The length and breadth of a rectangular plot are 96m and 72m respectively. It consists of a rectangle inside it surrounded by a border of uniform width. If the area of the surrounding border is equal to the area of the inner rectangle , find the length and breadth of the inner rectangle.

Given: Length (l) and breadth (b) of rectangular plot are 96 m and 72 m

Let the thickness be x m

Then length and breadth of inner rectangle are (l –2x) and (b – 2x)

Thus area of rectangular plot = lb = 96 m × 72 m = 6912 m² 

and area of inner rectangle = (l – 2x) (b – 2x)

                    = lb – 2x (l + b) + 4x² = 6912 – 336x + 4x²

Area of border = lb – (lb – 2x (l + b) + 4x²)

                    = 2x (l + b) – 4x² = 336x – 4x²

Also area of inner rectangle = area of border

⇒ 6912 – 336x + 4x² = 336x – 4x² 

⇒ 8x² – 672x + 6912 = 0

⇒ x² – 84x + 864 = 0

⇒ x² – 72x  – 12x + 864 = 0

⇒ x(x – 72)  – 12 (x – 72) = 0

⇒ (x – 12) (x – 72) = 0

⇒ x = 12  or  x = 72

But when x = 72 length and breadth will be negative

∴ x = 12

and length and breadth of inner rectangle are 96 – 2 × 12 and 72 – 2 × 12

                                                                    = 72 m and 48 m