# 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*

*u _{1} = unit 1*

*u _{2} = unit 2*

*x = converting factor from u _{2} to u_{1}*

#### 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

$$ 308.5 (mm/ft) \ 2.5 (ft) = \underline{771.3} (mm) $$