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:
- The factors of conversion go in pairs; if you know one, you know the other by interchanging numerator and denominator.
- They are equal to \( 1 \) and may therefore be used in the conversion of units as 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.
View Solution
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:
Example 2
Given that \( 1 \text{ in} = 2.54 \text{ cm} \), define factors of conversion and convert:
a) \( 23.8 \text{ cm} \) to \( \text{in} \)
b) \( 11.7 \text{ in} \) to \( \text{cm} \)
View Solution
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, simplify, and evaluate:
b) Convert \( 11.7 \text{ in} \) 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), simplify, and evaluate:
Exercises
Use the given information to write factors of conversion and convert.
-
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 pound and \( \text{kg} \) is kilograms, both units of mass). -
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 mile and \( \text{km} \) is kilometers, both units of length). -
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 hectare and \( \text{sq. ft} \) is square feet, both units of area).
Solutions to the Above Exercises
-
Given \( 1 \text{ lb} = 0.4536 \text{ kg} \)
Factors of conversion:
\[ \displaystyle \frac{1 \text{ lb}}{0.4536 \text{ kg}} = 1 \quad (I) \quad \text{or} \quad \displaystyle \frac{0.4536 \text{ kg}}{1 \text{ lb}} = 1 \quad (II) \]a) Convert \( 2.5 \text{ kg} \) to \( \text{lb} \)
Use factor (I) to cancel \( \text{kg} \):
\[ 2.5 \text{ kg} \times \frac{1 \text{ lb}}{0.4536 \text{ kg}} = \frac{2.5 \times 1}{0.4536} \]\[ 2.5 \text{ kg} = 5.51 \text{ lb} \]b) Convert \( 12 \text{ lb} \) to \( \text{kg} \)
Use factor (II) to cancel \( \text{lb} \):
\[ 12 \text{ lb} \times \frac{0.4536 \text{ kg}}{1 \text{ lb}} = 12 \times 0.4536 \]\[ 12 \text{ lb} = 5.4432 \text{ kg} \] -
Given \( 1 \text{ mi} = 1.60934 \text{ km} \)
Factors of conversion:
\[ \displaystyle \frac{1 \text{ mi}}{1.60934 \text{ km}} = 1 \quad (I) \quad \text{or} \quad \displaystyle \frac{1.60934 \text{ km}}{1 \text{ mi}} = 1 \quad (II) \]a) Convert \( 17.5 \text{ mi} \) to \( \text{km} \)
Use factor (II) to cancel \( \text{mi} \):
\[ 17.5 \text{ mi} \times \frac{1.60934 \text{ km}}{1 \text{ mi}} = 17.5 \times 1.60934 \]\[ 17.5 \text{ mi} = 28.16345 \text{ km} \]b) Convert \( 11.06 \text{ km} \) to \( \text{mi} \)
Use factor (I) to cancel \( \text{km} \):
\[ 11.06 \text{ km} \times \frac{1 \text{ mi}}{1.60934 \text{ km}} = \frac{11.06}{1.60934} \]\[ 11.06 \text{ km} = 6.87238 \text{ mi} \] -
Given \( 1 \text{ ha} = 107639 \text{ sq. ft} \)
Factors of conversion:
\[ \displaystyle \frac{1 \text{ ha}}{107639 \text{ sq. ft}} = 1 \quad (I) \quad \text{or} \quad \displaystyle \frac{107639 \text{ sq. ft}}{1 \text{ ha}} = 1 \quad (II) \]a) Convert \( 22000 \text{ sq. ft} \) to \( \text{ha} \)
Use factor (I) to cancel \( \text{sq. ft} \):
\[ 22000 \text{ sq. ft} \times \frac{1 \text{ ha}}{107639 \text{ sq. ft}} = \frac{22000}{107639} \]\[ 22000 \text{ sq. ft} = 0.20438 \text{ ha} \]b) Convert \( 1.3 \text{ ha} \) to \( \text{sq. ft} \)
Use factor (II) to cancel \( \text{ha} \):
\[ 1.3 \text{ ha} \times \frac{107639 \text{ sq. ft}}{1 \text{ ha}} = 1.3 \times 107639 \]\[ 1.3 \text{ ha} = 139930.7 \text{ sq. ft} \]