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Unit Factor Method

Convert between units with the unit factor or factor-label method

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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

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

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