Question

The total cost of a certain length of a piece of cloth is Rs 200. If the piece was 5 m longer and each metre of cloth costs Rs 2 less, the cost of the piece would have remained unchanged. How long is the piece and what is its original rate per metre ?

Answer 1

Let the length of the piece of cloth be x m.
Total cost of the piece of cloth = Rs 200
Then, cost per metre= \[\frac{200}{x}\]

New length = (x + 5) m
Since, the cost of piece of cloth remains unchanged.
∴ New cost per metre = Rs \[\frac{200}{x + 5}\]

According to the question,

\[\frac{200}{x} - \frac{200}{x + 5} = 2\]

⇒\[\frac{200}{x} - \frac{200}{x + 5} = 2\]

\[\Rightarrow \frac{200x + 1000 - 200x}{x^2 + 5x} = 2\]

\[\Rightarrow 2\left( x^2 + 5x \right) = 1000\]
\[ \Rightarrow x^2 + 5x - 500 = 0\]
\[ \Rightarrow x^2 + 25x - 20x - 500 = 0\]

\[\Rightarrow x\left( x + 25 \right) - 20\left( x + 25 \right) = 0\]
\[ \Rightarrow \left( x - 20 \right)\left( x + 25 \right) = 0\]

\[\Rightarrow x = 20 \text{or}\ x = - 25\]

As x cannot be negative, so x is 20.

∴ Cost per metre = Rs\[\frac{200}{20}\]

Hence, the length of the piece of cloth is 20 m and original rate is Rs 10 per metre.