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

Given, cost of cloth = ₹ 200.

Let original cloth be x metres.

So,

Cost of each meter = ₹ $\dfrac{200}{x}$

For new cloth,

Length = (x + 5) meters

Cost of each meter = ₹ $\dfrac{200}{x} - 2$

Given, cost does not change on the above alteration.

$\therefore (x + 5) \times \Big(\dfrac{200}{x} - 2\Big) = 200 \\[1em] \Rightarrow x \times \dfrac{200}{x} - 2x + 5 \times \dfrac{200}{x} - 5 \times 2 = 200 \\[1em] \Rightarrow 200 - 2x + \dfrac{1000}{x} - 10 = 200 \\[1em] \Rightarrow 2x - \dfrac{1000}{x} + 200 - 200 + 10 = 0 \\[1em] \Rightarrow 2x - \dfrac{1000}{x} + 10 = 0 \\[1em] \Rightarrow \dfrac{2x^2 - 1000 + 10x}{x} = 0 \\[1em] \Rightarrow \dfrac{2}{x}(x^2 - 500 + 5x) = 0 \\[1em] \Rightarrow x^2 + 5x - 500 = 0 \\[1em] \Rightarrow x^2 + 25x - 20x - 500 = 0 \\[1em] \Rightarrow x(x + 25) - 20(x + 25) = 0 \\[1em] \Rightarrow (x - 20)(x + 25) = 0 \\[1em] \Rightarrow x - 20 = 0 \text{ or } x + 25 = 0 \\[1em] \Rightarrow x = 20 \text{ or } x = -25.$

Since, length cannot be negative.

∴ x = 20.

Original rate = ₹ $\dfrac{200}{20}$ = ₹ 10/metre.

Hence, length of piece = 20 meters and cost = ₹ 10/metre.