A trader has 50 kg of pulses, part of which he sells at 8% profit and rest at 18% profit. He gains 14% on the whole.What is the quantity sold at 18% profit?

Options

A. 30 kg
B. 35 kg
C. 40 kg
D. 60 kg

Answer

To solve the problem step by step, we will use the concept of alligation to find the quantity sold at 18% profit.

### Step 1: Define the variables
Let:
- \( x \) = quantity sold at 8% profit
- \( y \) = quantity sold at 18% profit

From the problem, we know that:
\[ x + y = 50 \] (total quantity of pulses)

### Step 2: Calculate the overall profit
The overall profit percentage is given as 14%. This means that the total selling price (SP) of the pulses can be calculated as:
\[ \text{Total Cost Price (CP)} = 50 \text{ kg} \times \text{CP per kg} \]
Assuming the cost price per kg is \( C \), then:
\[ \text{Total CP} = 50C \]
The total selling price (SP) at a 14% profit is:
\[ \text{Total SP} = \text{Total CP} + 14\% \text{ of Total CP} = 50C + 0.14 \times 50C = 50C \times 1.14 = 57C \]

### Step 3: Calculate the selling prices for both parts
The selling price for the quantity sold at 8% profit is:
\[ \text{SP at 8%} = x \times C \times 1.08 \]
The selling price for the quantity sold at 18% profit is:
\[ \text{SP at 18%} = y \times C \times 1.18 \]

### Step 4: Set up the equation for total selling price
The total selling price can also be expressed as:
\[ \text{Total SP} = \text{SP at 8%} + \text{SP at 18%} \]
Substituting the expressions we derived:
\[ 57C = x \times C \times 1.08 + y \times C \times 1.18 \]

### Step 5: Simplify the equation
Dividing the entire equation by \( C \) (assuming \( C \neq 0 \)):
\[ 57 = 1.08x + 1.18y \]

### Step 6: Substitute \( y \) from the first equation
From the first equation \( x + y = 50 \), we can express \( y \) as:
\[ y = 50 - x \]
Substituting this into the selling price equation:
\[ 57 = 1.08x + 1.18(50 - x) \]

### Step 7: Solve for \( x \)
Expanding the equation:
\[ 57 = 1.08x + 59 - 1.18x \]
Combine like terms:
\[ 57 = 59 - 0.10x \]
Rearranging gives:
\[ 0.10x = 59 - 57 \]
\[ 0.10x = 2 \]
\[ x = \frac{2}{0.10} = 20 \text{ kg} \]

### Step 8: Find \( y \)
Now, substituting back to find \( y \):
\[ y = 50 - x = 50 - 20 = 30 \text{ kg} \]

### Conclusion
The quantity sold at 18% profit is **30 kg**.

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