Question

One side of a rectangle is $20\text{cm}$ larger than the other side.

If you make the smaller side two times larger and the larger side three times larger, then the perimeter of the new rectangle would be $240\text{cm}$.

Find the lengths of the sides of the original rectangle.

Answer 1

Step-1: Formula of the perimeter of the rectangle:

The perimeter of the rectangle is expressed as $P=2\left(l+w\right)$, where $l,w$ are the length and width respectively.

Step-2: Apply the formula of the perimeter of the rectangle to find the value of $x$.

Suppose the one side of the rectangle is $x\text{cm}$

It is given that the other side is $20\text{cm}$ larger than the one side.

Thus the other side is $\left(20+x\right)$.

The new length of a side when the smaller side is two times larger is $2x$

The new length of a side when the larger side is three times larger is $3\left(x+20\right)$,

Since, the perimeter of new rectangle is $240\text{cm}$.

Thus,

$\begin{array}{rcl}2\left(2x+3\left(x+20\right)\right)& =& 240\left[\mathrm{Simplify}\mathrm{the}\mathrm{left}\mathrm{side}\right]\\ & \Rightarrow & 2\left(2x+3x+60\right)=240\\ & \Rightarrow & 2\left(5x+60\right)=240\\ & \Rightarrow & 10x+120=240\left[\mathrm{Subtract}120\mathrm{from}\mathrm{both}\mathrm{sides}\right]\\ & \Rightarrow & 10x=240-120\\ & \Rightarrow & 10x=120\left[\mathrm{Divide}\mathrm{both}\mathrm{sides}\mathrm{by}10\right]\\ & \Rightarrow & x=\frac{120}{10}\\ \therefore x& =& 12\end{array}$

The larger side of original rectangle will be $20+12=32$

Hence, the lengths of sides of the original rectangle are $12\text{cm},32\text{cm}$.