# Factor of Conversion of Units

 $\require{cancel}$ $\require{bbox}$

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, ...

## Factor of Conversion of Units

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

Use the given information to write factors of conversion and convert.

1. Given that $1 \text { lb} = 0.4536 \text { kg}$ convert
a) $2.5 \text{ kg}$ to $\text { lb}$
b) $12 \text{ lb}$ to $\text { kg}$
(Hint: $\text{ lb}$ is the abbreviation for pound and $\text{ kg}$ is the abbreviation for kilograms and both are units of mass).

2. Given that $1 \text { mi} = 1.60934 \text { km}$ convert
a) $17.5 \text{ mi}$ to $\text{ km}$
b) $11.06 \text{ km}$ to $\text{ mi}$
(Hint: $\text{ mi}$ is the abbreviation for mile and $\text{ km}$ is the abbreviation for kilometers and both are units of length).

3. Given that $1 \text { ha} = 107639 \text { sq. ft}$ convert
a) $22000 \text{ sq. ft}$ to $\text{ ha}$
b) $1.3 \text{ ha}$ to $\text{ sq. ft}$
(Hint: $\text{ ha}$ is the abbreviation for hectare and $\text{ sq. ft}$ is the abbreviation for square feet and both are units of area).