Standard and Compound Units
This topic covers the fundamental skill of working with standard and compound units. It is essential for correctly setting up and solving quantitative problems, ensuring that calculations are physically and mathematically consistent.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Always ensure all quantities in a calculation use a consistent set of base units before you begin (e.g., convert everything to metres, kilograms, and seconds).
- Compound units are formed by dividing or multiplying standard units. For example, speed (m/s) is length divided by time, and pressure (N/m2) is force divided by area.
- A crucial conversion for volume and capacity is 1 cm3 = 1 ml. This directly links linear dimensions to liquid measures.
- Be careful with area and volume conversions: 1 m2 = (100 cm)2 = 10,000 cm2, and 1 m3 = (100 cm)3 = 1,000,000 cm3.
- Units can be written with a slash (e.g., kg/m3) or using negative indices (e.g., kg m-3).
Formulae
Speed = Distance / Time To calculate the rate of travel of an object.
Density = Mass / Volume To find the mass per unit volume of a substance.
Pressure = Force / Area To determine the force exerted over a specific surface area.
Unit Price = Total Cost / Number of Items To find the cost of a single item when sold in a batch.
Definitions
- Standard Unit
- A defined, conventional measure of a physical quantity, such as the metre (m) for length, the kilogram (kg) for mass, or the second (s) for time.
- Compound Unit
- A unit formed from a combination of two or more standard units, typically through multiplication or division. Examples include speed (m/s), density (kg/m3), and pressure (N/m2).
- Density
- The measure of mass contained within a unit of volume. It describes how tightly packed a substance is.
- Pressure
- The measure of force applied perpendicularly over a unit of area.
Worked example
A rectangular block of metal has dimensions 20 cm, 5 cm, and 4 cm. Its mass is 3.2 kg. Calculate its density in g/cm3.
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First, calculate the volume of the block in the required units (cm3).
Volume = length × width × height = 20 cm × 5 cm × 4 cm = 400 cm3 - 2
Next, convert the mass into the required units (g).
Since 1 kg = 1000 g, the mass is 3.2 kg × 1000 = 3200 g - 3
Apply the density formula using the consistent units.
Density = Mass / Volume = 3200 g / 400 cm3 - 4
Simplify the fraction to find the final answer.
3200 / 400 is the same as 32 / 4, which equals 8.
Answer: 8 g/cm3
Common mistakes
- ×Forgetting to convert all quantities to a consistent set of units before performing a calculation. For instance, calculating density using mass in kg and volume in cm3 will give an incorrect answer.
- ×Making arithmetic errors during unit conversion, such as multiplying by 100 instead of 1000 when converting from kg to g, or from m to mm.
- ×Incorrectly converting squared or cubed units. A common mistake is assuming 1 m2 = 100 cm2, when it is actually 100 cm x 100 cm = 10,000 cm2.
No-calculator tips
- ✓When dealing with fractions involving large numbers with trailing zeros, cancel the zeros first. For example, 45000 / 150 becomes 4500 / 15, which is much easier to compute.
- ✓To multiply a decimal by 10, 100, or 1000, simply move the decimal point to the right by 1, 2, or 3 places, respectively. Do the opposite for division.
- ✓Before you start a calculation, do a quick mental check. For instance, when calculating density, if you put volume over mass by mistake, you'll often get a very small decimal, which should signal that you've made an error.