Direct and Inverse Proportion | 正比例与反比例

The constant k tells us the rate at which y changes with respect to x. In direct proportion, k = y/x. If you know one pair of values (other than 0,0), you can find k, and then use the equation y = kx to find unknown values.

常数 k 告诉了我们 y 相对于 x 的变化率。在正比例中，k = y/x。如果你知道一对数值（不是 0,0），你就能求出 k，然后用方程 y = kx 来求未知值。

For example, if 5 notebooks cost £7.50, then k = 7.50 ÷ 5 = 1.50. The cost equation is y = 1.50x. To find the cost of 8 notebooks, substitute x = 8: y = 1.50 × 8 = £12.00.

例如，如果 5 个笔记本花费 7.50 英镑，那么 k = 7.50 ÷ 5 = 1.50。费用方程为 y = 1.50x。要求 8 个笔记本的费用，代入 x = 8：y = 1.50 × 8 = 12.00 英镑。

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

As mentioned, the graph of y = kx is a straight line through the origin. The larger the value of k, the steeper the line. If k is negative, the line slopes downwards, still passing through the origin — but this is a special case often explored later.

如前所述，y = kx 的图像是一条通过原点的直线。k 的值越大，直线越陡峭。如果 k 为负数，直线向下倾斜，仍然通过原点——但这是一种特殊情况，通常以后才会探讨。

Key features: (1) The graph is always a straight line. (2) The line always passes through (0, 0). (3) The gradient equals the constant k.

关键特征：(1) 图像总是一条直线。(2) 直线总是通过 (0, 0)。(3) 斜率等于常数 k。

5. What Is Inverse Proportion? | 什么是反比例？

Two quantities are inversely proportional if one quantity increases and the other decreases in such a way that their product remains constant. For example, the time taken to complete a job is inversely proportional to the number of workers, assuming they all work at the same rate.

如果两个量中一个增加，另一个就以乘积保持不变的方式减少，那么它们成反比例。例如，完成一项工作所需的时间与工人数量成反比，假设所有工人工作速率相同。

y = k/x   or   xy = k

Unlike direct proportion, the graph of y = k/x (for x > 0) is a curve, not a straight line. This shape is called a hyperbola. As x increases, y decreases, but it never reaches zero.

与正比例不同，y = k/x（当 x > 0 时）的图像是一条曲线，而不是直线。这种形状叫做双曲线。随着 x 增大，y 减小，但永远不会达到零。

6. Identifying Inverse Proportion from Tables | 从表格中识别反比例

To check for inverse proportion, calculate the product xy for each pair of values. If the product is constant (or nearly constant), the relationship is inversely proportional.

要检查是否成反比例，计算每一对数值的乘积 xy。如果乘积恒定（或近似恒定），那么这种关系就是反比例。

The product xy is always 24, so the time is inversely proportional to the number of workers. The equation is y = 24/x.

乘积 xy 始终是 24，所以时间与工人数量成反比。方程为 y = 24/x。

7. The Constant of Inverse Proportionality | 反比例常数

In inverse proportion, k = xy. Once you find k, you can express y as y = k/x. Always be careful: if x is tripled, y becomes one-third, not one-ninth. This is because y is multiplied by the reciprocal of the change factor.

在反比例中，k = xy。一旦你求出 k，就可以将 y 表示为 y = k/x。要始终注意：如果 x 变为原来的三倍，y 就变为原来的三分之一，而不是九分之一。这是因为 y 要乘以变化因子的倒数。

Example: A car travels a fixed distance. Speed and time are inversely proportional. If speed doubles, travel time halves. If speed is multiplied by ¾, time is multiplied by 4/3.

例子：汽车行驶固定距离。速度和时间成反比。如果速度加倍，行驶时间减半。如果速度乘以 ¾，时间则乘以 4/3。

8. Comparing Direct and Inverse Proportion | 正比例与反比例的对比

It is important not to confuse direct and inverse proportion. A simple way to distinguish them is to ask: “As one quantity doubles, does the other double (direct) or halve (inverse)?”

不要混淆正比例和反比例，这很重要。一个简单的区分方法是问：“当一个量加倍时，另一个量是加倍（正比例）还是减半（反比例）？”

This table summarises the key differences you need to remember for your studies and examinations.

这张表总结了你需要在学习和考试中记住的关键区别。

9. Solving Proportion Word Problems | 解决比例应用题

Word problems often describe a proportional situation. The first step is to decide whether the relationship is direct or inverse. Look for clues: “at the same rate” or “total remains the same” can hint at constant ratio or constant product.

应用题通常会描述一种比例情境。第一步是判断这种关系是正比例还是反比例。寻找线索：“以相同的速率”或“总量保持不变”可能提示比值恒定或乘积恒定。

Once you identify the type, write down the equation (y = kx or y = k/x), use given values to find k, and then substitute to find the unknown. Always check that your answer makes sense in the context.

一旦确定了类型，写出方程（y = kx 或 y = k/x），用给定的数值求出 k，然后代入求出未知量。一定要检查你的答案在情境中是否合理。

10. Common Misconceptions and Tips | 常见误区与提示

Misconception 1: Assuming all straight-line graphs indicate direct proportion. A graph like y = 2x + 3 is a straight line, but it does NOT pass through the origin. Therefore, it is not a direct proportion; it is a linear relationship with an intercept.

误区 1：以为所有直线图像都表示正比例。像 y = 2x + 3 这样的图像是一条直线，但它不通过原点。因此，它不是正比例关系；它是一个带有截距的线性关系。

Misconception 2: Thinking that if x increases and y decreases, it must be inverse proportion. This is only true if xy is constant. x + y = 10 shows a decrease but is not inverse proportion.

误区 2：认为如果 x 增加而 y 减少，就一定是反比例。只有当 xy 恒定时这才成立。x + y = 10 表明了一种递减关系，但不是反比例。

Tip: Always calculate the ratio or product to confirm the type of proportion. Do not rely only on the description of increasing or decreasing.

提示：始终计算比值或乘积来确认比例类型。不要仅仅依赖增加或减少的描述。

11. Practice Questions | 练习题

Try these questions to test your understanding:

试着用这些问题来测试你的理解：

• 5 pens cost £4. How much do 12 pens cost? (Direct proportion)
• If 3 workers can paint a house in 8 days, how long will it take 6 workers? (Inverse proportion)
• Is the relationship y = 7/x direct or inverse proportion? What is the constant?
• A map has a scale where 1 cm represents 5 km. Write an equation linking map distance (x) and real distance (y), and state the constant.

• 5 支笔花费 4 英镑。12 支笔要花多少钱？（正比例）
• 如果 3 个工人粉刷一栋房子需要 8 天，那么 6 个工人需要多少天？（反比例）
• 关系式 y = 7/x 是正比例还是反比例？常数是多少？
• 一张地图的比例尺是 1 厘米代表 5 千米。写出联系地图距离 (x) 和实际距离 (y) 的方程，并说明常数。

12. Summary and Key Points | 总结与要点

Remember: direct proportion means y = kx, constant ratio, straight line through origin. Inverse proportion means y = k/x, constant product, hyperbolic curve. Being able to find k and apply the correct equation is an essential skill that will support your progress in KS3 mathematics and beyond.

记住：正比例意味着 y = kx，比值恒定，图像为通过原点的直线。反比例意味着 y = k/x，乘积恒定，图像为双曲线。能够求出 k 并应用正确的方程是一项重要的技能，它将支持你在 KS3 及更高阶段数学学习中的进步。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）