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The question is:

A merchant sells 22 pens of a particular type. He sells the first pen at ₹10, the second at ₹11, the third at ₹12, and so on. After selling the first 11 pens (priced up to ₹20), he makes no profit and no loss. How much profit will he make after selling the first 22 pens?

### Solution:
1. **Understanding the Selling Prices:**
   - The selling price of the pens follows an arithmetic sequence: 10, 11, 12, ..., 20 (for the first 11 pens).
   - The 12th pen is sold at ₹21, the 13th at ₹22, ..., up to the 22nd pen at ₹31.

2. **Calculating Total Selling Price for the First 11 Pens:**
   - The selling prices form an arithmetic sequence where the first term \( a = 10 \) and the common difference \( d = 1 \).
   - Sum of the first 11 terms, \( S_{11} \), is calculated as:
     \[
     S_{11} = \frac{11}{2} \times (2 \times 10 + (11 - 1) \times 1) = \frac{11}{2} \times (20 + 10) = \frac{11}{2} \times 30 = 165 \, \text{(₹)}
     \]

3. **Understanding No Profit and No Loss After 11 Pens:**
   - Since the merchant makes no profit or loss after selling the first 11 pens, it means the total cost price of these 11 pens is ₹165.

4. **Calculating Total Selling Price for All 22 Pens:**
   - For the remaining 11 pens, the selling prices follow another arithmetic sequence with:
     - First term \( a = 21 \)
     - Common difference \( d = 1 \)
   - Sum of the next 11 terms (12th to 22nd pen):
     \[
     S_{12-22} = \frac{11}{2} \times (2 \times 21 + (11 - 1) \times 1) = \frac{11}{2} \times (42 + 10) = \frac{11}{2} \times 52 = 286 \, \text{(₹)}
     \]

   - Total selling price for all 22 pens:
     \[
     S_{22} = 165 + 286 = 451 \, \text{(₹)}
     \]

5. **Calculating Profit:**
   - The cost price for all 22 pens is twice the cost price of the first 11 pens, as the profit starts only from the 12th pen onwards.
     \[
     \text{Total Cost Price} = 2 \times 165 = 330 \, \text{(₹)}
     \]
   - Profit:
     \[
     \text{Profit} = \text{Total Selling Price} - \text{Total Cost Price} = 451 - 330 = 121 \, \text{(₹)}
     \]

6. **Calculating Profit Percentage:**
   - Profit percentage:
     \[
     \text{Profit Percentage} = \left( \frac{121}{330} \right) \times 100 = 36.67\%
     \]

Hence, the merchant's profit after selling all 22 pens is approximately **36.67%**.

If you have any questions or need further details, feel free to ask!

### Related Questions:
1. How do you find the sum of an arithmetic series?
2. What is the general formula for profit percentage?
3. What happens to the profit if the merchant sells 5 more pens at a higher price?
4. How would the profit change if the cost of production increases by 10%?
5. Can you calculate the new profit if the selling price of each pen is reduced by ₹2?

### Tip:
To calculate the profit or loss percentage, always ensure you have a clear understanding of both the selling price and cost price involved in the transaction.

A merchant sells 22 pens of a particular type. He sells the first pen at ₹10, second at ₹11, third at ₹12, ..., 11th at ₹20, and so on. After selling the first 11 pens, he makes no profit and no loss. How much profit will he make after selling the first 22 pens?

Learn how to calculate the profit a merchant makes after selling 22 pens with prices following an arithmetic progression. The problem involves calculating the sum of an arithmetic series and determining profit and profit percentage. Suitable for students in Grades 9-11.

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