The length of a rectangle exceeds its breadth by 4 cm. If length and breadth are each increased by 3 cm, the area of the new rectangle will be 81 sq. cm more than that of the given rectangle. Find the length and breadth of the given rectangle. Check your solution.

Let the breadth of the rectangle be x cm.
Then the length of the rectangle will be (x + 4) cm.
Therefore, the area of the given rectangle = length ´ breadth
= (x + 4)x
= (x² + 4x) cm²
By the given condition, we have:
Area of the new rectangle = Area of the given rectangle + 81 cm²
\\ Area of the new rectangle = New length ´ New breadth
(x² + 4x) + 81 = [(x + 4) + 3] (x + 3)
x² + 4x + 81 = (x +7)(x +3)
x² + 4x + 81 = x (x +3) + 7 (x + 3)
x² + 4x + 81 = x² + 3x + 7x + 21

Check:
(1) (14-10)cm = 4cm, i.e. the length of rectangle exceeds the breadth by 4 cm.
(2) New length = 14 + 7 = 17 cm and new breadth = 10 + 3 = 13 cm.
Therefore, area of the new rectangle – area of the given rectangle
= 17 ´ 13 – 14 ´ 10
= 221 – 140
= 81 cm.