Question

The area of a rectangle is 460 square meters. If the length is $15\% $ more than the breadth, what is the breadth of the rectangular field?
a.15 meters
b.26 meters
c.$34.5$ meters
d.Cannot be determined
e.None of these

Answer 1

Hint: Here, we will find the breadth of the rectangular field. We will use the condition to find the relation between the length and breadth of a rectangle and then find the length. Then we will substitute the obtained length and given breadth in the formula of the area of the rectangle and find the breadth of the rectangle. The area is defined as the space occupied by the two-dimensional shape.

Formula Used:
We will use the following formulas:
1.Amount of quantity is given by $x = \dfrac{P}{{100}} \times n$ where $x,P,n$ are the amount of quantity, percentage and the total amount respectively.
2.Area of a rectangular field is given by $A = l \times b$ where $l$ is the length of the rectangle and $b$ is the breadth of the rectangle.

Complete step-by-step answer:
Let $l$ be the length of a rectangle and $b$ be the breadth of the rectangle.
We are given that the length is $15\% $ more than the breadth and the area of a rectangle is 460 square meters.
Length$ = $ Breadth $ + 15\% \times $ Breadth
$ \Rightarrow l = b + 15\% \times b$
Now simplifying the percentage using the formula $x = \dfrac{P}{{100}} \times n$, we get
$ \Rightarrow l = b + \dfrac{{15}}{{100}} \times b$
By taking LCM on both the sides, we get
$ \Rightarrow l = b \times \dfrac{{100}}{{100}} + \dfrac{{15}}{{100}} \times b$
$ \Rightarrow l = \dfrac{{100b}}{{100}} + \dfrac{{15b}}{{100}}$
Adding the like terms, we get
$ \Rightarrow l = \dfrac{{115b}}{{100}}$
By dividing the numbers, we get
$ \Rightarrow l = \dfrac{{23b}}{{20}}$
We are given that the area of a rectangle is 460 square meters.
Substituting the value of length $l = \dfrac{{23}}{{20}}b$ in the formula $A = l \times b$, we get
$ \Rightarrow l \times b = 460$
$ \Rightarrow \dfrac{{23}}{{20}}b \times b = 460$
Multiplying the terms, we get
$ \Rightarrow \dfrac{{23}}{{20}}{b^2} = 460$
On cross-multiplication, we get
$ \Rightarrow {b^2} = 460 \times \dfrac{{20}}{{23}}$
$ \Rightarrow {b^2} = 20 \times 20$
By taking square root on both the sides, we get
$ \Rightarrow b = \sqrt {20 \times 20} $
$ \Rightarrow b = 20m$
Therefore, the breadth of a rectangular field is 20 m.
Thus option (e) is the correct answer.

Note: Rectangle is a two-dimensional geometric shape that has 4 sides where the opposite sides parallel to each other. Here, we should remember that the breadth has to be added with the $15\% $ of the breadth and not be multiplied with the breadth to find the length. This is because the length is $15\% $ more than breadth and not $15\% $ of the breadth.