A vendor sells at an actual profit of p%. If he instead sells at one-third of his actual selling price, he incurs a 40% loss. Find p (the actual profit percentage).

Introduction:
Link profit on cost to a hypothetical selling price that is one-third of the real selling price. The switch turns a gain into a loss, allowing us to back out the original profit percentage using a simple ratio relation.

Given Data / Assumptions:

• Let CP = C
• Actual SP = S = (1 + p) * C
• If SP becomes S/3, loss = 40% ⇒ (S/3 − C)/C = −0.4

Concept / Approach:
From (S/3) = 0.6 * C, deduce S = 1.8 * C. Since S = (1 + p) * C, compare coefficients to find p.

Step-by-Step Solution:
S/3 = 0.6 * C ⇒ S = 1.8 * CBut S = (1 + p) * C ⇒ 1 + p = 1.8 ⇒ p = 0.8 = 80%

Verification / Alternative check:
With C = 100, S = 180 (profit 80). At S/3 = 60, loss = 40 on C = 100 → −40% (matches).

Why Other Options Are Wrong:
72%, 120%, 60% do not satisfy S/3 = 0.6C.

Common Pitfalls:
Using 40% on SP; forgetting that loss% is always on CP.

Final Answer:
80%