Compound Measures | 复合度量

📚 Compound Measures | 复合度量

In mathematics, a compound measure is any measurement that combines two or more different units to express a rate or ratio. The most important compound measures in the IGCSE Edexcel Mathematics syllabus are speed, density and pressure. Questions involving compound measures often require unit conversion, formula manipulation and careful reading of word problems.

在数学中，复合度量是由两个或多个不同单位组合而成的测量结果，用来表示一种比率。Edexcel IGCSE 数学考纲中最重要的复合度量包括速度、密度和压强。涉及复合度量的题目通常需要单位换算、公式变形以及对应用题文字信息的细致理解。

1. What Are Compound Measures? | 什么是复合度量？

A compound measure is built from two different quantities. For example, speed is distance divided by time, so speed is a compound measure built from length and time. Density is mass divided by volume, and pressure is force divided by area.

复合度量由两个不同的量组合而成。例如，速度等于距离除以时间，因此速度是由长度和时间构成的复合度量。密度等于质量除以体积，压强等于力除以面积。

Speed uses distance and time.

速度使用距离和时间。
Density uses mass and volume.

密度使用质量和体积。
Pressure uses force and area.

压强使用力和面积。

Because compound measures involve two separate units, you must always check that both units are compatible before doing any calculation.

由于复合度量涉及两个独立的单位，在做任何计算前，你都应检查这两个单位是否匹配一致。

2. Speed, Distance and Time | 速度、距离和时间

The basic relationship is:

Speed = Distance ÷ Time

Distance = Speed × Time

Time = Distance ÷ Speed

Speed is usually given in metres per second (m/s), kilometres per hour (km/h) or miles per hour (mph). You must choose the correct units to match the question.

速度通常用米每秒（m/s）、千米每小时（km/h）或英里每小时（mph）表示。你必须根据题目要求选择正确的单位。

Example: A train travels 180 km at a constant speed of 120 km/h. Time = 180 ÷ 120 = 1.5 hours, which is 1 hour 30 minutes.

示例：一列火车以 120 km/h 的恒定速度行驶 180 km。时间 = 180 ÷ 120 = 1.5 小时，即 1 小时 30 分钟。

3. Density, Mass and Volume | 密度、质量和体积

Density describes how much mass is contained in a given volume. The formula is:

Density = Mass ÷ Volume

Mass = Density × Volume

Volume = Mass ÷ Density

Density can be measured in kg/m³ or g/cm³. The units must match the information given in the question.

密度的单位可以是 kg/m³ 或 g/cm³。单位必须与题目给出的信息保持一致。

Example: A metal block has mass 500 g and volume 250 cm³. Density = 500 ÷ 250 = 2 g/cm³.

示例：一块金属质量为 500 g，体积为 250 cm³。密度 = 500 ÷ 250 = 2 g/cm³。

If the mass is 2.7 kg and the volume is 1000 cm³, first convert 2.7 kg to 2700 g, then Density = 2.7 g/cm³. Always convert to compatible units before dividing.

如果质量为 2.7 kg，体积为 1000 cm³，先把 2.7 kg 换算成 2700 g，则密度 = 2.7 g/cm³。在进行除法前，务必先把单位换算为一致单位。

4. Pressure, Force and Area | 压强、力和面积

Pressure is the amount of force applied per unit area. The formula is:

Pressure = Force ÷ Area

Force = Pressure × Area

Area = Force ÷ Pressure

The standard unit of pressure is N/m², which is also called a pascal (Pa). Sometimes pressure is given in N/cm².

压强的标准单位是 N/m²，也称为帕斯卡（Pa）。有时压强也会用 N/cm² 表示。

Example: A force of 20 N is spread over an area of 4 m². Pressure = 20 ÷ 4 = 5 N/m² = 5 Pa.

示例：一个 20 N 的力分布在 4 m² 的面积上。压强 = 20 ÷ 4 = 5 N/m² = 5 Pa。

5. Other Compound Measures | 其他常见复合度量

Beyond speed, density and pressure, there are other compound measures that can appear in an IGCSE question.

除了速度、密度和压强之外，还有一些复合度量也可能出现在 IGCSE 题目中。

Population density = population ÷ area, often written as people per km².

人口密度 = 人口数 ÷ 面积，通常写为人/平方千米。
Flow rate = volume ÷ time, for example litres per minute.

流速/流量 = 体积 ÷ 时间，例如升/分钟。
Fuel consumption = distance ÷ fuel used, for example km per litre.

燃油消耗率 = 行驶距离 ÷ 燃油量，例如千米/升。

These measures follow the same general idea: divide one quantity by another to describe a rate.

这些度量遵循同样的基本思想：用一个量除以另一个量，从而描述一种比率。

6. Converting Compound Units | 复合单位的换算

Converting compound units is a common exam skill. The most important conversions are between km/h and m/s.

换算复合单位是一项常见的考试技能。最重要的换算是 km/h 与 m/s 之间的换算。

To convert km/h to m/s, multiply by 1000 and divide by 3600:

1 km/h = 1000 m ÷ 3600 s = 5/18 m/s

To convert m/s to km/h, multiply by 3600 and divide by 1000:

1 m/s = 3600 m ÷ 1000 s = 3.6 km/h

Example: Convert 72 km/h to m/s. 72 × 5/18 = 20 m/s.

示例：把 72 km/h 换算为 m/s。72 × 5/18 = 20 m/s。

Example: Convert 30 m/s to km/h. 30 × 3.6 = 108 km/h.

示例：把 30 m/s 换算为 km/h。30 × 3.6 = 108 km/h。

Original Unit	Convert to	Multiply by
km/h	m/s	5/18
m/s	km/h	3.6
g/cm³	kg/m³	1000

Density conversion: 1 g/cm³ = 1000 kg/m³. This is because 1 g = 0.001 kg and 1 cm³ = 10⁻⁶ m³, so 1 ÷ 10⁻⁶ = 1000.

密度换算：1 g/cm³ = 1000 kg/m³。这是因为 1 g = 0.001 kg，而 1 cm³ = 10⁻⁶ m³，因此 1 ÷ 10⁻⁶ = 1000。

7. Using the Formula Triangle | 公式三角形记忆法

A popular way to rearrange compound-measure formulas is to use a formula triangle. Write the quantity to be found at the top and the two remaining quantities at the bottom.

一个常用的简化公式变形方法是使用公式三角形。把要求的量写在三角形上方，另外两个量写在下方。

For example, in the density triangle, M (mass) goes at the top, and D (density) and V (volume) go at the bottom. Cover M, and you see D × V. Cover D, and you see M ÷ V. Cover V, and you see M ÷ D.

例如，在密度三角形中，M（质量）在上方，D（密度）和 V（体积）在下方。遮住 M，则看到 D × V；遮住 D，则看到 M ÷ V；遮住 V，则看到 M ÷ D。

The same idea works for speed (D, S, T) and pressure (P, F, A).

同样的思路也适用于速度三角形（D、S、T）和压强三角形（P、F、A）。

Triangle: cover the quantity you want; the remaining layout tells you the calculation.

三角形：遮住你要求的量，剩余的位置关系就会告诉你该如何计算。

8. Average Speed over a Journey | 一段行程的平均速度

Average speed is not found by simply averaging the speeds from different parts of a journey. You must always use:

平均速度不能简单地用一段旅程中各段速度取平均来得到。你必须始终使用：

Average Speed = Total Distance ÷ Total Time

平均速度 = 总距离 ÷ 总时间

Example: A person walks 6 km at 4 km/h, then runs 6 km at 12 km/h. Total distance = 12 km. Time for the first part = 6 ÷ 4 = 1.5 hours. Time for the second part = 6 ÷ 12 = 0.5 hours. Total time = 2 hours. Average speed = 12 ÷ 2 = 6 km/h.

示例：某人以 4 km/h 的速度行走 6 km，然后以 12 km/h 的速度跑步 6 km。总距离 = 12 km。第一部分时间 = 6 ÷ 4 = 1.5 小时。第二部分时间 = 6 ÷ 12 = 0.5 小时。总时间 = 2 小时。平均速度 = 12 ÷ 2 = 6 km/h。

If you incorrectly averaged the speeds you would get (4 + 12) ÷ 2 = 8 km/h, which is wrong because the person spent different amounts of time at each speed.

如果错误地直接取速度平均，你会得到 (4 + 12) ÷ 2 = 8 km/h，这是错误的，因为此人在两种速度下所用的时间不同。

9. Distance-Time Graphs | 距离-时间图像

A distance-time graph shows how distance changes with time. The gradient of the graph represents speed.

距离-时间图像表示距离随时间变化的情况。图像的斜率代表速度。

If the graph is a horizontal line, the object is stationary. If the graph is steeper, the speed is higher.

如果图像是一条水平线，说明物体静止不动。如果图像越陡，则速度越大。

Example: A line increases from (0, 0) to (2, 60). Gradient = 60 ÷ 2 = 30 km/h, assuming the distance is in km and time is in hours.

示例：一条直线从点 (0, 0) 上升到点 (2, 60)。斜率 = 60 ÷ 2 = 30 km/h，这里假设距离单位为 km，时间单位为小时。

For a curve, the instantaneous speed is found by drawing a tangent at that point and finding the tangent’s gradient.

对于曲线，瞬时速度可以通过在该点作切线，并求切线的斜率来得到。

10. Exam Tips and Common Errors | 考试技巧与常见错误

Always check that the units are compatible before substituting values.

在代入数值前，务必检查单位是否一致匹配。
Convert time correctly: 45 minutes = 0.75 hours, not 0.45 hours.

正确换算时间：45 分钟 = 0.75 小时，而不是 0.45 小时。
Do not average speeds directly. Use total distance ÷ total time.

不要直接对各段速度取平均。应使用总距离 ÷ 总时间。
When converting area or volume units, remember that 1 m² = 10⁴ cm² and 1 m³ = 10⁶ cm³.

换算面积或体积单位时，记住 1 m² = 10⁴ cm²，1 m³ = 10⁶ cm³。
Show your working clearly so that you gain method marks even if your final answer has a small error.

清晰写出计算过程，这样即使最终答案有微小错误，也可能获得步骤分。
Read the question to see whether the answer should be in m/s, km/h, g/cm³ or another unit.

仔细阅读题目，确认答案应该使用 m/s、km/h、g/cm³ 还是其他单位。

11. Summary | 小结

Compound measures appear throughout the IGCSE syllabus because they connect two real-world quantities in a single expression. Speed, density and pressure all follow the same pattern: divide one measure by another, and rearrange the formula when needed.

复合度量贯穿整个 IGCSE 考纲，因为它们将两个现实中的量用一个式子连接起来。速度、密度和压强都遵循相同的模式：用一个量除以另一个量，并在需要时对公式进行变形。

The most reliable strategies are to write the formula first, check the units, use the correct conversion factor and find average values from total quantities.

最可靠的解题策略是：先写出公式，检查单位，使用正确的换算系数，并用总量来计算平均值。

Practising a range of past-paper questions will help you recognise common question styles and avoid the common pitfalls described above.

多做历年真题可以帮助你识别常见题型，并避免上述常见陷阱。

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Compound Measures | 复合度量