Math101 · Independent Ontario learning library
Math101Unit 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.
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.
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?
