A trader had 2000 kgs of rice. He sold a part of it at 10% profit and the rest at 16% profit, so that he made a total profit of 14.2%. How much rice (in kg) did he sell at 10% profit?

Options

A. 1400
B. 600
C. 800
D. 1000

Answer

To solve the problem step by step, we can use the method of allegations.

### Step 1: Understand the given data
- Total quantity of rice = 2000 kg
- Profit from part sold at 10% = 10%
- Profit from part sold at 16% = 16%
- Overall profit = 14.2%

### Step 2: Set up the allegation method
We can set up the allegation method to find the ratio of the quantities sold at different profits.

1. Calculate the difference between the overall profit and the lower profit:
\[ \text{Difference 1} = 14.2\% - 10\% = 4.2\% \]

2. Calculate the difference between the higher profit and the overall profit:
\[ \text{Difference 2} = 16\% - 14.2\% = 1.8\% \]

### Step 3: Set up the ratio
Now we can set up the ratio of the quantities sold at 10% profit to the quantities sold at 16% profit:
\[ \text{Ratio} = \frac{\text{Difference 1}}{\text{Difference 2}} = \frac{4.2}{1.8} \]

### Step 4: Simplify the ratio
To simplify \( \frac{4.2}{1.8} \):
- Divide both the numerator and the denominator by 1.8:
\[ \frac{4.2 \div 1.8}{1.8 \div 1.8} = \frac{2.33}{1} \approx \frac{23}{10} \]
Thus, the ratio of the quantities sold at 10% to 16% is approximately \( 23:10 \).

### Step 5: Calculate the quantities sold
Let the quantity sold at 10% profit be \( 23x \) and the quantity sold at 16% profit be \( 10x \).
According to the total quantity:
\[ 23x + 10x = 2000 \]
\[ 33x = 2000 \]
\[ x = \frac{2000}{33} \approx 60.61 \]

### Step 6: Find the quantity sold at 10% profit
Now, we can find the quantity sold at 10% profit:
\[ \text{Quantity sold at 10%} = 23x = 23 \times 60.61 \approx 1399.99 \text{ kg} \]

### Final Answer
The trader sold approximately **1400 kg** of rice at 10% profit.

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