Question

The ratio of length and breadth of the rectangle is $5:2$ respectively. The respective ratio of its perimeter and area is $1:3$ (irrespective of the unit). What is the length of the rectangle?
$\left( a \right){\text{ 27 units}}$
$\left( b \right){\text{ 32 units}}$
$\left( c \right){\text{ 21 units}}$
$\left( d \right){\text{ 84 units}}$

Answer 1

Hint:
For solving this question we just need to apply the area and perimeter formula for the rectangle. First of all, we will assume the length and breadth be $5x{\text{ and 2x}}$ respectively. Then by using the formula, the perimeter of a rectangle $ = 2\left( {l + b} \right)$ and the area of a rectangle $ = l \times b$. Then on dividing, we can get the ratios easily.

Formula used:
Perimeter of rectangle $ = 2\left( {l + b} \right)$
Area of rectangle $ = l \times b$
Here,
$l$, will be the length of the rectangle
$b$, will be the breadth of the rectangle

Complete step by step solution:
 Let us assume the length be $5x$and the breadth of the rectangle be$2x$.
So now by using the formula,
Area of rectangle $ = l \times b$
On substituting the values, we get
$ \Rightarrow 5x \times 2x$
And on solving the above equation, we get
$ \Rightarrow 10{x^2}$
Now by using the formula for perimeter which is given by
Perimeter of rectangle $ = 2\left( {l + b} \right)$
On substituting the values, we get
$ \Rightarrow 2\left( {5x + 2x} \right)$
And on solving the above equation, we get
$ \Rightarrow 14x$
Therefore, the perimeter of the rectangle is $14x$
The area of the rectangle is $10{x^2}$
So now on calculating the ratios of these two we can write it as,
$ \Rightarrow \dfrac{{Perimeter{\text{ }}of{\text{ }}rectangle}}{{{\text{Area }}of{\text{ }}rectangle}}$
Now on substituting the values, we get
$ \Rightarrow \dfrac{{14x}}{{10{x^2}}}$
And on solving more we can get the values for$x$, which will be equal to
$ \Rightarrow x = 4.2$
Therefore, the length of the rectangle will be calculated as
$ \Rightarrow 5x = 5 \times 4.2$
On solving the above equation, we get
$ \Rightarrow 21units$

Therefore, the option $\left( c \right)$ is correct.

Note:
Here in this type of question there is only one concept needed which assumes the ratios to be in the terms of any constant and then by using the formula and can easily solve for it. There is also no need of taking care of units in this question.