A shopkeeper purchased 15 kg of variety A rice at ₹X per kg and 10 kg of variety B rice at ₹(X+5)per kg. The shopkeeper sold the whole quantity of variety A rice at `10%` profit and that of variety B rice at `20%` profit. The total selling price of A variety rice was 30 rs more than variety B rice. Had the two varieties been mixed and sold at an overall profit of `20%`, what would have been the selling price (per kg)?

Options

A. ₹26.40
B. ₹23.20
C. ₹24.20
D. ₹25.00

Answer

To solve the problem step-by-step, we will break it down into manageable parts.

### Step 1: Calculate the Cost Price of Each Variety
1. **Variety A**: The shopkeeper purchased 15 kg at ₹X per kg.
- Cost Price of Variety A = 15 kg * ₹X = ₹15X

2. **Variety B**: The shopkeeper purchased 10 kg at ₹(X + 5) per kg.
- Cost Price of Variety B = 10 kg * ₹(X + 5) = ₹10(X + 5) = ₹10X + ₹50

### Step 2: Calculate the Selling Price of Each Variety
1. **Selling Price of Variety A**: Sold at a 10% profit.
- Selling Price of Variety A = Cost Price + Profit
- Profit = 10% of Cost Price = 10% of ₹15X = ₹(15X * 0.10) = ₹1.5X
- Selling Price of Variety A = ₹15X + ₹1.5X = ₹16.5X

2. **Selling Price of Variety B**: Sold at a 20% profit.
- Selling Price of Variety B = Cost Price + Profit
- Profit = 20% of Cost Price = 20% of ₹(10X + 50) = ₹(10X + 50) * 0.20 = ₹2X + ₹10
- Selling Price of Variety B = ₹(10X + 50) + ₹(2X + 10) = ₹12X + ₹60

### Step 3: Set Up the Equation Based on Given Information
According to the problem, the total selling price of Variety A is ₹30 more than that of Variety B.
- Therefore, we can set up the equation:
\[ 16.5X = 12X + 60 + 30 \]
Simplifying this gives:
\[ 16.5X = 12X + 90 \]
\[ 16.5X - 12X = 90 \]
\[ 4.5X = 90 \]
\[ X = \frac{90}{4.5} = 20 \]

### Step 4: Calculate the Cost Prices with Found Value of X
1. **Cost Price of Variety A**:
- Cost Price of Variety A = ₹15X = ₹15 * 20 = ₹300

2. **Cost Price of Variety B**:
- Cost Price of Variety B = ₹(10X + 50) = ₹(10 * 20 + 50) = ₹200 + ₹50 = ₹250

### Step 5: Calculate the Total Cost Price
- Total Cost Price = Cost Price of Variety A + Cost Price of Variety B
\[ = ₹300 + ₹250 = ₹550 \]

### Step 6: Calculate the Selling Price for Overall Profit of 20%
- Total Selling Price for 20% profit:
\[ Selling Price = Cost Price + Profit = Cost Price + 20\% \text{ of Cost Price} \]
\[ = ₹550 + 0.20 * ₹550 = ₹550 + ₹110 = ₹660 \]

### Step 7: Calculate the Selling Price per kg
- Total weight of the rice = 15 kg (Variety A) + 10 kg (Variety B) = 25 kg
- Selling Price per kg = Total Selling Price / Total Weight
\[ = \frac{₹660}{25} = ₹26.4 \]

### Final Answer
The selling price per kg of the mixed rice is **₹26.4**.