Converting Standard and Compound Units
This topic covers the conversion of standard units (like length, mass, and time) and compound units (like speed and density). Mastering these conversions is crucial for ensuring consistency and accuracy in calculations across various scientific and engineering contexts.
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
- When converting area or volume units, the linear conversion factor must be squared or cubed, respectively. For example, since 1 m = 100 cm, it follows that 1 m2 = 1002 cm2 = 10,000 cm2.
- A critical link between volume and capacity is that 1 cm3 is equivalent to 1 ml. This allows easy conversion between units like m3 and litres.1 m3 = 1000 litres
- To convert compound units, treat the unit as a fraction and convert the numerator and denominator units independently. For example, to change km/h to m/s, convert km to m and hours to seconds.
- Always check if your conversion should result in a larger or smaller number. Converting from a larger unit to a smaller unit (e.g., metres to centimetres) will result in a larger numerical value, and vice versa.
Formulae
New Value = Old Value × Conversion Factor When converting from one unit to another. The conversion factor is a ratio of the two units, e.g., (1000 m / 1 km), arranged to cancel the original unit.
density = mass / volume A common physical relationship involving compound units (e.g., kg/m3). Useful for checking how unit conversions affect the final quantity.
speed = distance / time Another common relationship involving compound units (e.g., m/s or km/h). Used when converting speeds by changing the distance and time units separately.
Definitions
- Standard Unit
- A fundamental unit for 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 by the combination of two or more standard units, typically through multiplication or division. Examples include speed (m/s) and density (kg/m3).
- Conversion Factor
- The numerical multiplier or divisor required to change a measurement from one unit to an equivalent value in another unit.
Worked example
The density of a liquid is 50 g/L. What is its density in kg/m3?
- 1
State the initial quantity and the target units:
We need to convert 50 g/L into kg/m3.
- 2
First, convert the mass from grams (g) to kilograms (kg).
Since 1 kg = 1000 g, we divide by 1000:50 g = 0.05 kg - 3
Next, convert the volume from litres (L) to cubic metres (m3).
We know that 1000 L = 1 m3, so 1 L = 0.001 m3.
- 4
Now, substitute the converted values back into the density expression:
Density = (0.05 kg) / (0.001 m3) - 5
Calculate the final result.
Dividing by 0.001 is the same as multiplying by 1000:
0.05 × 1000 = 50The units are now kg/m3.
Answer: 50 kg/m3
Common mistakes
- ×Forgetting to square or cube the conversion factor for area and volume. For instance, incorrectly using 100 instead of 1002 for a m2 to cm2 conversion.
- ×When converting compound units, accidentally inverting the conversion factor for the unit in the denominator. For example, when converting from a 'per hour' unit to a 'per second' unit, you must divide by 3600, not multiply.
- ×Losing track of powers of 10 in multi-step conversions, especially when dealing with large conversion factors like the one between cm3 and m3 (1,000,000).
No-calculator tips
- ✓Use powers of 10 to manage large numbers. For example, instead of writing 1,000,000, use 106. This simplifies multiplication and division.
- ✓Perform a 'sanity check'. Converting from a large unit to a small one (e.g., hours to seconds) should yield a numerically larger answer. If not, you have likely divided when you should have multiplied.
- ✓To convert compound units, change one unit at a time. For km/h to m/s, first handle the km to m conversion, then separately handle the 'per hour' to 'per second' conversion. This breaks the problem into manageable steps.