If the length and breadth of a room are increased by 1m and area is increased by 21 sq.metre.If the length is increased by 1m and breadth is decreased by 1 m , then the area is decreased by 5 sq.mt. Find the perimeter and area of the room ?

The correct answer is: ∴ Area and Perimeter of the given rectangle are 96 m2 & 40 m, respectively.

Step-by-step solution:-
Let the length of the original rectangle be x units & Breadth be y units.
∴  Area of the original rectangle = length × breadth
∴ Area of the original rectangle = x × y = xy ........................................................... (Equation i)
As per the first given condition,
Length = x + 1m
Breadth = y + 1m
∴  Area of the new rectangle = length × breadth
∴  Area of the new rectangle = (x + 1) × (y + 1)
∴  Area of the new rectangle = xy + x + y + 1 ......................................................... (Equation ii)
It is given that-
New Area = Original Area + 21 m².
xy + x + y + 1 = xy + 21 ....................................................................................... (From Equations i & ii)
∴  x + y + 1 = 21
∴  x + y = 21 - 1
∴  x + y = 20 .............................................................................................. (Equation iii)
As per second condition,
Length = x + 1m
Breadth = y - 1m
∴ Area of the new rectangle = length × breadth
∴ Area of the new rectangle = (x + 1) × (y - 1)
∴ Area of the new rectangle = xy - x + y - 1 ......................................................... (Equation iv)
It is given that-
New Area = Original Area - 5 m².
xy - x + y - 1 = xy - 5 ....................................................................................... (From Equations i & iv)
∴  - x + y - 1 = -5
∴  - x + y = -5 + 1
∴  - x + y = -4
∴  x - y = 4 .............................................................................................. (Equation v)
Adding Equations iii & v, we get-
 x + y = 20
 x - y = 4
 2x = 24
∴  x = 12
∴  Length = x = 12 m ............................................................................................. (Equation vi)
Substituting x = 12 in Equation iii, we get-
 x + y = 20
 12 + y = 20
 y = 20 - 12
 y = 8
∴ Breadth = y = 8 m .......................................................................................... (Equation vii)
Area of the given rectangle = length × breadth
Area of the given rectangle = 12 × 8 …................................................................ (From Equations vi & vii)
Area of the given rectangle = 96 m²
Perimeter of the given rectangle = 2 (length + breadth)
Perimeter of the given rectangle = 2 (12 + 8) ...................................................... (From Equations vi & vii)
Perimeter of the given rectangle = 2 ×20
Perimeter of the given rectangle = 40 m
Final Answer:-
∴  Area and Perimeter of the given rectangle are 96 m² & 40 m, respectively.