Bargain impact on expected profit — An item is marked at ₹ M. A customer bargains and buys it at M/2. This reduces the trader’s expected profit by 66.66% (i.e., two-thirds) relative to selling at the marked price. What is the percentage discount realized by the customer?

Introduction / Context:
The problem links absolute discount with a relative change in profit. The trader’s expected profit at marked price is reduced to one-third of its original value when the customer pays M/2. We can infer the cost from this relationship and then confirm the discount percentage, which is naturally computed on the marked price.

Given Data / Assumptions:

• Marked price = M; actual selling price = M/2.
• Let cost be C; expected profit at M is (M − C).
• Actual profit at M/2 is (M/2 − C).
• Actual profit = one-third of expected profit (66.66% diminution).

Concept / Approach:
Set up (M/2 − C) = (1/3) * (M − C) to determine C in terms of M. Then the discount percentage is (M − M/2) / M * 100 irrespective of C. This confirms whether the bargain is 50%.

Step-by-Step Solution:

Verification / Alternative check:
With C = M/4, expected profit at M is 3M/4; at M/2 it is M/4, which is one-third of 3M/4, matching the 66.66% reduction.

Why Other Options Are Wrong:
33.33% and 66.66% are the reductions in profit or misapplied bases, not the price discount; “none” and 40% are inconsistent with the equation.

Common Pitfalls:
Comparing discount to cost or confusing “profit reduced by 66.66%” with “price discounted by 66.66%”. Always compute discounts on the marked price.

Final Answer:
50%