Profit percentage equals the cost price — solve quadratic to find CP A merchant sells goods for Rs 75 at a profit percentage numerically equal to the cost price (in rupees). What is the cost price?

Introduction / Context:
Here the profit percentage (a number) equals the cost price (also a number in rupees). This leads to a quadratic equation because SP = CP + (profit% of CP) and profit% equals CP itself as a number.

Given Data / Assumptions:

• Selling price SP = 75.
• Let CP = x rupees.
• Profit% = x% ⇒ Profit = (x/100) × x = x^2/100.

Concept / Approach:
Use SP = CP + Profit ⇒ 75 = x + x^2/100. This is a quadratic in x; solve for a positive, meaningful CP.

Step-by-Step Solution:
75 = x + x^2/100 ⇒ x^2 + 100x − 7500 = 0.Discriminant = 100^2 + 4×7500 = 10,000 + 30,000 = 40,000 ⇒ √D = 200.x = (−100 + 200)/2 = 50 (positive root).

Verification / Alternative check:
With CP = 50, profit% = 50%. Profit = 50% of 50 = 25; SP = 50 + 25 = 75 (consistent).

Why Other Options Are Wrong:
40, 60, 70, 45 do not satisfy x + x^2/100 = 75.

Common Pitfalls:
Interpreting “profit percent equals cost price” as profit in rupees equals CP; the statement is about the numerical percent value.

Final Answer:
Rs 50