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# 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

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

## Related Topics

• Basics - The SI-system, unit converters, physical constants, drawing scales and more.

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