By selling an umbrella at price P there is a profit of x%. If selling at two-thirds of that price (2/3 P) results in a 10% loss, what is the profit percentage at price P?

Introduction / Context:
We relate the selling price to cost using the loss information at a fractional price. Once we express P in terms of cost, the profit% at P follows directly.

Given Data / Assumptions:

• If sold at (2/3)P, loss = 10% ⇒ (2/3)P = 0.90 * C.
• We need profit% at price P.

Concept / Approach:
From (2/3)P = 0.90C, solve for P in terms of C, then compute profit% = (P − C)/C * 100.

Step-by-Step Solution:
(2/3)P = 0.90C ⇒ P = 0.90 * (3/2) * C = 1.35CProfit at P = (1.35C − C)/C * 100 = 35%

Verification / Alternative check:
Check the loss case: at 0.666...P = 0.9C indeed corresponds to 10% below cost, as used.

Why Other Options Are Wrong:
25% or 30% do not match the linear relationship between P and C implied by the (2/3)P loss condition.

Common Pitfalls:
Misplacing the 2/3 factor or applying the 10% to SP instead of CP.

Final Answer:
35%