A man purchases a certain number of oranges at 25 a rupee and the same number at 20 a rupee. He mixes them together and sells them at 45 for two rupees. How much percent does he gain or lose in the transaction?

To solve this problem, let's break it down into steps:

### 1. **Calculate the Number of Oranges Purchased and Their Costs**

Assume the man buys \( n \) oranges at the rate of 25 oranges per rupee, and the same number \( n \) at the rate of 20 oranges per rupee.

- **Cost of oranges bought at 25 per rupee:**
  \[
  \text{Number of oranges} = n \Rightarrow \text{Cost} = \frac{n}{25} \text{ rupees}
  \]

- **Cost of oranges bought at 20 per rupee:**
  \[
  \text{Number of oranges} = n \Rightarrow \text{Cost} = \frac{n}{20} \text{ rupees}
  \]

### 2. **Calculate the Total Cost and Total Number of Oranges**

- **Total cost of oranges:**
  \[
  \text{Total cost} = \frac{n}{25} + \frac{n}{20}
  \]

  To combine the fractions, find the common denominator:

  \[
  \text{Total cost} = \frac{4n + 5n}{100} = \frac{9n}{100} \text{ rupees}
  \]

- **Total number of oranges:**
  \[
  \text{Total oranges} = n + n = 2n
  \]

### 3. **Calculate the Selling Price**

The man sells the mixed oranges at the rate of 45 oranges for 2 rupees.

- **Selling price per orange:**
  \[
  \text{Selling price per orange} = \frac{2 \text{ rupees}}{45 \text{ oranges}}
  \]

- **Total selling price for 2n oranges:**
  \[
  \text{Total selling price} = 2n \times \frac{2}{45} = \frac{4n}{45} \text{ rupees}
  \]

### 4. **Calculate the Gain or Loss**

- **Profit or loss amount:**
  \[
  \text{Profit/Loss} = \text{Selling Price} - \text{Cost Price}
  \]
  \[
  \text{Profit/Loss} = \frac{4n}{45} - \frac{9n}{100}
  \]

  Find a common denominator (LCM of 45 and 100 is 900):

  \[
  \text{Profit/Loss} = \frac{4n \times 20}{900} - \frac{9n \times 9}{900}
  \]
  \[
  \text{Profit/Loss} = \frac{80n}{900} - \frac{81n}{900}
  \]
  \[
  \text{Profit/Loss} = \frac{-1n}{900}
  \]
  \[
  \text{Profit/Loss} = -\frac{n}{900} \text{ rupees}
  \]

This negative value indicates a loss.

### 5. **Calculate the Percentage Loss**

\[
\text{Percentage Loss} = \left(\frac{\text{Loss}}{\text{Cost Price}}\right) \times 100
\]
\[
\text{Percentage Loss} = \left(\frac{\frac{-n}{900}}{\frac{9n}{100}}\right) \times 100
\]
\[
\text{Percentage Loss} = \left(\frac{-1}{900} \times \frac{100}{9n} \times 100\right)
\]
\[
\text{Percentage Loss} = \frac{-100}{9} \approx -11.11\%
\]

### Conclusion

The man incurs a loss of approximately **11.11%** in the transaction.

---

Do you want further details or have any questions?

Here are some related questions you might be interested in:

1. What if the selling rate was 40 oranges for 2 rupees instead of 45?
2. How much would the gain or loss be if the purchase rate was different?
3. What is the break-even selling price per orange in this scenario?
4. How does changing the number of oranges purchased affect the profit/loss?
5. What if the man buys different quantities at different rates?

**Tip:** Always calculate the total cost and selling price before comparing to find profit or loss percentage.

Learn how to calculate the gain or loss percentage when a man buys oranges at different rates and sells them at a mixed price. This problem involves understanding cost price, selling price, and percentage calculations. Suitable for students in Grades 6-8.

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