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Hint: To solve the question, we have to mathematically represent the given information to get a relation between the length of original cloth and the cost per meter.

Complete step-by-step answer:

Let x, y be the length of the piece of cloth and the cost of the piece per meter respectively.

Given

The cost of a piece of cloth is equal to Rs. 200.

The calculated cost of a piece of cloth = xy.

\[\Rightarrow 200=xy\] …. (1)

The cost remains unchanged when the cloth is 5 meters longer and the cost of each meter of cloth is Rs. 2 less.

\[\Rightarrow 200=(x+5)(y-2)\]

Thus, we get \[xy=(x+5)(y-2)\]

By solving the above equation, we get

\[xy=xy+5y-2x-10\]

\[5y=2x+10\]

By substituting the above equation in equation (1) we get

\[5\left( \dfrac{200}{x} \right)=2x+10\]

\[1000=x(2x+10)\]

\[2{{x}^{2}}+10x-1000=0\]

\[{{x}^{2}}+5x-500=0\]

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

\[\Rightarrow x=20,-25\]

The measurement of the cloth cannot be negative. Thus, x = 20 meters.

By using equation (1) we get

\[200=20y\]

\[\Rightarrow y=10\] rupees per meter.

Thus, the original piece of cloth is 20 meters long in length and the rate per meter of the cloth is 10 rupees per meter.

Note: The possibility of the mistake can be at forming the right equations based on the given information. The other possibility of mistake can be, using all the calculated value obtained from solving the quadratic expression, cost and the length of cloth can never be negative. The alternative method of solving can be using factorization method for 200, where cost per metre and length of cloth can be guessed as 20, 10. By substituting values in the second equation derived from the given information and using a hit and trial method we can arrive at an answer.

Complete step-by-step answer:

Let x, y be the length of the piece of cloth and the cost of the piece per meter respectively.

Given

The cost of a piece of cloth is equal to Rs. 200.

The calculated cost of a piece of cloth = xy.

\[\Rightarrow 200=xy\] …. (1)

The cost remains unchanged when the cloth is 5 meters longer and the cost of each meter of cloth is Rs. 2 less.

\[\Rightarrow 200=(x+5)(y-2)\]

Thus, we get \[xy=(x+5)(y-2)\]

By solving the above equation, we get

\[xy=xy+5y-2x-10\]

\[5y=2x+10\]

By substituting the above equation in equation (1) we get

\[5\left( \dfrac{200}{x} \right)=2x+10\]

\[1000=x(2x+10)\]

\[2{{x}^{2}}+10x-1000=0\]

\[{{x}^{2}}+5x-500=0\]

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

\[\Rightarrow x=20,-25\]

The measurement of the cloth cannot be negative. Thus, x = 20 meters.

By using equation (1) we get

\[200=20y\]

\[\Rightarrow y=10\] rupees per meter.

Thus, the original piece of cloth is 20 meters long in length and the rate per meter of the cloth is 10 rupees per meter.

Note: The possibility of the mistake can be at forming the right equations based on the given information. The other possibility of mistake can be, using all the calculated value obtained from solving the quadratic expression, cost and the length of cloth can never be negative. The alternative method of solving can be using factorization method for 200, where cost per metre and length of cloth can be guessed as 20, 10. By substituting values in the second equation derived from the given information and using a hit and trial method we can arrive at an answer.