A piece of cloth costs ₹120 in total. If the piece had been 2 metres longer and the price per metre ₹2 less, the total cost would remain ₹120 unchanged. Find the original rate per metre of the cloth.

Introduction / Context:
This question mixes two unknowns—original length and original price per metre—under a constant total cost. Changing both length and rate but keeping the product constant leads to a system that can be solved using algebraic manipulation and a simple quadratic equation.

Given Data / Assumptions:

• Total cost (original) = ₹120.
• Original length = L metres; original rate = r ₹/m.
• After change: length = L + 2, rate = r − 2.
• Total cost remains ₹120 after the change.

Concept / Approach:
Model the total cost as rate * length. Then rL = 120 and (r − 2)(L + 2) = 120. Subtract the two equal-cost expressions to eliminate rL and get a relation between r and L. Use that relation with rL = 120 to solve for r (the required rate).

Step-by-Step Solution:

Verification / Alternative check:
Original: 10 m × ₹12/m = ₹120. New: 12 m × ₹10/m = ₹120. Unchanged total cost confirmed.

Why Other Options Are Wrong:

• ₹15, ₹10, ₹8, ₹18: Do not satisfy both equations simultaneously when paired with corresponding lengths; only ₹12 works.

Common Pitfalls:

• Setting up (r − 2)(L + 2) = rL but then incorrectly expanding or cancelling terms.
• Forgetting to reject negative length solutions.

Final Answer:
₹ 12