Profit percentage numerically equal to cost price: A person sells a watch for ₹ 75 and the percentage profit equals the cost price (in rupees). Find the cost price of the watch.

Introduction / Context:
Some puzzles equate a numerical percentage to a rupee value. Here, the profit percentage number equals the cost price itself, leading to a quadratic relation between SP and CP.

Given Data / Assumptions:

• SP = ₹ 75.
• Let CP = x. Profit% = x (numerically).

Concept / Approach:
Profit% = (SP − CP)/CP * 100. Setting this equal to x gives: (75 − x)/x * 100 = x ⇒ 75 − x = x^2/100 ⇒ x^2 + 100x − 7,500 = 0.

Step-by-Step Solution:
Solve x^2 + 100x − 7,500 = 0Discriminant = 100^2 + 4*7,500 = 40,000 ⇒ √D = 200x = (−100 + 200)/2 = 50 (reject negative root)

Verification / Alternative check:
If CP = ₹ 50, profit = ₹ 25. Profit% = 25/50 * 100 = 50, which equals the CP numerically—condition satisfied.

Why Other Options Are Wrong:

• ₹ 40 / ₹ 45 / ₹ 55: do not satisfy the quadratic relation with SP = 75.

Common Pitfalls:

• Treating “profit% equals CP” as profit equals CP; the statement is about the numeric percent equaling the rupee CP.

Final Answer:
₹ 50