📚 Proportion | 比例
Proportion is a fundamental topic in mathematics that describes how two quantities change together. In Edexcel IGCSE Mathematics, you will solve problems using direct proportion, inverse proportion, and the constant of proportionality.
比例是数学中的一个基础主题，它描述了两个量如何一同变化。在爱德思 IGCSE 数学中，你将使用正比例、反比例和比例常数来解决问题。
1. What is Proportion? | 什么是比例？
Proportion means that two quantities are related by a constant factor. When one quantity changes, the other changes in a predictable way. There are two main types: direct proportion and inverse proportion.
比例意味着两个量通过一个常数因子相互关联。当一个量变化时，另一个量以可预测的方式变化。有两种主要类型：正比例和反比例。
For example, if you buy apples at a fixed price per kilogram, the total cost is directly proportional to the mass. If you travel at a constant speed, the time taken is inversely proportional to the speed for a fixed distance.
例如，如果你按每公斤固定价格购买苹果，总成本与质量成正比例。如果你以恒定速度行驶，对于固定距离，所需时间与速度成反比例。
2. Direct Proportion | 正比例
Two variables y and x are in direct proportion if their ratio is constant. This means that as x increases, y increases by the same factor. The mathematical statement is y ∝ x, which means “y is proportional to x”.
如果两个变量 y 和 x 的比值恒定，则它们成正比例。这意味着当 x 增大时，y 按相同的倍数增大。数学表达为 y ∝ x，意思是 “y 与 x 成正比例”。
To change the statement into an equation, we introduce a constant k:
要将这个语句转化为方程，我们引入一个常数 k：
y = kx
Here k is called the constant of proportionality. It is the same for all pairs of values. For example, if y ∝ x and y = 6 when x = 2, then k = 6 ÷ 2 = 3. Therefore y = 3x.
这里的 k 称为比例常数，它对所有值对都是相同的。例如，如果 y ∝ x 且当 x = 2 时 y = 6，那么 k = 6 ÷ 2 = 3。因此 y = 3x。
3. Inverse Proportion | 反比例
Two variables y and x are in inverse proportion if their product is constant. As x increases, y decreases, and vice versa. The statement is y ∝ 1/x, meaning “y is inversely proportional to x”.
如果两个变量 y 和 x 的乘积恒定，则它们成反比例。当 x 增大时，y 减小，反之亦然。其表达为 y ∝ 1/x，意思是 “y 与 x 成反比例”。
The equation form is:
其方程形式为：
y = k/x
For example, if y is inversely proportional to x and y = 4 when x = 9, then k = xy = 9 × 4 = 36. So y = 36/x. When x = 3, y = 36 ÷ 3 = 12.
例如，如果 y 与 x 成反比例，且当 x = 9 时 y = 4，则 k = xy = 9 × 4 = 36。所以 y = 36/x。当 x = 3 时，y = 36 ÷ 3 = 12。
4. The Constant of Proportionality | 比例常数
The constant k is essential for solving proportion problems. To find k, substitute the known pair of values into the equation. For direct proportion, k = y/x; for inverse proportion, k = xy.
常数 k 对解决比例问题至关重要。要找到 k，将已知的一对值代入方程。对于正比例，k = y/x；对于反比例，k = xy。
Sometimes the relationship involves squares, cubes, or square roots. For instance, y ∝ x² means y = kx². If y = 20 when x = 2, then k = 20 ÷ 2² = 20 ÷ 4 = 5, so y = 5x².
有时关系涉及平方、立方或平方根。例如，y ∝ x² 表示 y = kx²。如果当 x = 2 时 y = 20，则 k = 20 ÷ 2² = 20 ÷ 4 = 5，所以 y = 5x²。
Always state the equation after finding k. This helps you solve for any other value.
找到 k 后一定要写出方程。这有助于你求解任何其他值。
5. Proportions with Powers and Roots | 幂与根的比例
IGCSE questions often include y ∝ x², y ∝ x³, y ∝ √x, or y ∝ 1/x². The pattern is the same: replace ∝ with “= k times”, then substitute values.
IGCSE 题目常涉及 y ∝ x²、y ∝ x³、y ∝ √x 或 y ∝ 1/x²。模式相同：将 ∝ 替换为 “= k 乘以”，然后代入数值。
For inverse square proportion, y = k/x². As x doubles, y becomes one quarter of its previous value, because 1/2² = 1/4.
对于平方反比，y = k/x²。当 x 加倍时，y 变为原来的四分之一，因为 1/2² = 1/4。
For example, y is inversely proportional to the square of x. If y = 5 when x = 2, then k = yx² = 5 × 4 = 20. So y = 20/x². When x = 4, y = 20/16 = 1.25.
例如，y 与 x 的平方成反比例。如果当 x = 2 时 y = 5，则 k = yx² = 5 × 4 = 20。所以 y = 20/x²。当 x = 4 时，y = 20/16 = 1.25。
6. Graphs of Proportion | 比例图像
The graph of direct proportion y = kx is a straight line that passes through the origin. Its gradient is k. If y ∝ x², the graph is a parabola through the origin. The shape of the graph helps you recognise the type of proportion.
正比例 y = kx 的图像是一条经过原点的直线，其斜率为 k。如果 y ∝ x²，图像是通过原点的抛物线。图像形状有助于你识别比例类型。
The graph of inverse proportion y = k/x is a hyperbola with two branches. It approaches the x-axis and y-axis but never touches them. For y = k/x², the curve is steeper and lies only in the first quadrant for positive x.
反比例 y = k/x 的图像是双曲线，有两条分支。它趋近 x 轴和 y 轴但永不触碰。对于 y = k/x²，曲线更陡峭，且对于正 x 仅位于第一象限。
In an exam, you may be asked to identify a graph. A line through the origin indicates direct proportion; a rectangular hyperbola indicates inverse proportion.
在考试中，你可能会被要求识别图像。过原点的直线表示正比例；双曲线表示反比例。
7. Solving Word Problems | 解决文字题
Follow these steps for proportion word problems.
对于比例文字题，请遵循以下步骤。
‘Write the proportion, find k, form the equation, substitute, answer with units.’
Step 1: Write the proportion statement using ∝. Step 2: Change to an equation with k. Step 3: Substitute given values to find k. Step 4: Write the full equation. Step 5: Substitute the new value and solve. Always include units in your final answer.
步骤 1：用 ∝ 写出比例语句。步骤 2：改成含 k 的方程。步骤 3：代入已知值求 k。步骤 4：写出完整方程。步骤 5：代入新值并求解。最终答案中始终包含单位。
Example: The cost C of hiring a taxi varies directly with the distance d. A 5 km journey costs £12. Find the cost of a 12 km journey. Since C ∝ d, C = kd. From C = 12 at d = 5, k = 12/5 = 2.4. So C = 2.4d. When d = 12, C = 2.4 × 12 = 28.8. The cost is £28.80.
示例：出租车费用 C 随距离 d 成正比例变化。5 公里行程花费 12 英镑。求 12 公里行程的费用。因为 C ∝ d，所以 C = kd。由 d = 5 时 C = 12，得 k = 12/5 = 2.4。所以 C = 2.4d。当 d = 12 时，C = 2.4 × 12 = 28.8。费用为 28.80 英镑。
8. Proportion and Ratio | 比例与比率
Ratio compares two quantities, such as 2 : 3. Proportion states that two ratios are equal. For example, 2 : 3 = 4 : 6 is a proportion. In direct proportion, the ratio of corresponding values remains constant.
比率比较两个量，如 2 : 3。比例表示两个比率相等。例如，2 : 3 = 4 : 6 是一个比例。在正比例中，对应值的比率保持不变。
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