The area of a rectangle increases by 200 sq. m, if the length is increased by 8 m and the breadth by 3 m. The area increases by 255 sq. m, if the length is increased by 3 m and breadth by 8 m. Find the length and the breadth of the rectangle.

Let 'l' be the length of the rectangle and 'b' be the breadth of the rectangle.

Area of the rectangle = length x breadth = lb

When the length is increased by 8m and breadth by 3 m, area is increases by 200 sq.m,

⇒ (l + 8)(b + 3) = lb + 200

⇒ l(b + 3) + 8(b + 3) = lb + 200

⇒ lb + 3l + 8b + 24 = lb + 200

⇒ 3l + 8b + 24 = 200

⇒ 3l + 8b = 200 - 24

⇒ 3l + 8b = 176……………..(1)

When the length is increased by 3m and breadth by 8m, area is increases by 255 sq.m,

⇒ (l + 3)(b + 8) = lb + 255

⇒ l(b + 8) + 3(b + 8) = lb + 255

⇒ lb + 8l + 3b + 24 = lb + 255

⇒ 8l + 3b + 24 = 255

⇒ 8l + 3b = 255 - 24

⇒ 8l + 3b = 231 ……………(2)

Solving the equation (1) and (2), we get

⇒ (3l + 8b = 176) x 8

(8l + 3b = 231) x 3

$\begin{matrix} & 24l & + & 64b & = & 1,408 \ & 24l & + & 9b & = & 693 \ & - & - & & & - \ \hline & & & 55b & = & 1,408 - 693 \ \Rightarrow & & & 55b & = & 715 \end{matrix}$

⇒ b = $\dfrac{715}{55}$

⇒ b = 13

From equation (1), we get:

⇒ 3l + 8 x 13 = 176

⇒ 3l + 104 = 176

⇒ 3l = 176 - 104

⇒ 3l = 72

⇒ l = $\dfrac{72}{3}$

⇒ l = 24

Hence, the length of the rectangle = 24 m and the breadth of the rectangle = 13 m.