Factor of Conversion of Units

The factor of conversion between units is defined and presented with examples on how it is used in conversions. More exercises with solutions are presented at the bottom of the page.
The methods of conversion using the factor of conversion suggested here may be applied to solve more challenging problems of conversion such as in Physics, Engineering, Chemistry, ...

It is known that \( 1 \text { km} = 1000 \text { m} \).
The above is of the form \[ A = B \] which may be written as \[ \displaystyle \frac{A}{B} = 1 \quad \text { or } \quad \displaystyle \frac{B}{A} = 1 \] Using the above, the equality \( 1 \text { km} = 1000 \text { m} \) given above may be written as \[ \displaystyle \frac{1 \text { km}}{1000 \text{ m}} = 1 \quad (I) \] or \[ \displaystyle \frac{1000 \text{ m}} {1 \text { km}} = 1 \quad (II) \] The above rates in (I) and (II) are called factors of conversion.
Note that
1) the factors of conversion go in pairs if you know one, you know the other by interchanging numerator and denominator
2) they are equal to \( 1 \) and may therefore be used in conversion of units as will be explained in the examples below.

use of Factor of Conversion

Example 1
Convert \( 12300 \text { m} \) (meters) into \( \text { km} \) (kilometers) using the rate of conversion in (I) or (II) defined above.
Solution to Example 1
We explain in details the method adopted to carry out the conversion.
We are given meters and we need to convert them to \( \text { km} \).
Start by writing that \[ 12300 \text { m} = 12300 \text { m} \times 1 \] Substitute \( 1 \) by the factor of conversion in (I) above which is equal to \( 1 \). \[ 12300 \text { m} = 12300 \text { m} \times \displaystyle \frac{1 \text { km}}{1000 \text{ m}} \] Cancel \( \text { m} \) on the right side \[ 12300 \text { m} = 12300 \cancel{\text { m}} \times \displaystyle \frac{1 \text { km}}{1000 \cancel{\text { m}} } \] Simplify and rewrite as \[ 12300 \text { m} = 12300 \times \displaystyle \frac{1 \text { km}}{1000 } \] Evaluate \[ \bbox[10px, border: 2px solid red] {12300 \text { m} = \displaystyle \frac{12300 \times 1 \text { km}}{1000 } = 12.3 \text { km}} \]

Example 2
Given that \( 1 \text{ in} = 2.54 \text{ cm} \), define factors of conversion and convert
a) \( \quad 23.8 \text{ cm} \) to \( \text{ in} \)
b) \( \quad 11.7 \text{ in} \) to \( \text{ cm} \)

Solution to Example 2
Using the fact that \( 1 \text{ in} = 2.54 \text{ cm} \), we may write two factors of conversion \[ \displaystyle \frac{1 \text { in}}{2.54 \text{ cm}} = 1 \quad (I) \] and \[ \displaystyle \frac{2.54 \text{ cm}} {1 \text { in}} = 1 \quad (II) \] a)
Convert \( 23.8 \text{ cm} \) to \( \text{ in} \)
Start with \[ 23.8 \text{ cm} = 23.8 \text{ cm} \times 1 \] We have \( \text{ cm} \) that we need to convert to \( \text{ in} \), hence use the factor of conversion (I) because it has \( \text{ cm} \) , which we need to cancel , in the denominator.
Substitute \( 1 \) by the factor of conversion in (I) \[ 23.8 \text{ cm} = 23.8 \text{ cm} \times \displaystyle \frac{1 \text { in}}{2.54 \text{ cm}} \] Cancel \( \text{ cm} \) on the right side \[ 23.8 \text{ cm} = 23.8 \cancel{\text{ cm} }\times \displaystyle \frac{1 \text { in}}{2.54 \cancel{\text{ cm} } } \] Simplify and rewrite as \[ 23.8 \text{ cm} = 23.8 \times \displaystyle \frac{1 \text { in}}{2.54 } \] Evaluate \[ \bbox[10px, border: 2px solid red]{ 23.8 \text{ cm} = \displaystyle \frac{23.8 \times 1 \text { in}}{2.54} = 9.37 \text { in} } \]

b)
Convert \( \quad 11.7 \text{ in} \) to \( \text{ cm} \)
Start with \[ 11.7 \text{ in} = 11.7 \text{ in} \times 1 \] We have \( \text{ in} \) that we need to convert to \( \text{ cm} \), hence use the factor of conversion (II) because it has \( \text{ in} \) in the denominator which we need to cancel.
Substitute \( 1 \) by the factor of conversion in (II) \[ 11.7 \text{ in} = 11.7 \text{ in} \times \displaystyle \frac{2.54 \text{ cm}} {1 \text { in}} \] Cancel \( \text{ in} \) \[ 11.7 \text{ in} = 11.7 \cancel{\text{ in}} \times \displaystyle \frac{2.54 \text{ cm}} {1 \cancel{\text { in}}} \] Simplify and rewrite as \[ 11.7 \text{ in} = \displaystyle \frac{11.7 \times 2.54 \text{ cm}} {1 } \] Evaluate \[ \bbox[10px, border: 2px solid red]{ 11.7 \text{ in} = 11.7 \times 2.54 \text{ cm} = 29.718 \text{ cm}} \]

Exercises

Solutions to the Above Exercises

Given \( 1 \text { lb} = 0.4536 \text { kg}\),

Using \( \quad 1 \text { lb} = 0.4536 \text { kg} \quad \) we write factors of conversion as follows \[ \displaystyle \frac{1 \text { lb} }{0.4536 \text { kg}} = 1 \quad (I) \] or \[ \displaystyle \frac{0.4536 \text { kg}} {1 \text { lb} } = 1 \quad (II) \]
a) Convert \( 2.5 \text{ kg} \) to \( { lb} \)
Use the factor of conversion (I) because it has \( \text { kg} \) in the denominator and we are converting \( \text { kg} \) to \( \text{ lb} \). Hence \[ \quad 2.5 \text{ kg} = 2.5 \text{ kg} \times \frac{1 \text { lb} }{0.4536 \text { kg}} \] Cancel \( \text{ kg} \) on the right \[ \quad 2.5 \text{ kg} = 2.5 \cancel{\text{ kg}} \times \frac{1 \text { lb} }{0.4536 \cancel{\text{ kg}} } \] Simplify and evaluate \[ \quad 2.5 \text{ kg} = \frac{2.5 \times 1 \text { lb} }{0.4536 } = 5.51 \text { lb}\]

b) Convert \( 12 \text{ lb} \) to \( { kg} \)
We are given \( \text{ lb} \) and therefore factor of conversion (I) will be used in the conversion because it has \( \text { lb} \) in the denominator. Hence \[ \quad 12 \text{ lb} = 12 \text{ lb} \times \frac{0.4536 \text { kg}} {1 \text { lb} } \] Cancel \( \text{ lb} \) on the right \[ \quad 12 \text{ lb} = 12 \cancel {\text{ lb}} \times \frac{0.4536 \text { kg}} {1 \cancel {\text{ lb}} } \] Simplify and evaluate \[ \quad 12 \text{ lb} = \frac{ 12 \times 0.4536 \text { kg}} {1 } = 5.4432 \text{ kg} \]
Given \( 1 \text { mi} = 1.60934 \text { km} \)

Use \( \quad 1 \text { mi} = 1.60934 \text { km} \quad \) to write the factors of conversion as follows \[ \displaystyle \frac{1 \text { mi}}{1.60934 \text { km}} = 1 \quad (I) \] or \[ \displaystyle \frac {1.60934 \text { km}} {1 \text { mi}} = 1 \quad (II) \]
a) Convert \( 17.5 \text{ mi} \) to \( \text{ km} \)
Use the factor of conversion (II) because it has \( \text { mi} \) in the denominator and we are converting \( \text { mi} \). Hence \[ \quad 17.5 \text{ mi} = 17.5 \text{ mi} \times \displaystyle \frac {1.60934 \text { km}} {1 \text { mi}} \] Cancel \( \text{ mi} \) on the right \[ \quad 17.5 \text{ mi} = 17.5 \cancel{\text{ mi}} \times \displaystyle \frac {1.60934 \text { km}} {1 \cancel{\text{ mi}}} \] Simplify and evaluate \[ \quad 17.5 \text{ mi} = \displaystyle \frac{ 17.5 \times 1.60934 \text { km} }{1} = 28.16345 \text { km} \]

b) \( 11.06 \text{ km} \) to \( \text{ mi} \)
We are given \( \text{ km} \) and therefore factor of conversion (I) will be used in the conversion because it has \( \text { km} \) in the denominator. Hence \[ \quad 11.06 \text{ km} = 11.06 \text{ km} \times \frac{1 \text { mi}}{1.60934 \text { km}} \] Cancel \( \text{ km} \) on the right \[ \quad 11.06 \text{ km} = 11.06 \cancel{\text{ km}} \times \frac{1 \text { mi}}{1.60934 \cancel{\text{ km}}} \] Simplify and evaluate \[ \quad 11.06 \text{ km} = \frac{ 11.06 \times 1 \text { mi}}{1.60934} = 6.87238 \text { mi} \]
Given \( 1 \text { ha} = 107639 \text { sq. ft} \)

Using \( \quad 1 \text { ha} = 107639 \text { sq. ft} \quad \), the factors of conversion are given by \[ \displaystyle \frac{1 \text { ha}}{107639 \text { sq. ft}} = 1 \quad (I) \] or \[ \displaystyle \frac{107639 \text { sq. ft}} {1 \text { ha}} = 1 \quad (II) \]
a) Convert \( 22000 \text{ sq. ft} \) to \( \text{ ha} \)
Use the factor of conversion (I) because it has \( \text { sq. ft} \) in the denominator and we are converting away from \( \text { sq. ft} \). Hence \[ \quad 22000 \text{ sq. ft} = 22000 \text{ sq. ft} \times \displaystyle \frac{1 \text { ha}}{107639 \text { sq. ft}} \] Cancel \( \text { sq. ft} \) on the right \[ \quad 22000 \text{ sq. ft} = 22000 \cancel { \text { sq. ft} } \times \displaystyle \frac{1 \text { ha}}{107639 \cancel { \text { sq. ft} }} \] Simplify and evaluate \[ \quad 22000 \text{ sq. ft} = \displaystyle \frac{22000 \times 1 \text { ha}}{107639 } = 0.20438 \text{ ha} \]

b) \( 1.3 \text{ ha} \) to \( \text{ sq. ft} \)
We are given \( \text{ ha} \) and therefore factor of conversion (II) will be used in the conversion because it has \( \text { ha} \) in the denominator. Hence \[ \quad 1.3 \text{ ha} = 1.3 \text{ ha} \times \frac{107639 \text { sq. ft}} {1 \text { ha}} \] Cancel \( \text{ ha} \) on the right \[ \quad 1.3 \text{ ha} = 1.3 \cancel{\text{ ha}} \frac{107639 \text { sq. ft}} {1 \cancel{\text { ha}}} \] Simplify and evaluate \[ \quad 1.3 \text{ ha} = \frac{ 1.3 \times 107639 \text { sq. ft}}{1} = 139930.7 \text{ sq. ft} \]

Factor of Conversion of Units