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Math101learn.math101.caUnit Conversion
Unit conversion multiplies by equivalence ratios equal to one so a quantity changes form without changing size.
Units can be handled like algebraic factors: arrange conversion ratios so unwanted units cancel.
Equivalent quantities
A conversion factor comes from an equality such as
Therefore both
and its reciprocal equal $1$ in value. Multiplying by the appropriate ratio changes units without changing the physical quantity.
Dimensional-analysis method
- Write the given value with its unit.
- multiply by a conversion factor.
- place the unwanted unit opposite so it cancels.
- continue until the desired unit remains.
- calculate and check magnitude.
Keeping units visible prevents most direction errors.
Worked example: length
The physical length has not changed.
Multi-step conversion
Convert $72$ km/h to m/s:
Kilometres and hours cancel in opposite positions.
Metric prefixes
Metric prefixes represent powers of ten:
- kilo: $1000$;
- centi: $1/100$;
- milli: $1/1000$.
Place-value movement can be a shortcut, but conversion factors make direction and compound units clearer.
Area conversions
Square the entire length conversion:
Multiplying by $100$ only once would convert a length, not an area.
Volume conversions
Cube the length conversion:
Useful capacity relationships include
Rates and compound units
Convert numerator and denominator units separately. Fuel economy, density, unit price, and speed all use compound units.
For example, converting dollars per kilogram to dollars per gram changes the denominator scale and therefore the numerical rate in the opposite intuitive direction.
Temperature conversions
Temperature scales are not simple multiplicative units because their zeros differ. Use the full formulas:
The added/subtracted $32$ is essential.
Reasonableness
Converting to a smaller unit should usually produce a larger numerical value for the same positive measurement. $2$ m becoming $200$ cm fits; $0.02$ cm does not.
Estimate magnitude and keep units through the calculation.
Common mistakes
Using a conversion factor upside down. Arrange it so unwanted units cancel.
Changing the number but dropping the unit trail. Units justify the operation.
Using a linear factor once for area or volume. Square or cube it.
Converting only one part of a compound unit. Treat numerator and denominator.
Using only multiplication for temperature. Offset scales need full formulas.
Quick self-check
- What unit is given and what unit is wanted?
- Does the conversion ratio equal one?
- Are unwanted units positioned to cancel?
- Are area/volume powers applied to the conversion factor?
- Are all parts of a rate converted?
- Does the final magnitude and unit make physical sense?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Convert 72 km/h to m/s.
- 72 × 1000/3600 = 20.
- Therefore 72 km/h = 20 m/s.
End of lesson
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