The area of a rectangle gets reduced by 8 m², if its length is reduced by 5 m and breadth increased by 3 m. If we increase the length by 3 m and breadth by 2 m, the area is increased by 74 m². Find the length and breadth of the rectangle.

Let x meters be the length and y meters be the breadth of the rectangle.

Given,

If the length is reduced by 5 m and breadth is increased by 3 m, then the area reduces by 8 m².

⇒ (x - 5)(y + 3) = xy - 8

⇒ (xy + 3x - 5y - 15) = xy - 8

⇒ 3x - 5y - 15 + 8 = xy - xy

⇒ 3x - 5y - 7 = 0

⇒ 3x = 5y + 7

⇒ x = $\dfrac{5y + 7}{3}$     ……..(1)

Given,

If the length is increased by 3 m and breadth by 2 m, then the area increases by 74 m².

⇒ (x + 3)(y + 2) = xy + 74

⇒ (xy + 2x + 3y + 6) = xy + 74

⇒ 2x + 3y + 6 - 74 = xy - xy

⇒ 2x + 3y - 68 = 0     …….(2)

Substituting value of x from equation (1) in (2), we get :

$\Rightarrow 2 \Big(\dfrac{5y + 7}{3}\Big) + 3y - 68 = 0 \\[1em] \Rightarrow \Big(\dfrac{10y + 14}{3}\Big) + 3y = 68 \\[1em] \Rightarrow \Big(\dfrac{10y + 14 + 9y}{3}\Big) = 68 \\[1em] \Rightarrow 19y + 14 = 68 \times 3 \\[1em] \Rightarrow 19y + 14 = 204 \\[1em] \Rightarrow 19y = 204 - 14 \\[1em] \Rightarrow 19y = 190 \\[1em] \Rightarrow y = \dfrac{190}{19} = 10.$

Substituting value of y in equation (1), we get :

$\Rightarrow x = \dfrac{5y + 7}{3} \\[1em] \Rightarrow x = \dfrac{5 \times 10 + 7}{3} \\[1em] \Rightarrow x = \dfrac{57}{3} = 19.$

Hence, the length and breadth of rectangles are 19 m and 10 m respectively.