A man purchases a certain number of toffees at 6 per rupee and the same number of toffees at 7 per rupee. He mixes the toffees and sells them at 6 per rupee. What is his gain or loss per cent?

Options

A. `6(9)/(13) %` loss
B. `7(9)/(13)% ` gain
C. `7(6)/(13)%`
D. `7(5)/(13)%` gain

Answer

To solve the problem step by step, we will follow a structured approach:

### Step 1: Understand the Problem
A man buys toffees at two different rates and sells them at a uniform rate. We need to calculate his gain or loss percentage.

### Step 2: Define Variables
Let’s assume the man buys \( n \) toffees at each rate.

### Step 3: Calculate Cost Price (CP)
1. **First Purchase**:
- Cost Price for toffees bought at 6 per rupee:
\[ \text{CP}_1 = \frac{n}{6} \text{ rupees} \]

2. **Second Purchase**:
- Cost Price for toffees bought at 7 per rupee:
\[ \text{CP}_2 = \frac{n}{7} \text{ rupees} \]

3. **Total Cost Price (CP)**:
\[ \text{Total CP} = \text{CP}_1 + \text{CP}_2 = \frac{n}{6} + \frac{n}{7} \]
To add these fractions, we find a common denominator (42):
\[ \text{Total CP} = \frac{7n}{42} + \frac{6n}{42} = \frac{13n}{42} \text{ rupees} \]

### Step 4: Calculate Selling Price (SP)
1. **Selling Price**:
- The man sells all \( 2n \) toffees at 6 per rupee:
\[ \text{SP} = \frac{2n}{6} = \frac{n}{3} \text{ rupees} \]

### Step 5: Calculate Profit
1. **Profit**:
\[ \text{Profit} = \text{SP} - \text{Total CP} = \frac{n}{3} - \frac{13n}{42} \]
To subtract these fractions, we find a common denominator (42):
\[ \text{Profit} = \frac{14n}{42} - \frac{13n}{42} = \frac{n}{42} \text{ rupees} \]

### Step 6: Calculate Profit Percentage
1. **Profit Percentage**:
\[ \text{Profit Percentage} = \left(\frac{\text{Profit}}{\text{Total CP}}\right) \times 100 = \left(\frac{\frac{n}{42}}{\frac{13n}{42}}\right) \times 100 \]
Simplifying this:
\[ \text{Profit Percentage} = \left(\frac{1}{13}\right) \times 100 = \frac{100}{13} \approx 7.69\% \]

### Step 7: Final Answer
The gain percentage is approximately \( 7 \frac{9}{13} \% \).