# The width of a rectangular box is 20% of the length. If the perimeter is 192 cm, then what are the length and width of the box?

Jul 6, 2017

length$= 80 c m$

#### Explanation:

Let the width be $w$
Let the length be $L$
Let perimeter be $p$

Drop the units of measurement for now

Then $p = 2 w + 2 L = 2 \left(w + L\right)$

but $w = \frac{20}{100} L$ so by substitution we have:

$p = 192 = 2 \left(\frac{20}{100} L + L\right)$

Factor out $L$

$192 = 2 L \left(\frac{20}{100} + 1\right)$

but $\frac{20}{100} + 1$ is the same as $\frac{20}{100} + \frac{100}{100} = \frac{120}{100}$

$192 = {\cancel{2}}^{1} L \left(\frac{120}{{\cancel{100}}^{50}}\right)$

$L = 192 \times \frac{50}{120} = 80$

$L = 80 c m$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
check

width = $\frac{20}{100} L = \frac{20}{100} \times 80 = 16$

$2 w + 2 L \to 2 \left(80\right) + 2 \left(16\right) = 192$ as required