The area of rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. However, if the length of this rectangle increases by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.

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

Given,

Area of rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units.

⇒ (x - 5)(y + 3) = xy - 9

⇒ xy + 3x - 5y - 15 = xy - 9

⇒ xy - xy + 3x - 5y = -9 + 15

⇒ 3x - 5y = 6 ……..(1)

Given,

If the length of this rectangle increases by 3 units and the breadth by 2 units, the area increases by 67 square units.

⇒ (x + 3)(y + 2) = xy + 67

⇒ xy + 2x + 3y + 6 = xy + 67

⇒ xy - xy + 2x + 3y = 67 - 6

⇒ 2x + 3y = 61 ……..(2)

Multiplying equation (1) by 2, we get :

⇒ 2(3x - 5y) = 2 × 6

⇒ 6x - 10y = 12 ………(3)

Multiplying equation (2) by 3, we get :

⇒ 3(2x + 3y) = 3 × 61

⇒ 6x + 9y = 183 ………(4)

Subtracting equation (3) from (4), we get :

⇒ (6x + 9y) - (6x - 10y) = 183 - 12

⇒ 6x - 6x + 9y + 10y = 171

⇒ 19y = 171

⇒ y = $\dfrac{171}{19}$ = 9.

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

⇒ 3x - 5(9) = 6

⇒ 3x - 45 = 6

⇒ 3x = 45 + 6

⇒ 3x = 51

⇒ x = $\dfrac{51}{3}$ = 17.

Hence, length = 17 units and breadth = 9 units.