If the length of a rectangle is reduced by 5 units and its breadth is increased by 3 units, then the area of the rectangle is reduced by 8 square units. If length is reduced by 3 units and breadth is increased by 2 units, then the area of rectangle will increase by 67 square units. Then find the length and breadth of the rectangle.

Let the length be 'x' and breadth be 'y'

Area of rectangle = length × breadth

Area of rectangle = xy

First case:

Length = x - 5

Breadth = y + 3

As, area is reduced by 8 sq. units

⇒ xy - (x - 5)(y + 3) = 8

⇒ xy - (xy + 3x - 5y - 15) = 8

⇒ xy - xy - 3x + 5y + 15 = 8

⇒ 3x - 5y = 7 eq.[1]

Second case:

Length = x - 3

Breadth = y + 2

As, the area is increased by 67 units

⇒ (x - 3)(y + 2) - xy = 67

⇒ xy + 2x - 3y - 6 - xy = 67

⇒ 2x - 3y = 73 eq.[2]

Multiplying eq.[1] by 2 both side, and Multiplying eq.[2] by 3 both side, we get

⇒ 6x - 10y = 14 eq.[3]

⇒ 6x - 9y = 219 eq.[4]

Subtracting eq.[3] from eq.[4]

⇒ 6x - 9y - 6x + 10y = 219 - 14

⇒ y = 205

Using this in eq.[1]

⇒ 3x - 5(205) = 7

⇒ 3x - 1025 = 7

⇒ 3x = 1032

⇒ x = 344

Hence, length = x = 344 units

Breadth = y = 219 units.