Question

The total cost of certain piece of cloth was ₹ 2,100. During special sale time, the shopkeeper offered \(2 \text{ m}\) extra cloth for free thus reducing the price of cloth per metre by ₹ 120. What was the original per metre price of cloth and its length?

Hint:

In cost-quantity problems, the equation \(\text{Rate}_1 - \text{Rate}_2 = \text{Difference}\) is a standard template that leads to a quadratic equation in quantity.

Answer 1

Step 3: Detailed Explanation:
Let original length be \(L\) metres and original price be \(P\) per metre.
\(L \times P = 2100 \Rightarrow P = 2100 / L\).
New length = \(L + 2\).
New price per metre = \(\frac{2100}{L + 2}\).
Given: \(P_{new} = P_{old} - 120\).
\[ \frac{2100}{L} - \frac{2100}{L + 2} = 120 \]
Divide by 120:
\[ \frac{17.5}{L} - \frac{17.5}{L + 2} = 1 \]
\[ 17.5 \left( \frac{L + 2 - L}{L(L+2)} \right) = 1 \]
\[ \frac{35}{L^2 + 2L} = 1 \Rightarrow L^2 + 2L - 35 = 0 \]
Factorizing: \((L + 7)(L - 5) = 0\).
Since length cannot be negative, \(L = 5 \text{ m}\).
Original price \(P = 2100 / 5 = ₹ 420\).
Step 4: Final Answer:
Original price was ₹ 420 per metre and length was 5 m.