Unit Factor Method
Convert between units with the unit factor or factor-label method
Unit Factor Method
Unit converting from one system to an other can be done with the unit factor method - also called dimensional analysis.
The relationship between two different units can be expressed as
$$ 1 (u_1) = x (u_2) \tag{1} $$
where
u1 = unit 1
u2 = unit 2
x = converting factor from u2 to u1
Example - inches vs. mm
$$ 1 (in) = 25.4 (mm) $$
Dividing the relationship (1) with the two sides - two unit factors for the relationship can be expressed as
$$ \frac {1 (u_1) }{ x (u_2)} = 1 \tag{2} $$
$$ \frac {x (u_2) }{ 1 (u_1)} = 1 \tag{3} $$
Example - inches vs. mm
$$ \frac {1 (in) }{ 25.4 (mm)} = 1 $$
$$ \frac {25.4 (mm) }{ 1 (in)} = 1 $$
Converting a value from one unit to an other can then simply by done by multiplying both sides of eq. 2 or 3 with the value.
$$ \frac {1 (u_1) \ a (u_2)}{ x (u_2)} = a (u_2) \tag{2a} $$
$$ \frac {x (u_2) \ b (u_1)}{ 1 (u_1)} = b (u_1) \tag{3a} $$
Example - Converting 10 mm to inches
$$ \frac {1 (in) \ 10 (mm)}{ 25.4 (mm)} = 10 (mm) $$
$$ 0.39 (in) = 10 (mm) $$
Factor-label Method
The factor-label method is an sequential combination of conversion factors where numerator and denominator of the conversion factor fractions can be cancelled out to achieve a desired set of dimensional units.
$$ \frac {1 (u_1) }{ x (u_2)} \ \frac {y (u_2) }{ 1 (u_3)} \ \frac {1 (u_3) }{ z (u_4)} = \frac {y (u_1) }{ x z (u_4)} \tag{4} $$
Example - Converting 2.5 feet to mm
Th unit factor for mm to feet can be calculated using eq. 4 as:
$$ \frac {25.4 (mm) }{ 1 (in)} \ \frac {12 (in) }{ 1 (ft)} = 304.8 (\frac {mm}{ft}) $$
10 mm can be converted to feet as
$$ 308.5 (mm/ft) \ 2.5 (ft) = \underline{771.3} (mm) $$