If the length of a rectangle is increased by 10 cm and the breadth decreased by 5 cm, the area remains unchanged. If the length is decreased by 5 cm and the breadth is increased by 4 cm, even then the area remains unchanged. Find the dimensions of the rectangle.

Let length be = l cm and Breadth be = b cm

Area = length × breadth = lb

In first condition :

Length is increased by 10 cm

Breadth is decreased by 5 cm.

But area remains unchanged:

∴ lb = (l + 10)(b - 5)

⇒ lb = lb - 5l + 10b - 50

⇒ 0 = 10b - 5l - 50

⇒ 50 = 10b - 5l

Dividing by 5,

⇒ 10 = 2b - l

⇒ l = 2b - 10 ……….(1)

In second condition:

Length is decreased by 5 cm

Breadth is increased by 4 cm

Here also area remains unchanged.

∴ lb = (l - 5)(b + 4)

⇒ lb = lb + 4l - 5b - 20

⇒ 0 = 4l - 5b - 20

⇒ 4l - 5b = 20 …….(2)

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

⇒ 4(2b - 10) - 5b = 20

⇒ 8b - 40 - 5b = 20

⇒ 3b = 20 + 40

⇒ 3b = 60

⇒ b = $\dfrac{60}{3}$ = 20 cm.

⇒ l = 2b - 10

= 2(20) - 10

= 40 - 10 = 30 cm.

Hence, length = 30 cm and breadth = 20 cm.