If the length of a rectangle is increased by 5 metres and breadth decreased by 3 metres, the area would decrease by 5 square metres. If the length is increased by 3 metres and breadth increased by 2 metres, the area would increase by 50 square metres. What are the length and breadth?

Let the length be ‘x’ and breadth be ‘y’.

Area of rectangle = length × breadth

= x × y

= xy

According to the question,

The length of a rectangle is increased by 5 metres and breadth decreased by 3 metres, the area would decrease by 5 square metres

⇒ (x + 5) × (y – 3) = xy – 5

⇒ x (y – 3) + 5(y – 3) = xy – 5

⇒ xy – 3x + 5y – 15 = xy – 5

⇒ xy – 3x + 5y – xy = –5 + 15

⇒ –3x + 5y = 10 … (1)

the length is increased by 3 metres and breadth increased by 2 metres, the area would increase by 50 square metres

⇒ (x + 3) × (y + 2) = xy + 50

⇒ x (y + 2) + 3(y + 2) = xy + 50

⇒ xy + 2x + 3y + 6 = xy + 50

⇒ xy + 2x + 3y – xy = 50 – 6

⇒ 2x + 3y = 44 … (2)

Equating equation (1) and (2)

–3x + 5y = 10 … (1)

2x + 3y = 44 … (2)

Multiplying equation (1) by 2 and equation (2) by 3 and equate

y = 8

Put y = 8 in Equation (3)

–6x + 10 × 8 = 20

⇒ –6x + 80 = 20

⇒ –6x = 20 – 80

⇒ –6x = –60

⇒ 6x = 60

⇒

⇒ x = 10

∴ The length of rectangle is 10 metres and breadth are 8 metres.