Question

The length of a rectangular plot of a land exceeds its breadth by \[23{\text{ }}m.\] If the length is decreased by \[15{\text{ }}m\] and the breadth is increased by \[7{\text{ }}m,\] the area is reduced by \[360{\text{ }}{m^2}.\] Find the length and breadth of the plot.

Answer 1

Hint: To solve this question, we will assume the breadth of rectangular plot be \[x{\text{ }}m.\] Then we will get length of the rectangular plot as \[\left( {x + 23} \right){\text{ }}m\]. Now we have been given few conditions, by taking one by one, we will make equations, and solve to get our required answer, i.e., length and breadth of the plot.

Complete step-by-step answer:
We have been given the length of a rectangular plot of a land that exceeds its breadth by \[23{\text{ }}m.\] It is given that the length is decreased by \[15{\text{ }}m\] and the breadth is increased by \[7{\text{ }}m,\] and the area is reduced by \[360{\text{ }}{m^2}.\] We need to find the length and breadth of the plot.
Let the breadth of rectangular plot be \[x{\text{ }}m.\]
$\Rightarrow$ Therefore, according to the question, length of the rectangular plot \[ = {\text{ }}\left( {x + 23} \right){\text{ }}m\]
Now, let us draw a figure using this given information, i.e., breadth of rectangular plot be \[x{\text{ }}m\] and the length of the rectangular plot be \[\left( {x + 23} \right)m.\]

We know that, area of rectangle $ = length \times breadth$
$\Rightarrow$ Therefore, area of rectangular plot \[ = {\text{ }}x\left( {x + 23} \right)\]
We have been given that the length is decreased by \[15{\text{ }}m\] and the breadth is increased by \[7{\text{ }}m,\] and the area is reduced by \[360{\text{ }}{m^2}.\]
So, length of the rectangular plot \[ = {\text{ }}\left( {x + 23} \right){\text{ }} - {\text{ }}15{\text{ }} = {\text{ }}\left( {x + 8} \right)m\]
And, breadth of the rectangular plot \[ = {\text{ }}\left( {x + 7} \right)m\]
Therefore, the area of the rectangular plot \[ = \left[ {\left( {x + 8} \right){\text{ }}\left( {x + 7} \right)} \right]\]
\[ = {\text{ }}{x^2} + {\text{ }}15x{\text{ }} + {\text{ }}56\]
Now, it is given that the area is reduced by \[360{\text{ }}{m^2}.\]
Therefore on equating both the area, we get
 \[\Rightarrow x\left( {x + 23} \right){\text{ }}-{\text{ }}360{\text{ }} = {\text{ }}{x^2} + {\text{ }}15x{\text{ }} + {\text{ }}56\]
\[
\Rightarrow {x^2} + {\text{ }}23x{\text{ }}-{\text{ }}360{\text{ }} = {\text{ }}{x^2} + {\text{ }}15x{\text{ }} + {\text{ }}56 \\
\Rightarrow 23x{\text{ }} - {\text{ }}15x{\text{ }} = {\text{ }}56{\text{ }} + {\text{ }}360 \\
\Rightarrow 8x{\text{ }} = {\text{ }}416
 \]
\[\Rightarrow x{\text{ }} = {\text{ }}\dfrac{{416}}{8}\]
\[\Rightarrow x{\text{ }} = {\text{ }}52\]
$\Rightarrow$ Therefore, breadth of the rectangular plot \[ = {\text{ }}52m\]
And length of the rectangular plot \[ = {\text{ }}x{\text{ }} + {\text{ }}23{\text{ }} = {\text{ }}75{\text{ }}m\]
So, now let us see the figure of a rectangular plot.

Thus, the length and breadth of the plot is \[75m\] and \[52m\] respectively.

Note: Students should carefully read the question. And take each sentence one by one. At first it is given that the length of a rectangular plot of a land exceeds its breadth by \[23{\text{ }}m.\] So, we have assumed breadth here. Then we are given that the length is decreased by \[15{\text{ }}m\] and the breadth is increased by \[7{\text{ }}m.\] So, we have taken this condition along with the first condition.