A merchant had 50 kg of pulse.He sells ane part at a profit of 10% and other at 5% loss. Overall he had a gain of 7%. Find the quantity of pulses, which he sold at 10% profit and 5% loss.

Options

A. 40 kg, 10 kg
B. 40 kg, 15 kg
C. 40 kg, 12 kg
D. 40kg, 9 kg

Answer

To solve the problem, we need to find the quantities of pulses sold at a 10% profit and a 5% loss. Let's denote:

- \( x \) = quantity sold at 10% profit
- \( y \) = quantity sold at 5% loss

From the problem, we know that:

1. The total quantity of pulses is 50 kg:
\[ x + y = 50 \quad \text{(1)} \]

2. The overall gain is 7%. To express this mathematically, we can use the formula for overall profit based on the selling prices and costs.

Let's assume the cost price of 1 kg of pulse is \( C \). Therefore, the total cost price of 50 kg is:
\[ \text{Total Cost Price} = 50C \]

The selling price for the part sold at 10% profit is:
\[ \text{Selling Price at 10% profit} = x \times C \times 1.10 = 1.10xC \]

The selling price for the part sold at 5% loss is:
\[ \text{Selling Price at 5% loss} = y \times C \times 0.95 = 0.95yC \]

The overall selling price is the sum of both selling prices:
\[ \text{Total Selling Price} = 1.10xC + 0.95yC \]

Given that the overall gain is 7%, we can express the total selling price in terms of the total cost price:
\[ \text{Total Selling Price} = \text{Total Cost Price} + 7\% \text{ of Total Cost Price} \]
\[ 1.10xC + 0.95yC = 50C + 0.07 \times 50C \]
\[ 1.10xC + 0.95yC = 50C + 3.5C \]
\[ 1.10xC + 0.95yC = 53.5C \]

Dividing the entire equation by \( C \) (assuming \( C \neq 0 \)):
\[ 1.10x + 0.95y = 53.5 \quad \text{(2)} \]

Now we have a system of equations:
1. \( x + y = 50 \) (1)
2. \( 1.10x + 0.95y = 53.5 \) (2)

### Step 1: Solve for \( y \) in terms of \( x \) using equation (1).
From equation (1):
\[ y = 50 - x \]

### Step 2: Substitute \( y \) in equation (2).
Substituting \( y \) in equation (2):
\[ 1.10x + 0.95(50 - x) = 53.5 \]
\[ 1.10x + 47.5 - 0.95x = 53.5 \]
\[ (1.10 - 0.95)x + 47.5 = 53.5 \]
\[ 0.15x + 47.5 = 53.5 \]

### Step 3: Isolate \( x \).
Subtract 47.5 from both sides:
\[ 0.15x = 53.5 - 47.5 \]
\[ 0.15x = 6 \]
Now, divide by 0.15:
\[ x = \frac{6}{0.15} = 40 \]

### Step 4: Find \( y \).
Using \( y = 50 - x \):
\[ y = 50 - 40 = 10 \]

### Final Answer:
The merchant sold 40 kg of pulses at a 10% profit and 10 kg at a 5% loss.