We need to determine the ratio in which two varieties of rice are mixed to achieve a selling price of $54 per kg while making a 20% profit. Here's the solution:

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Step 1: Calculate the cost price of the mixture

The shopkeeper is selling the mixture at $54 per kg and making a 20% profit. Therefore, the cost price of the mixture per kg can be calculated as:

$\text{Cost Price of Mixture} = \frac{\text{Selling Price}}{1 + \text{Profit Percentage}}$

$\text{Cost Price of Mixture} = \frac{54}{1 + 0.2} = \frac{54}{1.2} = 45 \, \text{per kg.}$

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Step 2: Use the rule of alligation

The cost prices of the two varieties of rice are $60 per kg and $40 per kg. The cost price of the mixture is $45 per kg. Using the rule of alligation:

$\text{Ratio} = \frac{\text{Higher Price} - \text{Mixture Price}}{\text{Mixture Price} - \text{Lower Price}}$

$\text{Ratio} = \frac{60 - 45}{45 - 40} = \frac{15}{5} = 3:1.$

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Final Answer:

The two varieties of rice are mixed in the ratio 3:1.

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Do you have any questions or need further explanation? Here are 5 related questions to explore:

1. What would the selling price be if the shopkeeper mixed the rice in a different ratio?
2. How does the profit percentage affect the cost price and selling price relationship?
3. Can we generalize this method for mixtures with more than two components?
4. How would the result change if one of the rice varieties cost less than $40 per kg?
5. What is the role of the rule of alligation in other types of mixture problems?

Tip: The rule of alligation is a quick way to find ratios in mixing problems involving costs or concentrations.