Unit Analysis

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.

$4\frac{1}{2}\text{ft}\times \frac{12\text{in}}{1\text{ft}}=\frac{9\text{ft}}{2}\times \frac{12\text{in}}{1\text{ft}}$

$=\frac{9\text{ft}\times 12\text{in}}{2\times 1\text{ft}}$

$=\frac{108\text{in}}{2}$

$=54\text{in}$

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.

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

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

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

$\frac{80\text{mi}}{\$6}\times \frac{\$2.50}{1\text{gal}}\times \frac{1\text{gal}}{3.79\text{L}}\times \frac{1\text{km}}{0.62\text{mi}}$

$=\frac{80\text{mi}\times \$2\text{.50}\times 1\text{gal}\times 1\text{km}}{\$6\times 1\text{gal}\times 3.79\text{L}\times 0.62\text{mi}}$

$=\frac{80\cancel{\text{mi}}\times \cancel{\$}2.50\times 1\cancel{\text{gal}}\times 1\text{km}}{\cancel{\$}6\times 1\cancel{\text{gal}}\times 3.79\text{L}\times 0.62\cancel{\text{mi}}}$

$=\frac{200\text{km}}{14.0988\text{L}}$

$\approx 14.2\text{km}/\text{L}$