A shopkeeper expects a gain of (45/2)% on his cost price. If his actual sale value is Rs. 392, what is the amount of profit he makes?

Introduction / Context:
This problem involves a fractional percentage gain on the cost price and a known selling price. The percentage gain is given as 45/2 percent, which equals 22.5%. The selling price is known, and we are asked to find the absolute profit. To do this, we must first compute the cost price using the relationship between cost price, profit percentage and selling price, and then subtract the cost price from the selling price.

Given Data / Assumptions:

• Expected gain = (45/2)% = 22.5% on cost price.
• Selling price (SP) = Rs. 392.
• Gain percentage is based on cost price.
• We assume the shopkeeper actually achieves this expected gain.
• We must find the profit in rupees.

Concept / Approach:
If profit is p% on cost price, then selling price equals cost price multiplied by (1 + p/100). Here, p = 22.5. Let cost price be C. Then SP = C * (1 + 22.5/100) = 1.225C. Since SP is given, we can write 392 = 1.225C and solve for C. Finally, profit is SP - C. Working with the decimal 1.225 is straightforward if we treat it as the fraction 49/40 or 1225/1000 depending on convenience.

Step-by-Step Solution:
Step 1: Gain percentage = 45/2% = 22.5%. Step 2: Let cost price be C rupees. Step 3: Selling price SP = C * (1 + 22.5/100) = C * 1.225. Step 4: Given SP = 392, so 1.225C = 392. Step 5: Express 1.225 as a fraction: 1.225 = 1225/1000 = 49/40. Step 6: Thus, (49/40) * C = 392. Step 7: Multiply both sides by 40: 49C = 392 * 40. Step 8: Compute 392 * 40 = 15,680. Step 9: So C = 15,680 / 49 = 320. Step 10: Profit = SP - CP = 392 - 320 = 72.

Verification / Alternative check:
With CP = Rs. 320 and gain percentage = 22.5%, profit should be 22.5% of 320. Compute 10% of 320 = 32, 20% = 64, 2.5% = 8, so 22.5% = 64 + 8 = 72. Selling price = 320 + 72 = 392, which matches the given SP. This confirms that the profit of Rs. 72 is correct.

Why Other Options Are Wrong:
Profits of Rs. 70, Rs. 74, Rs. 76 or Rs. 68 would not correspond to a gain of exactly 22.5% on the implied cost price. When recomputed, they would give slightly different selling prices or slightly different percentages. Only a profit of Rs. 72 maintains the exact 22.5% gain with an SP of Rs. 392.

Common Pitfalls:
Many learners misread 45/2% as 45.2% or incorrectly convert it to a decimal. Others treat 22.5% as if it is applied to the selling price rather than the cost price. Converting the percentage to a clear fraction and writing SP = (1 + p/100) * CP are reliable ways to avoid confusion.

Final Answer:
The shopkeeper's profit is Rs. 72.