This problem involves finding the ratio in which two ingredients (varieties of rice) are mixed, given their costs, the selling price of the mixture, and the profit percentage.

Calculating Mixture Cost Price

First, we need to determine the cost price (CP) of the mixture. We know the selling price (SP) is Rs. 55/kg and the profit is 10%.

The formula relating SP, CP, and profit percentage is:

$ SP = CP \times (1 + \text{Profit Percentage}) $

Substituting the given values:

$ 55 = CP \times (1 + 0.10) $

$ 55 = CP \times 1.10 $

Solving for CP:

$ CP = \frac{55}{1.10} = 50 $

So, the cost price of the mixture is Rs. 50/kg.

Applying Rule of Allegation for Ratio

Now, we use the rule of allegation to find the ratio in which the two rice varieties were mixed. This method compares the cost prices of the individual varieties with the cost price of the mixture.

• Cost Price of Variety 1 (CP1): Rs. 40/kg
• Cost Price of Variety 2 (CP2): Rs. 60/kg
• Mean Cost Price (Mixture CP): Rs. 50/kg

The rule states that the ratio of quantities is inversely proportional to the difference of their prices from the mean price.

Difference for Variety 1 = Mean CP - CP1 = \( 50 - 40 = 10 \)

Difference for Variety 2 = CP2 - Mean CP = \( 60 - 50 = 10 \)

The ratio in which the two varieties are mixed is:

$ \text{Ratio} = \frac{\text{Difference for Variety 2}}{\text{Difference for Variety 1}} = \frac{60 - 50}{50 - 40} = \frac{10}{10} $

$ \text{Ratio} = 1:1 $

Final Ratio

The ratio in which the two types of rice are mixed is 1:1.