📚 Compound Measures | 复合单位
Compound measures are quantities that combine two or more base units to describe a physical relationship. In IGCSE Mathematics, you will work primarily with speed, density, and pressure — each defined as a rate involving distance and time, mass and volume, or force and area. Understanding compound measures means you can convert between related units, rearrange formulas, and interpret real-world graphs such as distance–time and speed–time graphs.
复合单位是由两个或更多基本单位组合而成的量，用于描述物理关系。在 IGCSE 数学中，你主要接触速度、密度和压强——它们分别定义为距离与时间、质量与体积、或力与面积的比率。理解复合单位意味着你可以在相关单位之间进行换算、灵活变形公式，并能解读现实中的距离–时间图和速度–时间图等图表。
1. What are Compound Measures? | 什么是复合单位？
When we measure something like length or mass, we use a single unit: metres or kilograms. A compound measure, however, involves a rate or a combination of units. For example, speed is not a simple length; it is the distance travelled per unit time. Similarly, density is mass per unit volume. The key idea is that one quantity is 'per' another, often expressed using division or multiplication of units.
当我们测量长度或质量这类量时，使用单一单位：米或千克。而复合单位涉及一个比率或多个单位的组合。例如，速度不是简单的长度，而是单位时间内走过的距离。同样，密度是单位体积内的质量。核心概念是“每”某个单位的量，通常通过单位的除法或乘法来表达。
The three main compound measures in the IGCSE Edexcel syllabus are:
IGCSE Edexcel 考纲中的三个主要复合单位是：
- Speed (or velocity) — distance per unit time
- 速度 — 单位时间的距离
- Density — mass per unit volume
- 密度 — 单位体积的质量
- Pressure — force per unit area
- 压强 — 单位面积的力
Each follows a simple formula which you must be able to use in all rearrangements and in unit conversion problems.
每个复合单位都遵循一个简单的公式，你必须能够灵活变形并用于单位换算问题。
2. Speed Formula and Units | 速度公式与单位
Speed is the distance travelled divided by the time taken. The standard formula is:
速度是行驶路程除以所用时间。标准公式是：
Speed = Distance ÷ Time
d km/h. The relationship relies on the facts that 1 km = 1000 m and 1 hour = 3600 seconds.
常见速度单位包括米每秒 (m/s)、千米每小时 (km/h) 和英里每小时 (mph)。考试中可能需要在它们之间进行换算，尤其是 m/s 和 km/h 之间。换算基于 1 km = 1000 m 和 1 hour = 3600 s。
econds, depending on the desired speed unit. For instance, 2 hours 30 minutes = 2.5 hours.
当所给时间包含小时和分钟时，务必将其统一换算成小时（小数形式）或秒，取决于所需的速度单位。例如，2 小时 30 分 = 2.5 小时。
3. Density Formula and Units | 密度公式与单位
Density measures how much mass is contained in a given volume. The formula is:
密度衡量给定体积内包含多少质量。公式是：
Density = Mass ÷ Volume
hese. Note that 1 kg = 1000 g and 1 m³ = 1,000,000 cm³, so the conversion factor is large.
常见单位是克每立方厘米 (g/cm³) 和千克每立方米 (kg/m³)。在 IGCSE 题目中，经常需要在两者间换算。注意 1 kg = 1000 g，1 m³ = 1,000,000 cm³，因此换算因子较大。
Water has a density of 1 g/cm³ or 1000 kg/m³, a useful reference point. Objects with density greater than water sink; those with lower density float.
水的密度是 1 g/cm³ 或 1000 kg/m³，这是一个有用的参考值。密度大于水的物体会下沉，密度小于水的会上浮。
4. Pressure Formula and Units | 压强公式与单位
Pressure is the force applied perpendicular to a surface, divided by the area over which the force acts. The formula is:
压强是垂直作用在表面上的力除以力所作用的面积。公式是：
Pressure = Force ÷ Area
convert between N/cm² and N/m², remember that 1 cm² = 0.0001 m², so 1 N/cm² = 10,000 N/m².
压强的国际单位是帕斯卡 (Pa)，相当于牛顿每平方米 (N/m²)。你也会看到 N/cm²，尤其在涉及小面积时。换算时记住 1 cm² = 0.0001 m²，因此 1 N/cm² = 10,000 N/m²。
Pressure increases if the same force is concentrated on a smaller area; this explains why sharp objects cut more easily.
同样的力集中在更小面积上时压强增大；这就解释了为什么尖锐物体更容易切割。
5. Unit Conversion for Speed | 速度的单位换算
Converting speed between km/h and m/s is a core skill. To change km/h to m/s, multiply by (1000/3600), which simplifies to ÷ 3.6:
在 km/h 和 m/s 之间换算速度是一项核心技能。要将 km/h 换算为 m/s，需要乘以 (1000/3600)，这可以简化为除以 3.6：
Speed in m/s = Speed in km/h ÷ 3.6
Conversely, to change m/s to km/h, multiply by 3.6.
反之，将 m/s 换算为 km/h 则乘以 3.6。
Example: 90 km/h = 90 ÷ 3.6 = 25 m/s. This conversion is essential when using distance–time graphs where axes might have mixed units.
示例：90 km/h = 90 ÷ 3.6 = 25 m/s。当使用距离–时间图的坐标轴可能包含混合单位时，这类换算至关重要。
If a problem gives speed in km/h but distance in metres and time in seconds, convert all quantities to consistent units before calculation. Do not forget that time in minutes must become hours or seconds.
如果题目给出的速度是 km/h，而距离用米、时间用秒，那么要先统一单位再计算。别忘了时间以分钟给出时要转为小时或秒。
6. Unit Conversion for Density | 密度的单位换算
Density conversions between g/cm³ and kg/m³ often cause mistakes. The conversion factor is 1000:
在 g/cm³ 和 kg/m³ 之间进行密度换算经常出错。换算因子是 1000：
1 g/cm³ = 1000 kg/m³
Hence, to convert a density from g/cm³ to kg/m³, multiply by 1000. To convert from kg/m³ to g/cm³, divide by 1000.
因此，将密度由 g/cm³ 转为 kg/m³ 要乘以 1000；反之，由 kg/m³ 转为 g/cm³ 要除以 1000。
For example, aluminium has a density of 2.7 g/cm³, which is the same as 2700 kg/m³. In questions about the mass of an object given its volume in m³ and density in g/cm³, you must convert one set of units to match before using the formula.
例如，铝的密度为 2.7 g/cm³，也就是 2700 kg/m³。在已知物体体积单位为 m³ 而密度单位为 g/cm³ 时，使用公式前必须将一组单位统一。
A common exam trick: a cylinder has volume in cm³ and mass in kg. You cannot simply divide without converting mass to grams or volume to m³. Always check the unit requirement of the final answer.
常见的考试陷阱：一个圆柱体体积以 cm³ 给出而质量以 kg 给出。不能直接相除，必须将质量转为克，或将体积转为 m³。始终检查最终答案要求的单位。
7. Unit Conversion for Pressure | 压强的单位换算
/m², the two can be used interchangeably. The tricky conversion is between N/cm² and N/m²:
压强可以用 N/m²、N/cm² 或 Pa 表示。由于 1 Pa = 1 N/m²，两者可以互换使用。棘手的是 N/cm² 与 N/m² 之间的换算：
1 N/cm² = 10,000 N/m²
ing a force in newtons by 1 cm² yields a number 10,000 times larger than dividing by 1 m².
因为 1 cm = 0.01 m，所以 1 cm² = 0.0001 m²。因此，同样的力除以 1 cm² 得到的数值是除以 1 m² 的 10,000 倍。
In exam problems, you might be given the area of a piston in cm² and ask for pressure in Pa. Convert the area to m² first (multiply by 10⁻⁴) or calculate in N/cm² then multiply by 10,000. Both methods work if done consistently.
在考试问题中，可能给出活塞面积单位为 cm²，要求压强单位为 Pa。先将面积转化为 m²（乘以 10⁻⁴），或以 N/cm² 计算后再乘以 10,000。两种方法只要一致均可。
8. Using Compound Measures in Graphs | 在图表中使用复合单位
Compound measures appear heavily in graphs: distance–time graphs for speed and speed–time graphs for acceleration and distance. In a distance–time graph, the gradient represents speed. A straight line means constant speed; a horizontal line means stationary.
复合单位大量出现在图表中：距离–时间图用于速度，速度–时间图用于加速度和距离。在距离–时间图中，梯度代表速度。直线表示匀速，水平线表示静止。
For a speed–time graph, the gradient gives acceleration, and the area under the graph gives the distance travelled. Both gradient and area require careful handling of compound units. If speed is in m/s and time in seconds, area units are metres.
对速度–时间图，梯度表示加速度，曲线下的面积表示经过的距离。梯度和面积都必须谨慎处理复合单位。若速度单位是 m/s，时间单位是秒，则面积单位是米。
Always check axes labels. A graph might have distance in km and time in minutes. Speed calculated directly from the gradient would be in km/min, which may need converting to km/h or m/s. The same applies to speed–time area – if speed is in km/h and time in h, the area is in km.
始终检查坐标轴标签。图中可能距离用 km、时间用 min，则直接从梯度算出的速度是 km/min，可能需要转化为 km/h 或 m/s。速度–时间图的面积同理——若速度单位 km/h、时间单位 h，则面积单位是 km。
9. Problem Solving with Compound Measures | 复合单位的问题解决
IGCSE problems often involve multi-step processes: reading values, choosing the correct formula, converting units, and then interpreting the answer. A typical question might ask: "A car travels at 80 km/h for 45 minutes. How far does it travel? Give your answer in metres."
IGCSE 题目常涉及多步过程：读取数值、选择正确公式、换算单位，然后解释答案。典型问题可能问："一辆汽车以 80 km/h 行驶 45 分钟，它走了多远？答案以米为单位。"
t speed to m/s first: 80 ÷ 3.6 ≈ 22.22 m/s, time in seconds = 2700 s, distance ≈ 60,000 m.
步骤：将 45 分钟转为 0.75 小时；距离 = 速度 × 时间 = 80 × 0.75 = 60 km。再将 60 km 转为米：60 × 1000 = 60,000 m。或者先将速度转为 m/s：80 ÷ 3.6 ≈ 22.22 m/s，时间秒数 = 2700 s，距离 ≈ 60,000 m。
Another common style involves density and regular shapes. You might be given the dimensions of a cuboid and its mass, then asked for density. Compute volume from the dimensions, ensure unit consistency, then divide mass by volume.
另一种常见题型涉及密度和规则形状。题目可能给出一个长方体的尺寸和质量，要求求密度。根据尺寸计算体积，确保单位一致，然后质量除以体积。
For pressure, a question could describe a rectangular block resting on different faces. The pressure exerted depends on the contact area, so you must identify the correct area and possibly convert cm² to m².
对于压强，题目可能描述一个长方体以不同面放置。产生的压强取决于接触面积，因此必须找出正确的面积，并可能需要将 cm² 转换为 m²。
10. Common Pitfalls and Tips | 常见陷阱与技巧
Many errors come from mixing units. Always write down the units of each quantity before substituting into the formula. Use a consistent unit pathway: if distance is in km and time in hours, speed is in km/h. Never divide a km distance by a time in seconds without converting one of them.
许多错误源于单位混淆。始终在代入公式前写下每个量的单位。采用一致的路径：若距离为 km、时间为小时，速度就是 km/h。绝不要用秒为单位的时间直接除以 km 为单位的距离而不进行换算。
0 cm³. Area and volume conversions require squaring or cubing the linear conversion factor.
换算面积或体积单位时忘记平方或立方是另一个典型错误。记住：1 m² = 10,000 cm²，而不是 100 cm²；1 m³ = 1,000,000 cm³。面积和体积换算需要将长度换算因子平方或立方。
On graphs, students sometimes confuse the gradient of a distance–time graph (speed) with the area under a speed–time graph (distance). Use the mnemonic: 'Gradient = Speed' for distance–time and 'Area = Distance' for speed–time.
在图表方面，学生有时会混淆距离–时间图的梯度（速度）和速度–时间图下的面积（距离）。使用助记法：距离–时间图 “梯度 = 速度”，速度–时间图 “面积 = 距离”。
Always check that your final answer is reasonable. A car speed of 300 m/s is unrealistic; you may have forgotten to convert. A density of 0.0001 g/cm³ for a rock suggests a unit conversion error. Sensible checking helps avoid losing marks.
始终检查最终答案是否合理。一辆汽车的速度为 300 m/s 是不现实的；你可能忘了换算。一块岩石密度为 0.0001 g/cm³ 表明单位换算有误。合理的检查能避免失分。
thod or algebraic manipulation. However, be aware that triangles can fail if you are not careful with units; algebraic rearrangement is safer for multi-step problems.
最后，在变形公式如 质量 = 密度 × 体积 或 力 = 压强 × 面积 时，可以使用三角形法或代数变换。不过，注意三角形法若单位不小心容易出错；对于多步问题，代数变形更可靠。
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