Let's break the problem down step by step:

Information provided:

1. Varieties of rice:

   • First variety costs Rs. 20/kg.
   • Second variety costs Rs. 24/kg.
   • Third variety costs Rs. 30/kg.

2. The merchant sells the mixture at Rs. 30/kg and earns a profit of 20%.

3. The mixture contains 2 kg of the third variety (which costs Rs. 30/kg).

4. We need to find out how many kilograms of the second variety can be in the mixture.

Step 1: Determine the selling price of the mixture.

The mixture is sold at Rs. 30/kg, and the merchant earns a 20% profit. To find the cost price (C.P.) of the mixture, we use the formula for profit:

$\text{Selling Price (S.P.)} = \text{Cost Price (C.P.)} \times (1 + \text{Profit Percentage})$

Given:

• S.P. = Rs. 30/kg
• Profit Percentage = 20% = 0.2

$30 = \text{C.P.} \times (1 + 0.2)$

$30 = \text{C.P.} \times 1.2$

$\text{C.P.} = \frac{30}{1.2} = 25 \, \text{Rs./kg}$

Thus, the cost price of the mixture is Rs. 25/kg.

Step 2: Set up the equation for the cost price of the mixture.

Let:

• $x$ kg be the amount of the first variety (Rs. 20/kg),
• $y$ kg be the amount of the second variety (Rs. 24/kg),
• $2$ kg is the amount of the third variety (Rs. 30/kg) as given.

The total cost of the mixture is:

$\text{Total Cost} = (20x) + (24y) + (30 \times 2)$

Simplifying:

$\text{Total Cost} = 20x + 24y + 60$

The total weight of the mixture is $x + y + 2$ kg, and since the cost price of the mixture is Rs. 25/kg, we set up the equation for the total cost:

$\frac{20x + 24y + 60}{x + y + 2} = 25$

Step 3: Solve for $y$.

Multiply both sides of the equation by $x + y + 2$:

$20x + 24y + 60 = 25(x + y + 2)$

Expand the right-hand side:

$20x + 24y + 60 = 25x + 25y + 50$

Simplify the equation:

$20x + 24y + 60 - 25x - 25y - 50 = 0$

$-5x - y + 10 = 0$

Rearrange to get:

$y = -5x + 10$

Step 4: Find integer values of $x$ and $y$.

For $y$ to be an integer, $x$ must be an integer. Let’s test values of $x$:

• If $x = 1$, then $y = -5(1) + 10 = 5$.
• If $x = 2$, then $y = -5(2) + 10 = 0$.

Thus, the possible values are $(x = 1, y = 5)$ or $(x = 2, y = 0)$.

Step 5: Conclusion.

The number of kilograms of the second variety that can be in the mixture is 5 kg (when $x = 1$, $y = 5$).

Would you like a further breakdown of any part of the solution or additional details?

Related Questions:

1. What would the mixture cost per kg if the merchant didn't aim for any profit?
2. How would the answer change if the merchant wanted to earn a 10% profit instead of 20%?
3. What is the total weight of the mixture in the scenario above?
4. If the price of the second variety of rice increased to Rs. 26/kg, how would it affect the mixture's composition?
5. Can there be a scenario where all three varieties are mixed in equal proportions?

Tip:

When dealing with mixture problems, always balance the total cost and total weight to maintain consistency in unit price calculations.