A piece of cloth cost ₹300. If the piece was 5 metre longer and each metre of cloth cost ₹2 less, the cost of the piece would have remained unchanged. How long is the original piece of cloth and what is the rate per metre?

Let the length of original cloth be x metres

Since, cost of total piece = ₹300

So, cost of each metre = $\dfrac{300}{x}$

Given, new length is 5 metre more i.e. (x + 5) metres and cost of each metre is ₹2 less i.e. $\dfrac{300}{x} - 2$

Since total cost remains unchanged

$\therefore (x + 5) \times \big(\dfrac{300}{x} - 2 \big) = 300 \\[1em] \Rightarrow (x + 5)\big(\dfrac{300 - 2x}{x}\big) = 300 \\[1em] \Rightarrow (x + 5)(300 - 2x) = 300x \text{ (On cross multiplication) } \\[1em] \Rightarrow 300x - 2x^2 + 1500 - 10x - 300x = 0 \\[1em] \Rightarrow -2x^2 + 1500 - 10x = 0 \\[1em] \Rightarrow -2(x^2 + 5x - 750) = 0 \\[1em] \Rightarrow x^2 + 5x - 750 = 0 \\[1em] \Rightarrow x^2 + 30x - 25x - 750 = 0 \\[1em] \Rightarrow x(x + 30) - 25(x + 30) = 0 \\[1em] \Rightarrow (x + 30)(x - 25) = 0 \\[1em] \Rightarrow x + 30 = 0 \text{ or } x - 25 = 0 \\[1em] x = -30 \text{ or } x = 25$

Since, length cannot be negative hence, x ≠ -30

∴ x = 25 , $\dfrac{300}{x}$ = 12

Hence, the length of original cloth is 25 metre and cost per metre is ₹12.