A man purchases a certain number of apples at 3 per rupee and the same number of apples at 4 per rupee.He mixes them together and sells them at 3 per rupee. What is his gain or loss per cent?

Answer

To solve the problem step by step, we will calculate the cost price (CP), selling price (SP), and then determine the gain or loss percentage.

### Step 1: Determine the Cost Price (CP)
The man purchases apples in two batches:
- **First batch**: He buys apples at 3 apples per rupee. Let's assume he buys **N apples**. Therefore, the cost for this batch is:
\[ \text{CP}_1 = \frac{N}{3} \text{ rupees} \]
- **Second batch**: He buys the same number of apples (N) at 4 apples per rupee. Therefore, the cost for this batch is:
\[ \text{CP}_2 = \frac{N}{4} \text{ rupees} \]

Now, we can calculate the total cost price (CP) for all apples:
\[ \text{Total CP} = \text{CP}_1 + \text{CP}_2 = \frac{N}{3} + \frac{N}{4} \]

### Step 2: Find a Common Denominator
To add the fractions, we need a common denominator. The least common multiple of 3 and 4 is 12.
\[ \text{Total CP} = \frac{4N}{12} + \frac{3N}{12} = \frac{7N}{12} \text{ rupees} \]

### Step 3: Determine the Selling Price (SP)
The man sells the mixed apples at 3 apples per rupee. Since he has a total of \(2N\) apples (N from each batch), the selling price is:
\[ \text{Total SP} = \frac{2N}{3} \text{ rupees} \]

### Step 4: Calculate Gain or Loss
Now, we can find out if there is a gain or loss by comparing the total selling price (SP) with the total cost price (CP):
\[ \text{Loss} = \text{Total CP} - \text{Total SP} = \frac{7N}{12} - \frac{2N}{3} \]

### Step 5: Find a Common Denominator for Loss Calculation
The least common multiple of 12 and 3 is 12:
\[ \text{Total SP} = \frac{2N}{3} = \frac{8N}{12} \]
Thus, the loss can be calculated as:
\[ \text{Loss} = \frac{7N}{12} - \frac{8N}{12} = -\frac{N}{12} \text{ rupees} \]

### Step 6: Calculate Loss Percentage
To find the loss percentage, we use the formula:
\[ \text{Loss Percentage} = \left(\frac{\text{Loss}}{\text{Total CP}}\right) \times 100 \]
Substituting the values:
\[ \text{Loss Percentage} = \left(\frac{\frac{N}{12}}{\frac{7N}{12}}\right) \times 100 = \left(\frac{1}{7}\right) \times 100 \approx 14.28\% \]

### Final Answer
The man incurs a loss of approximately **14.28%**.