A trader had 22 quintals of wheat. He sold a part of it at 23% profit and the rest at 33% profit, so that he made a total profit of 27%. How much wheat did he sell at 33% profit?

Options

A. 1. 1320kg.
B. 2. 440kg.
C. 3. 880kg.
D. 4. 1760kg.

Answer

To solve the problem step by step, we can use the concept of allegation to find out how much wheat was sold at a 33% profit.

### Step 1: Understand the Given Information
- Total wheat = 22 quintals
- Profit on part of the wheat = 23%
- Profit on the remaining part = 33%
- Overall profit = 27%

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

1. Write down the profits:
- Profit 1 (P1) = 23%
- Profit 2 (P2) = 33%
- Overall Profit (P_avg) = 27%

2. Set up the allegation:
- The difference between the overall profit and the lower profit:
\[ P_avg - P1 = 27 - 23 = 4 \]
- The difference between the higher profit and the overall profit:
\[ P2 - P_avg = 33 - 27 = 6 \]

### Step 3: Determine the Ratio
The ratio of the quantities sold at 23% profit to those sold at 33% profit is given by the differences calculated above:
\[ \text{Ratio} = 6 : 4 \]
This simplifies to:
\[ 3 : 2 \]

### Step 4: Calculate the Parts of Wheat Sold
Let the quantity sold at 23% profit be \(3x\) and the quantity sold at 33% profit be \(2x\).

Since the total wheat is 22 quintals, we can write the equation:
\[ 3x + 2x = 22 \]
\[ 5x = 22 \]
\[ x = \frac{22}{5} = 4.4 \text{ quintals} \]

### Step 5: Find the Quantity Sold at 33% Profit
Now, we can find the quantity sold at 33% profit:
\[ \text{Quantity sold at 33% profit} = 2x = 2 \times 4.4 = 8.8 \text{ quintals} \]

### Step 6: Convert Quintals to Kilograms
Since 1 quintal = 100 kg, we convert the quantity sold at 33% profit into kilograms:
\[ 8.8 \text{ quintals} = 8.8 \times 100 = 880 \text{ kg} \]

### Final Answer
The amount of wheat sold at 33% profit is **880 kg**.

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