$k$ 的值越大，直线越陡。

Important: Not every straight line represents direct proportion. A line such as $y = 3x + 2$ does not pass through the origin, so $y$ and $x$ are not directly proportional. The equation of a directly proportional relationship must have no constant term added.

重要提示：并非每一条直线都表示正比例。像 $y = 3x + 2$ 这样的直线不经过原点，因此 $y$ 与 $x$ 不成正比例。正比例关系的方程中不能含有额外加的常数项。

y = kx passes through (0, 0) | y = kx + c does not (unless c = 0)

4. Solving Direct Proportion Problems | 解正比例问题

Examination questions often give you one pair of corresponding values and ask you to find another value. The standard method is to find $k$ first, then substitute the new value.

考试题目通常会给出一对对应的数值，然后要求你求另一个值。标准的解法是先求出 $k$，再将新的数值代入。

Step-by-step worked example:

分步解答示例：

Given that $y \propto x$, and $y = 15$ when $x = 3$, find $y$ when $x = 9$.

已知 $y \propto x$，且当 $x = 3$ 时 $y = 15$，求当 $x = 9$ 时 $y$ 的值。
Find the constant: $k = y \div x = 15 \div 3 = 5$. Therefore $y = 5x$.

求常数：$k = y \div x = 15 \div 3 = 5$。因此 $y = 5x$。
Substitute $x = 9$: $y = 5 \times 9 = 45$.

代入 $x = 9$：$y = 5 \times 9 = 45$。

You can check your answer by noticing that $x$ was multiplied by 3, so $y$ must also be multiplied by 3: $15 \times 3 = 45$.

你可以通过观察来检验答案：$x$ 乘以了 3，所以 $y$ 也必须乘以 3：$15 \times 3 = 45$。

5. Direct Proportion with Squares and Cubes | 平方与立方正比例

In the IGCSE syllabus, you also need to handle relationships such as $y \propto x^2$, $y \propto x^3$, and $y \propto \sqrt{x}$. Translating these into equations gives $y = kx^2$, $y = kx^3$, and $y = k\sqrt{x}$.

在 IGCSE 教学大纲中，你还需要处理 $y \propto x^2$、$y \propto x^3$ 以及 $y \propto \sqrt{x}$ 这类关系。将它们转化为方程得到 $y = kx^2$、$y = kx^3$ 和 $y = k\sqrt{x}$。

Example with a square: Given that $y \propto x^2$ and $y = 20$ when $x = 2$, find $y$ when $x = 5$.

平方关系的示例：已知 $y \propto x^2$，且当 $x = 2$ 时 $y = 20$，求当 $x = 5$ 时 $y$ 的值。

Since $y = kx^2$, substitute: $20 = k \times 2^2 = 4k$. Thus $k = 5$, giving $y = 5x^2$. When $x = 5$, $y = 5 \times 25 = 125$.

因为 $y = kx^2$，代入得：$20 = k \times 2^2 = 4k$。因此 $k = 5$，得到 $y = 5x^2$。当 $x = 5$ 时，$y = 5 \times 25 = 125$。

Notice that when $x$ is multiplied by 2.5, $y$ is multiplied by $2.5^2 = 6.25$. This is the key difference between linear proportion and quadratic proportion.

注意，当 $x$ 乘以 2.5 时，$y$ 乘以 $2.5^2 = 6.25$。这就是线性正比例与平方正比例之间的关键区别。

y ∝ x² ⇒ y = kx² | y ∝ x³ ⇒ y = kx³ | y ∝ √x ⇒ y = k√x

6. Real-Life Applications | 实际应用

Direct proportion appears in many practical contexts. You should be comfortable setting up an equation from a written problem.

正比例在许多实际情境中都会出现。你应该能够熟练地从文字问题中建立方程。

Currency conversion: The amount in another currency is directly proportional to the amount exchanged. If £1 = $1.25, then $ amount $= 1.25 \times$ £ amount.

货币兑换：兑换后的外币金额与兑换的本币金额成正比。若 £1 = $1.25，则美元金额 = 1.25 × 英镑金额。
Distance and time: At a constant speed, distance travelled is directly proportional to time taken. $d = vt$, where $v$ is the speed.

距离与时间：在匀速运动中，行驶距离与所用时间成正比。$d = vt$，其中 $v$ 是速度。
Physics – Hooke’s law: The extension of a spring is directly proportional to the force applied, $F = kx$.

物理——胡克定律：弹簧的伸长量与施加的力成正比，$F = kx$。
Recipes: The quantity of each ingredient is directly proportional to the number of servings.

食谱：每种配料的用量与用餐人数成正比。

When solving a word problem, first identify the two quantities, decide whether the relationship is linear, quadratic, or cubic, then find $k$ using the data given.

解文字题时，首先确定两个量，判断关系是线性、平方还是立方，然后利用给定数据求出 $k$。

7. Common Misconceptions | 常见错误概念

Students often make preventable mistakes with direct proportion. Here are the most frequent ones, with corrections.

学生在正比例问题上常常犯一些本可避免的错误。以下是最常见的几个，并附上纠正方法。

Confusing direct proportion with inverse proportion. Inverse proportion has the form $y = k \div x$; its graph is a curve, not a straight line through the origin.

混淆正比例与反比例。反比例的形式为 $y = k \div x$，其图像是一条曲线，而不是经过原点的直线。
Assuming every straight line is proportional. A line with a non-zero intercept, such as $y = 2x + 3$, is not direct proportion.

认为所有直线都是正比例。截距不为零的直线，例如 $y = 2x + 3$，不是正比例。
Using addition instead of multiplication. In direct proportion, values scale by multiplication, not by adding the same amount.

用加法代替乘法。在正比例中，数值通过乘法来缩放，而不是加上同一个量。
Forgetting that $x^2$ affects $k$ differently. When $y \propto x^2$, you must square $x$ before comparing with $y$.

忘记 $x^2$ 对 $k$ 的影响不同。当 $y \propto x^2$ 时，必须先对 $x$ 平方，再与 $y$ 比较。

Always check that your final equation makes sense: when $x = 0$, $y$ must also be 0 for a direct proportion relationship.

始终检查你得到的最终方程是否合理：对于正比例关系，当 $x = 0$ 时，$y$ 也必须是 0。

8. Recognising Direct Proportion from Tables | 从表格中识别正比例

IGCSE questions sometimes present a table of values and ask you to determine whether two quantities are directly proportional. The key test is checking whether $y \div x$ gives the same constant for every row.

IGCSE 题目有时会给出一个数值表格，要求你判断两个量是否成正比。关键的检验方法是检查每一行的 $y \div x$ 是否都得到同一个常数。

Consider the following table:

考虑下面的表格：

x	1	2	3	4
y	3.5	7	10.5	14

Calculate $y \div x$ for each column: $3.5 \div 1 = 3.5$, $7 \div 2 = 3.5$, $10.5 \div 3 = 3.5$, $14 \div 4 = 3.5$. Since the ratio is constant, $y \propto x$ with $k = 3.5$.

对每一列计算 $y \div x$：$3.5 \div 1 = 3.5$，$7 \div 2 = 3.5$，$10.5 \div 3 = 3.5$，$14 \div 4 = 3.5$。由于比值恒定，所以 $y \propto x$，且 $k = 3.5$。

If the ratio changes from row to row, the quantities are not directly proportional. In that case, you might need to test another power, such as $y \div x^2$.

如果各行之间的比值不同，则这两个量不构成正比例。在这种情况下，你可能需要检验其他幂次，例如 $y \div x^2$。

9. Exam-Style Question | 考试题型演练

Let us work through a typical Edexcel IGCSE question involving direct proportion with a square root.

让我们一起来解答一道典型的 Edexcel IGCSE 题目，其中涉及平方根的正比例关系。

Question: $T$ is directly proportional to $\sqrt{n}$. Given that $T = 12$ when $n = 9$, calculate the value of $T$ when $n = 16$.

题目：$T$ 与 $\sqrt{n}$ 成正比。已知当 $n = 9$ 时 $T = 12$，求当 $n = 16$ 时 $T$ 的值。

Solution: Write $T = k\sqrt{n}$. Substitute $T = 12$, $n = 9$:

解答：写出 $T = k\sqrt{n}$。代入 $T = 12$，$n = 9$：

12 = k√9 = 3k ⇒ k = 4 所以 T = 4√n

Now substitute $n = 16$: $T = 4 \times \sqrt{16} = 4 \times 4 = 16$.

现在代入 $n = 16$：$T = 4 \times \sqrt{16} = 4 \times 4 = 16$。

Always show the equation after finding $k$; method marks are often awarded for this step even if your final arithmetic is wrong.

求出 $k$ 后一定要写出方程；这一步通常能获得步骤分，即使你最后的计算结果有误。

10. Key Points Summary | 重点总结

Direct proportion is a short but highly examined topic in Edexcel IGCSE Mathematics. Keep the following points in mind.

正比例是 Edexcel IGCSE 数学中篇幅不长但考试频率很高的一个专题。请牢记以下要点。

$y \propto x$ means $y = kx$, and the graph is a straight line through the origin.

$y \propto x$ 表示 $y = kx$，其图像是经过原点的直线。
$k = y \div x$ is constant; use one pair of values to find it, then substitute the other value.

$k = y \div x$ 是常数；用一对数值求出 $k$，再代入另一个数值。
Handle squared and cubed relationships correctly: $y = kx^2$, $y = kx^3$, $y = k\sqrt{x}$.

正确处理平方和立方关系：$y = kx^2$、$y = kx^3$、$y = k\sqrt{x}$。
Check tables by verifying a constant ratio; if $y \div x$ is not constant, try $y \div x^2$.

通过检验恒定比值来检查表格；如果 $y \div x$ 不是常数，尝试 $y \div x^2$。
In word problems, identify quantities, set up the equation, find $k$, then answer the question with correct units if needed.

在文字题中，先确定量，建立方程，求出 $k$，然后回答问题，必要时带上正确的单位。

Practise translating $\propto$ into $= k$ and back again, and you will find direct proportion questions very predictable in the exam.

勤加练习将 $\propto$ 转化为 $= k$ 以及逆向转化，你就会发现正比例题目在考试中非常”套路化”。

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Direct Proportion | 正比例