Buying p apples for ₹ q and selling q apples for ₹ p (with p < q) — Over the entire deal, does the merchant gain or lose, and by what percentage (expressed in terms of p and q)?

Introduction / Context: This symbolic profit-and-loss problem compares unit cost and unit selling price derived from two reciprocal bundle quotes. Since p < q, the selling price per apple will be lower than the cost per apple, leading to a loss. The task is to express that loss percentage in terms of p and q.

Given Data / Assumptions:

• Buys p apples for ₹ q ⇒ cost per apple c = q / p.
• Sells q apples for ₹ p ⇒ selling price per apple s = p / q.
• Given p < q.

Concept / Approach: Compare s to c to find the factor s/c. The loss% is 100 * (1 − s/c). Here, s/c simplifies nicely to (p^2 / q^2) which is less than 1 due to p < q.

Step-by-Step Solution:

Verification / Alternative check: Numeric example: p = 2, q = 3 ⇒ buy at ₹ 1.5 each, sell at ₹ 0.666… each ⇒ loss% = (1 − 4/9)*100 = (5/9)*100 = 55.55…%.

Why Other Options Are Wrong: The other expressions mix signs or denominators incorrectly (e.g., using p^2 in the denominator or implying a gain when p < q implies a loss).

Common Pitfalls: Interpreting “p apples for ₹ q” as per-rupee pricing instead of per-apple cost, or forgetting to compare on the same base.

Final Answer: 100 * (q^2 − p^2) / q^2 % loss