# The perimeter of square A is 5 times greater than the perimeter of square B. How many times greater is the area of square A than the area of square B?

Jan 31, 2016

If the length of each side of a square is $z$ then its perimeter $P$ is given by:

$P = 4 z$

Let the length of each side of square $A$ be $x$ and let $P$ denote its perimeter. .
Let the length of each side of square $B$ be $y$ and let $P '$ denote its perimeter.

$\implies P = 4 x \mathmr{and} P ' = 4 y$

Given that: $P = 5 P '$

$\implies 4 x = 5 \cdot 4 y$
$\implies x = 5 y$
$\implies y = \frac{x}{5}$

Hence, the length of each side of square $B$ is $\frac{x}{5}$.

If the length of each side of a square is $z$ then its perimeter $A$ is given by:
$A = {z}^{2}$

Here the length of square $A$ is $x$
and the length of square $B$ is $\frac{x}{5}$

Let ${A}_{1}$ denote the area of square $A$ and ${A}_{2}$ denote the area of square $B$.

$\implies {A}_{1} = {x}^{2} \mathmr{and} {A}_{2} = {\left(\frac{x}{5}\right)}^{2} ^$

$\implies {A}_{1} = {x}^{2} \mathmr{and} {A}_{2} = {x}^{2} / 25$

Divide ${A}_{1}$ by ${A}_{2}$

$\implies {A}_{1} / {A}_{2} = {x}^{2} / \left({x}^{2} / 25\right)$

$\implies {A}_{1} / {A}_{2} = 25$

$\implies {A}_{1} = 25 {A}_{2}$

This shows that the area of square $A$ is $25$ times greater than the area of square $B$.