If the length and the breadth of a room are increased by 1 metre, the area is increased by 21 square metres. If the length is increased by 1 metre and breadth is decreased by 1 metre the area is decreased by 5 square metres. Find the perimeter of the room.

Let length = l metres and breadth = b metres.

Area = lb m²

Given,

If the length and the breadth of a room are increased by 1 metre, the area is increased by 21 square metres,

∴ (l + 1)(b + 1) - lb = 21

⇒ lb + l + b + 1 - lb = 21

⇒ l + b = 21 - 1

⇒ l + b = 20 ……….(1)

Given,

If the length is increased by 1 metre and breadth is decreased by 1 metre the area is decreased by 5 square metres.

∴ lb - (l + 1)(b - 1) = 5

⇒ lb - (lb - l + b - 1) = 5

⇒ lb - lb + l - b + 1 = 5

⇒ l - b = 4 ……….(2)

Adding equation (1) and (2) we get,

⇒ l + b + l - b = 20 + 4

⇒ 2l = 24

⇒ l = 12 m.

Substituting value of l in (2) we get,

⇒ 12 - b = 4

⇒ b = 12 - 4 = 8 m.

Perimeter of room = 2(l + b) = 2 × 20 = 40 m.

Hence, perimeter of room = 40 m.