A shopkeeper had 50 Kg. rice He sold one part of it at a profit of 10% and on the other, he losed 5%. His total profit was 7% on the whole transaction. How many Kg. rice was sold at a profit of 10% ?

Options

A. 40 kg
B. 30 kg
C. 15 kg
D. 20 kg

Answer

To solve the problem step by step, we need to determine how many kilograms of rice were sold at a profit of 10%.

Let's denote:
- \( x \) = the quantity of rice sold at a profit of 10%
- \( 50 - x \) = the quantity of rice sold at a loss of 5%

### Step 1: Calculate the cost price of the rice
Assuming the cost price of the rice is \( C \) per kg, the total cost price for 50 kg of rice is:
\[ \text{Total Cost Price} = 50C \]

### Step 2: Calculate the selling price for the part sold at a profit of 10%
The selling price for the part sold at a profit of 10% is:
\[ \text{Selling Price at 10% profit} = x \times C \times (1 + 0.10) = x \times C \times 1.10 = 1.10xC \]

### Step 3: Calculate the selling price for the part sold at a loss of 5%
The selling price for the part sold at a loss of 5% is:
\[ \text{Selling Price at 5% loss} = (50 - x) \times C \times (1 - 0.05) = (50 - x) \times C \times 0.95 = 0.95(50 - x)C \]

### Step 4: Calculate the total selling price
The total selling price for the rice is the sum of the selling prices from both parts:
\[ \text{Total Selling Price} = 1.10xC + 0.95(50 - x)C \]

### Step 5: Set up the equation for total profit
The total profit is given as 7% of the total cost price:
\[ \text{Total Profit} = 0.07 \times \text{Total Cost Price} = 0.07 \times 50C = 3.5C \]

### Step 6: Set the equation for selling price equal to cost price plus profit
Now we can set up the equation:
\[ 1.10xC + 0.95(50 - x)C = 50C + 3.5C \]
Simplifying this gives:
\[ 1.10xC + 0.95(50C - xC) = 53.5C \]

### Step 7: Expand and simplify the equation
Expanding the left side:
\[ 1.10xC + 47.5C - 0.95xC = 53.5C \]
Combining like terms:
\[ (1.10x - 0.95x)C + 47.5C = 53.5C \]
\[ 0.15xC + 47.5C = 53.5C \]

### Step 8: Isolate \( x \)
Subtract \( 47.5C \) from both sides:
\[ 0.15xC = 53.5C - 47.5C \]
\[ 0.15xC = 6C \]
Dividing both sides by \( 0.15C \):
\[ x = \frac{6C}{0.15C} = 40 \]

### Conclusion
The quantity of rice sold at a profit of 10% is:
\[ \boxed{40 \text{ kg}} \]