A shopkeeper has 50kg of rice. He sells a part of it at `20%` profit and the rest at `40%` profit. If he gains `25%` on the whole, find tile quantity sold at `20%` profit.

Options

A. 37.5 kg
B. 35.5 kg
C. 12.5 kg
D. 36 kg

Answer

To solve the problem step by step, we will use the allegation method to find the quantity of rice sold at a 20% profit.

### Step 1: Understand the given data
- Total quantity of rice = 50 kg
- Profit from the first part = 20%
- Profit from the second part = 40%
- Overall profit = 25%

### Step 2: Set up the allegation method
We will use the allegation method to find the ratio of the quantities sold at different profit percentages.

1. **Calculate the differences**:
- Difference between the higher profit (40%) and the overall profit (25%):
\[ 40\% - 25\% = 15\% \]
- Difference between the overall profit (25%) and the lower profit (20%):
\[ 25\% - 20\% = 5\% \]

### Step 3: Form the ratio
Using the differences calculated:
- The ratio of the quantities sold at 20% profit to the quantities sold at 40% profit is:
\[ \text{Ratio} = \frac{5}{15} = \frac{1}{3} \]
This means for every 1 part sold at 20% profit, 3 parts are sold at 40% profit.

### Step 4: Calculate the total parts
Let the quantity sold at 20% profit be \(x\) kg and the quantity sold at 40% profit be \(3x\) kg. Therefore:
\[ x + 3x = 50 \text{ kg} \]
\[ 4x = 50 \]
\[ x = \frac{50}{4} = 12.5 \text{ kg} \]

### Step 5: Calculate the quantity sold at 20% profit
Now, we need to find the quantity sold at 20% profit:
\[ \text{Quantity sold at 20% profit} = 1x = 12.5 \text{ kg} \]

### Step 6: Calculate the quantity sold at 40% profit
The quantity sold at 40% profit is:
\[ 3x = 3 \times 12.5 = 37.5 \text{ kg} \]

### Final Answer
The quantity of rice sold at 20% profit is:
\[ \text{Quantity sold at 20% profit} = 12.5 \text{ kg} \]