A piece of cloth costs Rs 200. If the piece were 5m longer and each meter of cloth cost Rs 2 less, the cost of the piece would have remained unchanged. How long is the piece and what is its original rate per meter.

Let the piece of cloth is x metre and it costs Rs. y/metre.

Then by the given conditions

As, the total cost of cloth = 200

⇒ xy = 200 ...... (i)

Also, given if length will increase by 5 m cost will decrease by Rs 2.

⇒ (x + 5)(y – 2) = 200

⇒ xy + 5y – 2x – 10 = 200

⇒ 200 + 5y – 2x – 10 = 200

⇒ 5y – 2x – 10 = 0

⇒ 5y = 2x + 10

Putting this in (i),

⇒ x(2x + 10) = 200×5

⇒ 2x² + 10x = 1000

⇒ x² + 5x – 500 = 0

⇒ x² + 25x – 20x – 500 = 0

⇒ x(x + 25) – 20(x + 25) = 0

⇒ (x + 25)(x – 20) = 0

So, either x = – 25 or x = 20

As the length of the cloth cannot be negative,

So, x = 20 metre

Putting in (i),

20y = 200

y = 10

So, the cloth is 20 m long and the original rate is Rs. 10/metre