What is the percentage of profit if the purchase price of one item is profit when buying 9 items?

To solve this problem, let's break it down step by step:

1. **Define the variables:**
   - Let $ P $ be the purchase price of one item.
   - Let $ S $ be the selling price of one item.

2. **Understand the given information:**
   - The profit made when selling 9 items is equal to the purchase price of 1 item.

3. **Calculate the total purchase price and total profit:**
   - The total purchase price for 9 items is $ 9P $.
   - The profit from selling 9 items is $ S \times 9 - 9P $.

4. **Set up the equation based on the given information:**
   - The profit from selling 9 items equals the purchase price of 1 item: 
     \[
     S \times 9 - 9P = P
     \]

5. **Solve for $ S $:**
   \[
   S \times 9 - 9P = P
   \]
   \[
   S \times 9 = 9P + P
   \]
   \[
   S \times 9 = 10P
   \]
   \[
   S = \frac{10P}{9}
   \]

6. **Calculate the percentage profit:**
   - The profit for one item is $ S - P $.
   - Using the value of $ S $ found, $ S - P = \frac{10P}{9} - P $.
   \[
   S - P = \frac{10P}{9} - \frac{9P}{9} = \frac{P}{9}
   \]

7. **Find the percentage profit:**
   \[
   \text{Percentage Profit} = \left( \frac{\frac{P}{9}}{P} \right) \times 100 = \left( \frac{1}{9} \right) \times 100 \approx 11.11\%
   \]

Therefore, the percentage of profit is approximately $ 11.11\% $.

Would you like more details or have any questions?

Here are some related questions to expand your understanding:

1. What would be the selling price if the purchase price is $100?
2. How does the percentage profit change if the number of items sold increases?
3. What is the formula to calculate profit percentage in general?
4. How can this percentage profit be affected by changes in market prices?
5. If the purchase price doubles, what would be the new selling price to maintain the same profit percentage?
6. What is the relationship between profit margin and profit percentage?
7. How would you represent this problem graphically?
8. Can you derive the profit percentage if the profit when buying 12 items equals the purchase price of 2 items?

**Tip:** When dealing with percentages in profit calculations, always clearly define your cost and selling price to avoid confusion.

Learn how to calculate the percentage profit when the purchase price of one item equals the profit from selling nine items. This problem involves understanding profit margins and percentage calculations in business contexts. Suitable for high school students and beyond.

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