The idea: multiply by one
Any quantity can be multiplied by 1 without changing its value. A conversion factor is just a way of writing 1 using different units. Because 1 kilometre is 1,000 metres, the fractions
1000 m / 1 km and 1 km / 1000 m
both equal one. Choose the version that cancels the unit you want to get rid of and leaves the unit you want. If you set up the fraction correctly, the unwanted units cancel on paper, and the algebra makes the mistake visible when you get it wrong. This is dimensional analysis in action, applied to the practical job of changing units.
A reliable procedure
- Write the starting quantity with its unit.
- Write the target unit.
- Find the relationship between the units, from a trusted reference.
- Write it as a fraction with the unit to remove on the bottom.
- Multiply, cancel units, and check that only the target unit remains.
- Sanity-check the size of the answer.
If the units do not cancel to the target unit, do not guess. Re-write the factor upside down, or add another step.
Example 1: speed
Convert 90 km/h to m/s.
90 km/h × (1000 m / 1 km) × (1 h / 3600 s)
The km cancels, the h cancels, leaving m/s. The numbers give 90 × 1000 ÷ 3600 = 90,000 ÷ 3,600 = 25.
So 90 km/h is 25 m/s. As a check, 36 km/h is 10 m/s, and 90 is 2.5 times 36, so 25 m/s fits.
Example 2: length
Convert 2.5 inches to millimetres. One inch is defined as exactly 25.4 mm.
2.5 in × (25.4 mm / 1 in) = 63.5 mm.
Example 3: pressure
Convert 30 psi to kilopascals. One psi is approximately 6.895 kPa.
30 psi × (6.895 kPa / 1 psi) ≈ 206.85 kPa, so about 207 kPa.
Note the word “approximately”. Many conversion factors are defined exactly (inch to millimetre); others, like psi, come from combining definitions and are usually quoted to a few figures. Keep as many digits as you need during the calculation and round at the end, following the guidance in significant figures.
Squares and cubes: apply the factor to the whole unit
The most common conversion error is forgetting that a factor must be squared or cubed for areas and volumes.
Since 1 m = 100 cm, then 1 m² = (100 cm)² = 10,000 cm², not 100 cm². Likewise 1 m³ = (100 cm)³ = 1,000,000 cm³.
In the procedure, this appears naturally: to convert 0.5 m² to cm², write
0.5 m² × (100 cm / 1 m)² = 0.5 × 10,000 cm² = 5,000 cm².
A useful related relationship is 1 litre = 1,000 cm³, so 1 m³ = 1,000 litres.
Compound units
Treat each unit separately. To convert a flow of 12 L/min to m³/h, handle the litre and the minute in turn:
12 L/min × (1 m³ / 1000 L) × (60 min / 1 h) = 12 × 60 ÷ 1000 m³/h = 0.72 m³/h.
Writing the cancellations explicitly guards against the common slip of multiplying by 60 when you should divide, or the reverse.
Temperature is different
Most conversions are pure scaling, but temperature scales also have different zero points. A reading in Celsius becomes kelvin by adding 273.15, so 20 °C is 293.15 K. Fahrenheit uses °F = °C × 9/5 + 32, so 100 °C is 212 °F.
But a temperature difference does not need the offset. A rise of 10 °C is a rise of 10 K. This matters in formulas that use differences, and it is worth stating clearly in your working which one you mean.
Mass, force and weight
Different unit systems treat these differently, and confusing them is a classic source of error. In SI, the kilogram measures mass and the newton measures force. Some older engineering conventions use pound-force and pound-mass, and a quantity “in pounds” may be either. Always check which is meant before converting. A widely reported case of unit confusion, the Mars Climate Orbiter loss in 1999, involved mixed force units between teams, which is why formal checks of units at interfaces are good practice.
Practical habits
- Convert early, calculate once. Convert all inputs to a single consistent system, then compute.
- Write factors out on paper or in the cell. A hidden factor of 3.6 in a long formula is hard to audit later.
- Use trusted sources for conversion factors, and record where they came from.
- Keep units in engineering spreadsheets, and label columns so that a pasted value in the wrong units stands out.
- Be careful with prefixes. Mixing up milli and mega is a thousand-million-fold error. A refresher on SI prefixes helps.
- Check the result is plausible. If a pump supposedly moves 0.72 cubic metres an hour, does that sound like a garden pump or a mine drainage system?
For work involving safety, compliance or contracts, follow the units and conversion rules required by the standards and customers that apply to you; these vary by country and industry.