📚 Direct Proportion | 正比例
In mathematics, proportion describes how two quantities change in relation to each other. Direct proportion is one of the most fundamental concepts in the IGCSE Edexcel syllabus, appearing in everything from simple recipes to currency conversions. Mastering it builds a strong foundation for more advanced topics like linear functions and rational equations. This guide will walk you through the definition, key formula, graphs, tables, and a wide range of exam-style applications.
在数学中，比例描述了两个量如何相对变化。正比例是 IGCSE Edexcel 大纲中最基本的概念之一，从简单的食谱到货币兑换都会出现。掌握好它，能为线性函数和有理方程等更高级的课题打下坚实基础。本指南将带你逐步了解定义、核心公式、图形、表格以及各种考试类型的应用题。
1. What is Direct Proportion? | 什么是正比例？
Two quantities are said to be in direct proportion if an increase in one causes a proportional increase in the other, and a decrease causes a proportional decrease. This means the ratio between them remains constant. In everyday language we say 'as one doubles, the other doubles; if one is halved, the other is halved.' For example, the more hours you work at a fixed hourly rate, the more you earn — doubling the hours exactly doubles the pay.
如果两个量中一个增加引起另一个成比例增加，一个减少引起另一个成比例减少，那么它们就成正比。这意味着它们之间的比值保持不变。日常生活中我们说“一个翻倍，另一个也翻倍；一个减半，另一个也减半”。例如，按固定小时费率工作的时间越长，收入就越多——工作时间翻倍，报酬也正好翻倍。
Direct proportion is often introduced in IGCSE through real-world scenarios: buying apples at a constant price per kilogram, distance travelled at a steady speed with constant time, or the cost of fuel and the number of litres pumped. All these obey the rule that when one variable is multiplied by a number, the other is multiplied by the same number.
IGCSE 中通常通过现实场景引入正比例：以每千克固定价格购买苹果、以恒定速度行驶固定时间的路程、或者燃油费用与加油升数。这些情景都遵循同一个规则：当一个变量乘以某个数时，另一个变量也乘以同一个数。
2. The Mathematical Definition | 数学定义
Here k is called the constant of proportionality or the constant of variation. The phrase 'y is directly proportional to x' is often written using the proportionality symbol: y ∝ x. However, to solve problems we always convert to the equation form y = kx.
从形式上讲，如果存在一个非零常数 k 使得 y = kx，那么 y 与 x 成正比。这里的 k 被称为比例常数或变分常数。短语“y 与 x 成正比” 通常用比例符号表示：y ∝ x。然而，在解决问题时，我们总是将其转换为等式形式 y = kx。
In IGCSE Edexcel exam questions you may see the wording 'y is proportional to x' or 'y varies directly as x' ou need to be comfortable switching between the verbal statement and the algebraic equation.
在 IGCSE Edexcel 试题中，你可能会看到“y 与 x 成比例”或“y 随 x 直接变化”。这两种表达意思完全相同：存在某个常数 k，使得 y = kx。你需要熟练地在文字叙述和代数方程之间进行转换。
3. The Core Formula: y = kx | 核心公式：y = kx
irect proportion. k can be any real number except zero. If k > 0, both quantities increase together; if k < 0, one decreases as the other increases, but the proportional relationship still holds. In most IGCSE contexts, k is positive because the quantities being modelled are positive (mass, cost, length, etc.).
方程 y = kx 是正比例的支柱。k 可以是除零以外的任意实数。若 k > 0，两个量同向增加；若 k < 0，一增一减，但比例关系仍然成立。在大多数 IGCSE 情境中，k 为正，因为所建模的量都是正数（质量、费用、长度等）。
y = kx
y ÷ x is always constant for directly proportional quantities. When given pairs of values, you can test for direct proportion by checking whether y/x gives the same number each time.
将方程变形可得 k = y/x。这表明对于成正比的量，y ÷ x 的比值始终恒定。给出一对数值时，你可以通过检验 y/x 是否每次都得到同一个数值来判断是否成正比。
4. Finding the Constant of Proportionality (k) | 求比例常数 (k)
ero. Substitute the values into k = y ÷ x. For instance, if you know that 5 notebooks cost £ k = 7.50 / 5 = 1.50. The constant of proportionality is 1.50, meaning each notebook costs £1.50.
要找到 k，你需要一组完整的非零对应值 (x, y)。将其代入 k = y ÷ x。例如，已知 5 本笔记本售价 7.50 英镑，则 k = 7.50 / 5 = 1.50。比例常数是 1.50，意味着每本笔记本 1.50 英镑。
g the equation y = 3x. You then use this equation to find y for a new x, or x for a new y.
在试题中，你往往需要先求出 k 才能回答题目主体。一旦知道了 k，就可以写出连接 y 和 x 的完整方程。例如，若 y 与 x 成正比，且当 x = 8 时 y = 24，则 k = 24 ÷ 8 = 3，得出方程 y = 3x。然后你就可以用这个方程为新的 x 值求 y，或者为新的 y 值求 x。
5. Direct Proportion and Tables of Values | 正比例与数值表
When data is presented in a table, you can identify direct proportion by calculating the ratio y/x for every column. If all ratios are equal, the variables are directly proportional. Consider the table:
当数据以表格形式呈现时,你可以通过计算每一列的比值 y/x 来判断是否成正比。如果所有比值相等，那么变量就成正比。请看下表：
| x | 2 | 5 | 10 | 12 |
|---|---|---|---|---|
| y | 7 | 17.5 | 35 | 42 |
is y = 3.5x. Any missing value can be found by multiplying x by 3.5 or dividing y by 3.5.
分别计算：7/2 = 3.5, 17.5/5 = 3.5, 35/10 = 3.5, 42/12 = 3.5。由于比值始终为 3.5，因此关系为 y = 3.5x。任何缺失值都可以通过 x 乘以 3.5 或 y 除以 3.5 求得。
If the ratios were not equal, the relationship would not be direct proportion. Sometimes exam questions ask you to complete a table using the proportional relationship; you apply the constant multiplier to x or divide y accordingly.
若比值不相等，则不是正比例关系。有些试题会要求你利用比例关系补全表格；你只需将常数乘数应用于 x，或对 y 做相应除法。
6. Graphs of Direct Proportion | 正比例的图形
ssing through the origin (0,0). The slope or gradient of the line is exactly the constant k. This gives a very quick visual check: if a straight line goes through the origin, the variables are in direct proportion. If it is a straight line but does not pass through the origin, the relationship is linear but not directly proportional.
y = kx 的图像是一条通过原点 (0,0) 的直线。该直线的斜率或梯度恰好就是常数 k。这提供了一个非常快捷的视觉判断：如果一条直线通过原点，变量就成正比。如果是直线但不通过原点，则关系是线性但不是正比例。
Graph: a straight line through (0,0) with gradient k
sketch the graph for a given k, ensuring the line passes through the origin and has the correct steepness.
在 IGCSE 考试中，可能会给出一张图，要求你通过选取线上一点来求常数 k。选择容易读取坐标的点 (x,y)，然后计算 k = y/x。也可能要求你为给定的 k 画出草图，确保直线过原点且倾斜度正确。
7. Using Direct Proportion to Solve Problems | 使用正比例解决问题
There are two main methods for solving direct proportion problems: the constant method and the unitary method. In the constant method, you find k first using a known pair, then apply it to find unknown values. In the unitary method, you find the value for one unit of x first, then scale up. Both are valid and will give identical results.
解决正比例问题主要有两种方法：常数法和单位法。常数法是先用一对已知值求出 k，然后应用它求未知值。单位法是先求出 x 的一个单位对应的值，然后按比例放大。两种方法都有效且结果一致。
Example: If 8 metres of fabric cost £28, how much will 12 metres cost? Using the constant method, k = 28/8 = 3.5 (£ per metre), then cost for 12 m = 3.5 × 12 = £42. Using the unitary method, cost per metre = £28 ÷ 8 = £3.50, then multiply by 12: £3.50 × 12 = £42. The unitary method is especially intuitive for currency and pricing questions.
例如：如果 8 米布料售价 28 英镑，那么 12 米需要多少钱？用常数法，k = 28/8 = 3.5（英镑每米），然后 12 米的价格 = 3.5 × 12 = 42 英镑。用单位法，每米价格 = 28 ÷ 8 = 3.50 英镑，再乘以 12：3.50 × 12 = 42 英镑。单位法在货币和定价问题中尤其直观。
8. Real-Life Application: Currency Conversion | 实际应用：货币兑换
o the amount in another, with the exchange rate serving as the constant k. For example, if £1 = $1.25, then dollars = 1.25 × pounds. Here k = 1.25. If you have £200, you get 1.25 × 200 = $250. Reversing the conversion involves dividing by k.
货币兑换是 IGCSE 中经典的直接比例场景。一种货币的金额与另一种货币的金额成正比，汇率充当常数 k。例如，若 £1 = $1.25，则美元 = 1.25 × 英镑。此时 k = 1.25。如果你有 200 英镑，你会得到 1.25 × 200 = 250 美元。逆向兑换则需要除以 k。
Exam questions may also involve a commission or fixed fee. In such cases, the relationship is no longer pure direct proportion because of the added constant. Be careful to read whether the transaction includes a flat charge — direct proportion applies only when the graph passes through the origin, meaning zero pounds gives zero dollars with no extra fees.
考试题目可能涉及手续费或固定费用。这类情况下，由于加入了常数，关系不再是纯正比例。要仔细读题，看交易是否包含固定费用——只有在图像过原点、即零英镑兑换零美元且无额外费用的前提下，才适用正比例。
9. Real-Life Application: Recipes and Scaling | 实际应用：食谱与缩放
eeding 200 × 1.5 = 300 g. The constant k for flour to servings is 200/4 = 50 g per person.
食谱是正比例的绝佳示例。每种配料的用量与用餐人数成正比。如果一份 4 人份的食谱需要 200 克面粉，那么 6 人份就按比例系数 6/4 = 1.5 放大，需要 200 × 1.5 = 300 克。面粉与人份的比例常数 k 为 200/4 = 50 克/人。
This type of question often appears in IGCSE as part of a functional maths context. You may be asked to adapt recipes for a different number of people, or to find how many people can be served with a certain amount of an ingredient. Remember to keep the ratio of ingredients constant — all scale by the same factor.
这类问题常出现在 IGCSE 中，作为功能性数学情境的一部分。你可能需要为不同人数调整食谱，或者根据某种配料的现有数量计算能供应多少人。请记住保持配料比例恒定——所有配料均按相同的倍数缩放。
10. Direct Proportion vs. Inverse Proportion | 正比例与反比例
irect proportion, as x increases, y increases; in inverse proportion, as x increases, y decreases. Students often confuse the two, especially when reading 'y is proportional to ...' statements. Remember that 'directly proportional' always leads to multiplication by k, while 'inversely proportional' leads to division.
正比例意味着 y = kx，而反比例意味着 y = k/x。正比例中 x 增大 y 也增大；反比例中 x 增大 y 反而减小。学生经常混淆二者，尤其是在阅读“y 与……成比例”这类表述时。请记住，“成正比”总是得出乘以 k 的形式，而“成反比”则得出除以 x 的形式。
Edexcel IGCSE exams often include one question that mixes up the vocabulary, asking you to decide whether a relationship is direct or inverse proportion from a table, graph, or description. Check the ratio y/x for direct proportion and the product xy for inverse proportion. If y/x is constant, it is direct; if xy is constant, it is inverse.
Edexcel IGCSE 考试常会在词汇上做文章，要求你根据表格、图形或文字描述判断一个关系是正比例还是反比例。检查比值 y/x 是否恒定来判断正比例，检查乘积 xy 是否恒定来判断反比例。若 y/x 恒定则为正比，若 xy 恒定则为反比。
11. Common Mistakes in Direct Proportion | 正比例的常见错误
nships of the exact form y = kx (with no added constant term) represent direct proportion.
一个非常常见的错误是认为所有线性关系都是正比例。方程 y = 2x + 3 是线性关系但不是正比例，因为它图像不通过原点，且 y/x 不是常数。只有形式严格为 y = kx（没有附加常数项）的关系才代表正比例。
Another mistake is misidentifying k when x and y are given in the 'wrong' order or when units are different. Always compute k = y/x with the dependent variable y as the numerator. If a question says 'the number of cakes, c, is directly proportional to the mass of flour, m', then write c = k m, so k = c/m. Don't inadvertently invert the fraction.
另一个错误是当 x 和 y 的顺序“反了”或者单位不同时，错误地识别 k。务必以因变量 y 作为分子计算 k = y/x。如果题目表述为“蛋糕数量 c 与面粉质量 m 成正比”，那么应写成 c = k m，因此 k = c/m。注意不要不小心将分数颠倒了。
Students also sometimes forget to keep the same factor when scaling up several quantities in a recipe. All ingredients must be multiplied by the same scale factor. Doubling the people means doubling every ingredient, not adding a fixed amount.
学生有时会在按照食谱放大几种配料时忘记保持同一倍数。所有配料必须乘以相同的比例系数。人数翻倍意味着每种配料都要翻倍，而不是增加一个固定数量。
12. Exam Tips and Practice Strategy | 应试技巧与练习策略
which, then set up y = kx with the correct letters. Always show your substitution clearly.
在 IGCSE Edexcel 中，正比例题目通常占 2 到 5 分，结构一般为：第(a)部分求 k，第(b)部分用公式求 y，第(c)部分给 y 求 x。仔细审题，判断哪个变量依赖于哪个，然后用正确的字母建立 y = kx。务必清晰地展示代入过程。
When drawing graphs, label axes, use a ruler, and extend the line through the origin. If asked to find k from a graph, pick a point that lies exactly on the grid intersections to avoid reading errors. Double-check your arithmetic when finding k: a small error early on affects all subsequent parts.
绘制图形时，记得标记坐标轴、使用直尺，并将直线延长通过原点。如果要求根据图形求 k，选取一个恰好落在网格交点上的点，以避免读数误差。求 k 时要再次核对计算：开头的小错误会影响后续所有部分。
h solid understanding of y = kx, you can handle any direct proportion problem confidently.
最后，利用 Edexcel IGCSE 数学 A 和 B 的历年真题进行练习。留意那些将正比例与单位换算、地图与比例尺，甚至时间固定下的速度-距离-时间关系相结合的题目。只要扎实理解 y = kx，你就能自信地应对任何正比例问题。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图