Given:
- Two varieties of rice are mixed in the ratio 2:5.
- The mixture is sold at Rs 12 per kg at a profit of 20%.
- The first variety costs Rs 7 per kg more than the second variety.

To find:
- The cost per kg of the first variety of rice.

Assumptions:
- Let the cost per kg of the second variety of rice be 'x'.
- The cost per kg of the first variety of rice will be 'x + 7'.

Solution:

Step 1: Determining the ratio
- Let the weight of the first variety of rice be '2x' kg.
- Let the weight of the second variety of rice be '5x' kg.
- The total weight of the mixture will be '2x + 5x' kg.

Step 2: Determining the cost of the mixture
- The total cost of the mixture can be calculated as follows:
Total cost = (cost per kg of the first variety * weight of the first variety) + (cost per kg of the second variety * weight of the second variety)
Total cost = (x + 7) * 2x + x * 5x
Total cost = 2x^2 + 14x + 5x^2
Total cost = 7x^2 + 14x

Step 3: Determining the selling price of the mixture
- The selling price of the mixture is given as Rs 12 per kg.
- The total selling price of the mixture can be calculated as follows:
Total selling price = selling price per kg * total weight of the mixture
Total selling price = 12 * (2x + 5x)
Total selling price = 12 * 7x
Total selling price = 84x

Step 4: Determining the profit percentage
- The profit percentage can be calculated using the formula:
Profit percentage = (Profit / Cost price) * 100
Profit percentage = ((Total selling price - Total cost) / Total cost) * 100
Profit percentage = ((84x - (7x^2 + 14x)) / (7x^2 + 14x)) * 100
Profit percentage = ((70x - 7x^2) / (7x^2 + 14x)) * 100
Profit percentage = (7x * (10 - x) / 7x * (x + 2)) * 100
Profit percentage = (10 - x) / (x + 2) * 100

Step 5: Solving the profit percentage equation
- As the profit percentage is given as 20%, we can write the equation as follows:
(10 - x) / (x + 2) * 100 = 20
10 - x = 20 * (x + 2)
10 - x = 20x + 40
21x = -30
x = -30 / 21
x = -10/7

Step 6: Finding the cost per kg of the first variety of