# Unit Analysis

Unit analysis means using the rules of multiplying and reducing fractions to solve problems involving different units.

Example 1:

A desk is $4\frac{1}{2}$ feet long. Find the length of the desk in inches.

There are $12$ inches in $1$ foot. Write this relationship as a fraction, and multiply it by the given length.

$4\frac{1}{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ft}\text{\hspace{0.17em}}\text{\hspace{0.17em}}×\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{12\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}}{1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ft}}=\frac{9\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ft}}{2\text{\hspace{0.17em}}\text{\hspace{0.17em}}}×\frac{12\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}}{1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ft}}$

$=\frac{9\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ft}\text{\hspace{0.17em}}\text{\hspace{0.17em}}×\text{\hspace{0.17em}}\text{\hspace{0.17em}}12\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}}{2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ft}}$

$=\frac{108\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}}{2}$

$=54\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}$

Note that you can cancel the unit "ft" just as though it were a number being multiplied.

Example 2:

A car can drive $80$ miles on $6$ worth of gasoline. If $1$ gallon of gasoline costs $2.50$ , find how far the car can travel in kilometers on $1$ liter of gasoline.

To solve this problem you need to know some conversions.

$1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{kilometer}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{km}\right)\approx 0.62\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{miles}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{mi}\right)$

$1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{gallon}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{gal}\right)\approx 3.79\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{liters}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(L\right)$

Write all the conversions as fractions, so that the unwanted units will cancel.

$\frac{80\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{mi}}{6}×\frac{2.50}{1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{gal}}×\frac{1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{gal}}{3.79\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{L}}×\frac{1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{km}}{0.62\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{mi}}$

$=\frac{80\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{mi}\text{\hspace{0.17em}}×\text{\hspace{0.17em}}2\text{.50}\text{\hspace{0.17em}}×\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{gal}\text{\hspace{0.17em}}×\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{km}}{6\text{\hspace{0.17em}}×\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{gal}\text{\hspace{0.17em}}×\text{\hspace{0.17em}}3.79\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{L}\text{\hspace{0.17em}}×\text{\hspace{0.17em}}0.62\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{mi}}$

$=\frac{80\text{\hspace{0.17em}}\text{\hspace{0.17em}}\overline{)\text{mi}}\text{\hspace{0.17em}}×\text{\hspace{0.17em}}\overline{)}2.50\text{\hspace{0.17em}}×\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\overline{)\text{gal}}\text{\hspace{0.17em}}×\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{km}}{\overline{)}6\text{\hspace{0.17em}}×\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\overline{)\text{gal}}\text{\hspace{0.17em}}×\text{\hspace{0.17em}}3.79\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{L}\text{\hspace{0.17em}}×\text{\hspace{0.17em}}0.62\text{\hspace{0.17em}}\text{\hspace{0.17em}}\overline{)\text{mi}}}$

$=\frac{200\text{\hspace{0.17em}}\text{km}}{14.0988\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{L}}$

$\approx 14.2\text{\hspace{0.17em}}\text{km}/\text{L}$