Step 1: Understanding the Question:
A retailer marks up his goods and then offers a discount, erroneously believing his profit will simply be the markup minus the discount ($50% - 25% = 25%$). We must calculate the true profit percentage using cost price, marked price, and selling price dynamics.
Step 2: Key Formula or Approach:
Assume a base Cost Price (CP) of 100, calculate the Marked Price (MP), apply the discount to find the Selling Price (SP), and then calculate the true profit percentage using $(SP - CP) / CP \times 100$.
Step 3: Detailed Explanation:
Let the Cost Price (CP) of the goods be Rs. 100.
The retailer marks up his goods by 50% above the cost price.
Marked Price (MP) = CP + 50% of CP = $100 + 50 = 150$ Rs.
He then offers a discount of 25% on this marked price.
Remember, the discount is strictly calculated on the Marked Price, not the Cost Price.
Discount amount = 25% of 150 = $0.25 \times 150 = 37.5$ Rs.
The Selling Price (SP) is the marked price minus the discount amount.
SP = MP - Discount = $150 - 37.5 = 112.5$ Rs.
Now we find the actual profit made on the sale.
Actual Profit = SP - CP = $112.5 - 100 = 12.5$ Rs.
Since our base CP is exactly 100, the absolute profit amount directly translates to the profit percentage.
Actual Profit Percentage = $\left( \frac{12.5}{100} \right) \times 100 = 12.5%$.
The retailer's faulty reasoning ignored the fact that the 25% discount was applied to the much larger 150 (making the deduction 37.5) rather than 100 (which would be 25). The true profit is exactly half of what he thought.
Step 4: Final Answer:
His actual profit on the sales is 12.5%.