Direct and Inverse Proportion | 正比例与反比例

📚 Direct and Inverse Proportion | 正比例与反比例

Proportion is one of the most practical topics in IGCSE Mathematics because it appears in money, speed, density, recipes, scale diagrams and many other real-life situations. Understanding whether two quantities change in the same direction or in opposite directions is a key skill for many examination questions.

比例是 IGCSE 数学中最实用的课题之一，因为它出现在金钱、速度、密度、食谱、比例图以及许多现实生活情境中。理解两个量是沿相同方向变化还是相反方向变化，是许多考试题目的关键技能。

1. Understanding Ratio and Proportion | 理解比与比例

A ratio compares two or more quantities in the same unit, while a proportion states that two ratios are equal. Ratios can be simplified by dividing all parts by a common factor.

比率用于比较两个或多个相同单位的量，而比例表示两个比率相等。比率可以通过将所有部分除以公因数来化简。

For example, if a recipe uses flour and sugar in the ratio 3 : 1, tripling the recipe keeps the same proportion, 9 : 3. This is different from simply adding 2 to each part, which would give 5 : 3 and change the taste.

例如，如果一份食谱使用面粉和糖的比例为 3 : 1，将食谱用量增加到三倍仍保持相同比例 9 : 3。这不同于简单地对每个部分加 2，那样会得到 5 : 3 并改变口味。

Ratio compares two or more quantities: a : b.
比率比较两个或多个量：a : b。
Proportion compares two equal ratios: a : b = c : d.
比例表示两个相等的比率：a : b = c : d。
Simplify ratios like fractions, but keep the same order.
像分数一样化简比率，但要保持相同顺序。

2. Direct Proportion: Definition and Notation | 正比例：定义与符号

When y is directly proportional to x, we write y ∝ x. This means that as x increases, y increases at the same constant rate, and as x decreases, y decreases at the same constant rate.

当 y 与 x 成正比时，我们写作 y ∝ x。这意味着随着 x 增大，y 以相同的恒定速率增大；随着 x 减小，y 也以相同的恒定速率减小。

The proportion statement y ∝ x can always be converted into an equation by introducing a constant k:

比例关系式 y ∝ x 总可以通过引入常数 k 转化为方程：

y = kx

Here k is non-zero and is called the constant of proportionality. If k = 3, then doubling x causes y to double as well; tripling x triples y.

这里的 k 是非零的，称为比例常数。如果 k = 3，那么 x 加倍时 y 也加倍；x 变成三倍时 y 也变成三倍。

3. The Constant of Proportionality | 比例常数

The constant k is the multiplier that connects x and y in direct proportion. To find k, divide the value of y by the corresponding value of x:

常数 k 是在正比例中连接 x 和 y 的乘数。要求 k，可以用 y 的值除以对应的 x 的值：

k = y ÷ x

For example, if y = 20 when x = 4, then k = 20 ÷ 4 = 5. The equation becomes y = 5x, so every unit increase in x increases y by 5.

例如，如果当 x = 4 时 y = 20，则 k = 20 ÷ 4 = 5。方程变为 y = 5x，因此 x 每增加 1，y 增加 5。

Always use a known pair of values to find k before solving for an unknown value in a later part of the question.

在解答题目后续部分的未知值之前，一定要先用一对已知值求出 k。

4. Direct Proportion in Tables and Graphs | 表格与图像中的正比例

In a table, direct proportion shows a constant ratio y/x. Each y-value is the same multiple of the corresponding x-value.

在表格中，正比例表现为 y/x 的比值恒定。每个 y 值都是对应 x 值的相同倍数。

x	1	2	3	4
y	5	10	15	20

Each pair gives the same value of k: 5 ÷ 1 = 5, 10 ÷ 2 = 5, 15 ÷ 3 = 5 and 20 ÷ 4 = 5. The graph of y = kx is a straight line passing through the origin.

每一对都给出相同的 k 值：5 ÷ 1 = 5，10 ÷ 2 = 5，15 ÷ 3 = 5，20 ÷ 4 = 5。y = kx 的图像是一条经过原点的直线。

The gradient of this line is the constant k. A larger k gives a steeper line, while a smaller positive k gives a gentler slope.

这条直线的斜率就是常数 k。k 越大，直线越陡；k 为正且越小，直线越平缓。

5. Inverse Proportion: Definition and Notation | 反比例：定义与符号

When y is inversely proportional to x, we write y ∝ 1/x. This means that as x increases, y decreases by the same factor, and as x decreases, y increases by the same factor.

当 y 与 x 成反比时，我们写作 y ∝ 1/x。这意味着随着 x 增大，y 以相同倍数减小；随着 x 减小，y 以相同倍数增大。

The proportion statement can be converted into an equation using the constant k:

比例关系式可以用常数 k 转化为方程：

y = k/x and xy = k

Here the product xy remains constant. If x doubles, y halves; if x is divided by 3, y is multiplied by 3.

这里的乘积 xy 保持恒定。如果 x 加倍，y 减半；如果 x 除以 3，y 则乘以 3。

6. Inverse Proportion in Tables and Graphs | 表格与图像中的反比例

For inverse proportion, the product xy is constant. In a table, each x-value multiplied by its paired y-value gives the same number.

对于反比例，乘积 xy 恒定。在表格中，每个 x 值与其配对的 y 值相乘都得到相同的数。

x	1	2	4	8
y	8	4	2	1

Each product xy equals 8, so this table shows inverse proportion with k = 8. The graph of y = k/x is a rectangular hyperbola.

每个乘积 xy 都等于 8，因此该表格表示 k = 8 的反比例。y = k/x 的图像是一条等轴双曲线。

The curve gets closer and closer to the x-axis and y-axis but never touches them. These lines are called asymptotes.

曲线会越来越接近 x 轴和 y 轴，但永远不会接触它们。这些线称为渐近线。

7. Finding the Proportionality Constant | 求解比例常数

To solve proportion problems, first identify the type of proportion and write the equation with k. Then use a known pair of values to find k, and finally substitute the new value to find the unknown quantity.

解比例题时，首先要判断比例类型并写出含 k 的方程。然后用一对已知值求出 k，最后代入新值来求未知量。

Example: y is inversely proportional to x. If y = 12 when x = 3, find y when x = 9.

例题：y 与 x 成反比。当 x = 3 时 y = 12，求当 x = 9 时 y 的值。

First find k: k = xy = 3 × 12 = 36.

首先求 k：k = xy = 3 × 12 = 36。

Then substitute x = 9: y = 36 ÷ 9 = 4.

然后代入 x = 9：y = 36 ÷ 9 = 4。

For direct proportion, use k = y ÷ x instead. Writing the equation first prevents many common errors.

对于正比例，则用 k = y ÷ x。先写出方程可以避免许多常见错误。

8. Real-Life Applications | 实际应用

Direct proportion appears in earnings: total pay = hourly rate × hours worked. If the hourly rate is fixed, more hours give more pay in the same ratio.

正比例出现在工资计算中：总工资 = 时薪 × 工作小时数。如果时薪固定，工作时间越多，工资按相同比例增加。

Inverse proportion

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Direct and Inverse Proportion | 正比例与反比例