The area of a rectangle gets reduced by 8 m², when its length is reduced by 5m and its breadth is increased by 3 m. If we increase the length by 3 m and breadth by 2 m, the area is increased by 74 m². Find the length and the breadth of the rectangle.

Let the length of a rectangle = x m

and the breadth of a rectangle = y m

Then, Area of rectangle = xy m²

Condition I :

Area is reduced by 8m², when length = (x – 5) m and breadth = (y + 3) m

Then, area of rectangle = (x – 5)×(y + 3) m²

According to the question,

xy – (x – 5)×(y + 3) = 8

⇒ xy – (xy + 3x – 5y – 15) = 8

⇒ xy – xy – 3x + 5y + 15 = 8

⇒ – 3x + 5y = 8 – 15

⇒ 3x – 5y = 7 …(i)

Condition II:

Area is increased by 74m², when length = (x + 3) m and breadth = (y + 2) m

Then, area of rectangle = (x + 3)×(y + 2) m²

According to the question,

(x + 3)×(y + 2) – xy = 74

⇒ (xy + 3y + 2x + 6) – xy = 74

⇒ xy + 2x + 3y + 6 – xy = 74

⇒ 2x + 3y = 74 – 6

⇒ 2x + 3y = 68 …(ii)

On multiplying Eq. (i) by 2 and Eq. (ii) by 3, we get

6x – 10y = 14 …(iii)

6x + 9y = 204 …(iv)

On subtracting Eq. (i) from Eq. (ii), we get

6x + 9y – 6x + 10y = 204 – 14

⇒ 19y = 190

⇒ y = 10

On putting the value of y = 10 in Eq. (i), we get

3x – 5 (10) = 7

⇒ 3x – 50 = 7

⇒ 3x = 57

⇒ x = 19

Hence, the length of the rectangle is 19m and the breadth of a rectangle is 10m