A man purchases a certain number of oranges at 4 a rupee and the same number of oranges at 5 a rupee. He mixes them together and sells them at 4 a rupee. What is his gain or loss per cent?

Options

A. `11(2)/(9)% ` gain
B. `11(2)/(9)` loss
C. `11(1)/(9)%` gain
D. `11(1)/(9)` loss

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the Cost Price of Oranges
- The man purchases oranges at two different rates:
- At 4 oranges for 1 rupee, the cost price (CP) of 1 orange = \( \frac{1}{4} \) rupee.
- At 5 oranges for 1 rupee, the cost price (CP) of 1 orange = \( \frac{1}{5} \) rupee.

### Step 2: Calculate the Total Cost Price for 'n' Oranges
- Let’s assume he buys 'n' oranges at each rate.
- The total cost price for 'n' oranges at the first rate:
\[ \text{CP for } n \text{ oranges} = n \times \frac{1}{4} = \frac{n}{4} \text{ rupees} \]
- The total cost price for 'n' oranges at the second rate:
\[ \text{CP for } n \text{ oranges} = n \times \frac{1}{5} = \frac{n}{5} \text{ rupees} \]

### Step 3: Calculate the Total Cost Price (CP)
- The total cost price for all the oranges (2n oranges):
\[ \text{Total CP} = \frac{n}{4} + \frac{n}{5} \]
- To add these fractions, we need a common denominator (which is 20):
\[ \text{Total CP} = \frac{5n}{20} + \frac{4n}{20} = \frac{9n}{20} \text{ rupees} \]

### Step 4: Calculate the Selling Price (SP)
- He sells the oranges at 4 oranges for 1 rupee, so the selling price of 1 orange = \( \frac{1}{4} \) rupee.
- For 2n oranges, the selling price is:
\[ \text{Total SP} = 2n \times \frac{1}{4} = \frac{2n}{4} = \frac{n}{2} \text{ rupees} \]

### Step 5: Calculate Profit
- Profit is calculated as:
\[ \text{Profit} = \text{SP} - \text{CP} = \frac{n}{2} - \frac{9n}{20} \]
- To subtract these fractions, we need a common denominator (which is 20):
\[ \text{Profit} = \frac{10n}{20} - \frac{9n}{20} = \frac{n}{20} \text{ rupees} \]

### Step 6: Calculate Profit Percentage
- Profit percentage is given by:
\[ \text{Profit Percentage} = \left( \frac{\text{Profit}}{\text{CP}} \right) \times 100 = \left( \frac{\frac{n}{20}}{\frac{9n}{20}} \right) \times 100 \]
- Simplifying this:
\[ \text{Profit Percentage} = \left( \frac{1}{9} \right) \times 100 = \frac{100}{9} \approx 11.11\% \]

### Final Answer
- The gain percentage is \( 11 \frac{1}{9} \% \).

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