Question

The ratio of the length and breadth of a 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?
$
  A{\text{ 27 units}} \\
  {\text{B 32 units}} \\
  {\text{C 21 units}} \\
  {\text{D 84 units}} \\
 $

Answer 1

Hint- Here we will proceed by assuming the length and breadth of the rectangle. Then we will use the formula of area of rectangle i.e. $l \times b$and perimeter of rectangle i.e. 2(l + b) to find the ratio of length and breadth of rectangle. Hence we will equate the calculated ratio of perimeter and area with the given ratio to find the length of the rectangle to get the desired result.

Complete step by step answer:

Let the length of the rectangle be 5x units.
Then the breadth of the rectangle is 2x units.
We know that perimeter of the rectangle = 2(l + b)
Now we will find the perimeter of the rectangle using the above formula = 2(5x + 2x)
 $ \Rightarrow 14x{\text{ units}}$
Here we will find the area of the rectangle by using the formula of area of rectangle $l \times b$.
Area of rectangle $ = \left( {5x} \right) \times \left( {2x} \right)$
$ \Rightarrow 10{x^2}units$
Now we are given the ratio of perimeter and area $ = \dfrac{1}{3}$
Here we will equate the calculated ratio of perimeter and area with the given ratio to find the length of the rectangle.
$ \Rightarrow \dfrac{{14x}}{{10{x^2}}} = \dfrac{1}{3}$
Now we will cross multiply both the ratios.
$ \Rightarrow 10{x^2} = 42x$
Or $10{x^2} - 42x = 0$
Or $x\left( {10x - 42} \right) = 0$
Either $x = 0$ or $x = \dfrac{{42}}{{10}}$
$ \Rightarrow $x = 0 or x = 4.2
As if we will put the value of x = 0 to find the length, length would be zero which is not possible.
So we will take x = 4.2 to find the length of the rectangle.
$ \Rightarrow 5x = 5 \times 4.2 = 21$
Therefore, the length of the rectangle is 21 units.
Hence option C is correct.
Note – While solving this question, one can also verify the answer by substituting the value of calculated length and breadth of the rectangle to find the ratio of perimeter and area of rectangle. Then we will get to know that the calculated ratio is equal to the given ratio which implies that the answer is right.