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The problem asks for the actual percentage profit made by a shopkeeper who buys an article for Rs. 400, marks it for sale with an 80% profit on the cost, and then gives a 15% discount on the marked price.

### Step-by-step solution:

1. **Cost Price (C.P.):**
   The shopkeeper buys the article at Rs. 400.
   \[
   \text{C.P.} = 400
   \]

2. **Marked Price (M.P.):**
   The shopkeeper marks the article at a price that gives an 80% profit on the cost price. So, the marked price is:
   \[
   \text{M.P.} = \text{C.P.} + 80\% \text{ of C.P.} = 400 + \left( 0.80 \times 400 \right) = 400 + 320 = 720
   \]

3. **Discount:**
   The shopkeeper gives a 15% discount on the marked price. Therefore, the selling price (S.P.) is:
   \[
   \text{S.P.} = \text{M.P.} - 15\% \text{ of M.P.} = 720 - \left( 0.15 \times 720 \right) = 720 - 108 = 612
   \]

4. **Profit:**
   The actual profit is the difference between the selling price and the cost price:
   \[
   \text{Profit} = \text{S.P.} - \text{C.P.} = 612 - 400 = 212
   \]

5. **Percentage Profit:**
   The percentage profit is calculated as:
   \[
   \text{Percentage Profit} = \left( \frac{\text{Profit}}{\text{C.P.}} \right) \times 100 = \left( \frac{212}{400} \right) \times 100 = 53\%
   \]

Thus, the actual percentage profit made by the shopkeeper is **53%**.

Would you like further details or have any questions?

### Related Questions:
1. How does a discount affect the actual profit percentage?
2. What is the effect of increasing the discount on the profit?
3. How can we calculate the selling price if a different profit percentage is marked?
4. If the discount were 10%, what would be the new profit percentage?
5. What happens to the marked price if the cost price changes?

### Tip:
Always apply the discount on the marked price after calculating the price with the intended profit for accurate results.

A shopkeeper buys an article for Rs. 400 and marks it for sale at a price that gives him 80% profit on his cost. He, however, gives a 15% discount on the marked price to his customer. Calculate the actual percentage profit made by the shopkeeper.

Solve this problem involving a shopkeeper who buys an article for Rs. 400, marks it for an 80% profit, and gives a 15% discount. Learn step-by-step how to calculate the percentage profit after applying the discount.

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