📚 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 ∝ x, which is read as "y is directly proportional to x". The symbol ∝ means "is proportional to".
我们将其写作 y ∝ x，读作"y 与 x 成正比"。符号 ∝ 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 ∝ 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 ∝ 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 = yx. This means that for every pair of corresponding values, the quotient y ÷ x must be the same.
为了求出 k，我们将方程变形为 k = yx。这意味着对于每一对对应的数值，y ÷ x 的商必须相同。
y = kx ⇔ k = y ÷ x ⇔ y ÷ x = k (constant)
Example: Suppose y ∝ x and y = 8 when x = 2. Then k = 8 ÷ 2 = 4, so the equation is y = 4x. If x = 7, then y = 4 × 7 = 28.
示例：设 y ∝ x，且当 x = 2 时 y = 8。则 k = 8 ÷ 2 = 4，因此方程为 y = 4x。若 x = 7，则 y = 4 × 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 ∝ x, and y = 15 when x = 3, find y when x = 9.
已知 y ∝ x，且当 x = 3 时 y = 15，求当 x = 9 时 y 的值。
Find the constant: k = y ÷ x = 15 ÷ 3 = 5. Therefore y = 5x.
求常数：k = y ÷ x = 15 ÷ 3 = 5。因此 y = 5x。
Substitute x = 9: y = 5 × 9 = 45.
代入 x = 9：y = 5 × 9 = 45。
You can check your answer by noticing that x was multiplied by 3, so y must also be multiplied by 3: 15 × 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 × 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 ∝ x2, y ∝ x3, and y ∝ √x. Translating these into equations gives y = kx2, y = kx3, and y = k√x.
在 IGCSE 教学大纲中，你还需要处理 y ∝ x2、y ∝ x3 以及 y ∝ √x 这类关系。将它们转化为方程得到 y = kx2、y = kx3 和 y = k√x。
Example with a square: Given that y ∝ x2 and y = 20 when x = 2, find y when x = 5.
平方关系的示例：已知 y ∝ x2，且当 x = 2 时 y = 20，求当 x = 5 时 y 的值。
Since y = kx2, substitute: 20 = k × 22 = 4k. Thus k = 5, giving y = 5x2. When x = 5, y = 5 × 25 = 125.
因为 y = kx2，代入得：20 = k × 22 = 4k。因此 k = 5，得到 y = 5x2。当 x = 5 时，y = 5 × 25 = 125。
Notice that when x is multiplied by 2.5, y is multiplied by 2.52 = 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 2.52 = 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 × £ 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 ÷ 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 ÷ 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 x2 affects k differently. When y ∝ x2, you must square x before comparing with y.
忘记 x2 对 k 的影响不同。当 y ∝ x2 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 ÷ 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 ÷ 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 ÷ x for each column: 3.5 ÷ 1 = 3.5, 7 ÷ 2 = 3.5, 10.5 ÷ 3 = 3.5, 14 ÷ 4 = 3.5. Since the ratio is constant, y ∝ x with k = 3.5.
对每一列计算 y ÷ x：3.5 ÷ 1 = 3.5，7 ÷ 2 = 3.5，10.5 ÷ 3 = 3.5，14 ÷ 4 = 3.5。由于比值恒定，所以 y ∝ 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 ÷ x2.
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 ÷ x2。
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 √n. Given that T = 12 when n = 9, calculate the value of T when n = 16.
题目：T 与 √n 成正比。已知当 n = 9 时 T = 12，求当 n = 16 时 T 的值。
Solution: Write T = k√n. Substitute T = 12, n = 9:
解答：写出 T = k√n。代入 T = 12，n = 9：
12 = k√9 = 3k ⇒ k = 4 所以 T = 4√n
Now substitute n = 16: T = 4 × √16 = 4 × 4 = 16.
现在代入 n = 16：T = 4 × √16 = 4 × 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 ∝ x means y = kx, and the graph is a straight line through the origin.
y ∝ x 表示 y = kx，其图像是经过原点的直线。
k = y ÷ x is constant; use one pair of values to find it, then substitute the other value.
k = y ÷ 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 = kx2, y = kx3, y = k√x.
正确处理平方和立方关系：y = kx2、y = kx3、y = k√x。
Check tables by verifying a constant ratio; if y ÷ x is not constant, try y ÷ x2.
通过检验恒定比值来检查表格；如果 y ÷ x 不是常数，尝试 y ÷ x2。
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 ∝ into = k and back again, and you will find direct proportion questions very predictable in the exam.
勤加练习将 ∝ 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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