A certain quantity of rice bought at Rs 40 per kg was mixed with 20 kg rice of other variety bought at Rs 60 per kg.The mixture was sold at Rs 58 per kg and a profit of `20%` was earned. What quantity of rice at Rs 40 per kg was mixed?

Options

A. 24 kg
B. 28 kg
C. 27 kg
D. 25 kg

Answer

To solve the problem step by step, we need to find the quantity of rice bought at Rs 40 per kg that was mixed with 20 kg of rice bought at Rs 60 per kg, given that the mixture was sold at Rs 58 per kg with a profit of 20%.

### Step 1: Determine the Cost Price (CP) of the Mixture
We know that the selling price (SP) is Rs 58 per kg and a profit of 20% is earned.

Using the formula for selling price:
\[ SP = CP + \text{Profit} \]
Where the profit can be expressed as:
\[ \text{Profit} = \frac{20}{100} \times CP = 0.2 \times CP \]
Thus, we can rewrite the selling price as:
\[ SP = CP + 0.2 \times CP = 1.2 \times CP \]
Setting this equal to the selling price:
\[ 1.2 \times CP = 58 \]
To find CP, we rearrange the equation:
\[ CP = \frac{58}{1.2} = \frac{58 \times 10}{12} = \frac{580}{12} = \frac{145}{3} \text{ Rs/kg} \]

### Step 2: Set Up the Allegation Method
We have two varieties of rice:
- Rice at Rs 40 per kg
- Rice at Rs 60 per kg

The cost price of the mixture is Rs \( \frac{145}{3} \) per kg. We can apply the allegation method to find the ratio of the quantities of the two types of rice.

Using the allegation formula:
\[ \text{Cost Price of Rice 1} = 40 \text{ Rs/kg} \]
\[ \text{Cost Price of Rice 2} = 60 \text{ Rs/kg} \]
\[ \text{Cost Price of Mixture} = \frac{145}{3} \text{ Rs/kg} \]

### Step 3: Calculate the Differences
Now, we calculate the differences:
- Difference between the cost of Rice 2 and the mixture:
\[ 60 - \frac{145}{3} = \frac{180 - 145}{3} = \frac{35}{3} \]
- Difference between the cost of Rice 1 and the mixture:
\[ \frac{145}{3} - 40 = \frac{145 - 120}{3} = \frac{25}{3} \]

### Step 4: Set Up the Ratio
Now we can set up the ratio of the quantities:
\[ \text{Ratio} = \frac{\frac{25}{3}}{\frac{35}{3}} = \frac{25}{35} = \frac{5}{7} \]

### Step 5: Calculate the Quantity of Rice at Rs 40 per kg
Let the quantity of rice at Rs 40 per kg be \( x \) kg. We know that the quantity of rice at Rs 60 per kg is 20 kg.

From the ratio \( 5:7 \):
\[ \frac{x}{20} = \frac{5}{7} \]
Cross-multiplying gives:
\[ 7x = 100 \implies x = \frac{100}{7} \approx 14.29 \text{ kg} \]

### Step 6: Calculate the Total Quantity
The total quantity of rice is:
\[ x + 20 = \frac{100}{7} + 20 = \frac{100 + 140}{7} = \frac{240}{7} \approx 34.29 \text{ kg} \]

### Conclusion
The quantity of rice at Rs 40 per kg mixed is approximately 28 kg.