A milkman buys pure milk at Rs. x per litre and sells the mixture at Rs. 2x per litre. He mixes 2 litres of water with every 6 litres of pure milk before selling. Assuming water is free, what percentage profit does he make on his cost price?

Introduction / Context:
This problem is a classic mixture and profit question that tests understanding of how adding a free component such as water affects overall profit percentage. The key idea is that the seller pays only for milk but charges customers for the entire mixture, including the water, which creates a higher effective selling price per unit of cost.

Given Data / Assumptions:

• Cost price of pure milk = Rs. x per litre.
• Selling price of the mixture = Rs. 2x per litre.
• For every batch, 6 litres of milk are mixed with 2 litres of water.
• Water is assumed to be free, so it has zero cost.
• We need to find the profit percentage on the cost of the milk used.

Concept / Approach:
Profit percentage is calculated as (Profit / Cost price) * 100. Here, the cost is only for the milk, while revenue is collected on the total mixture, which includes both milk and water. We first compute the total cost of the milk portion, then compute the total selling price of the mixture for one batch, find the profit amount, and finally convert this profit into a percentage of the cost price.

Step-by-Step Solution:
Step 1: For each batch, the milkman uses 6 litres of milk and 2 litres of water, so total mixture = 6 + 2 = 8 litres.Step 2: Cost price for 6 litres of milk = 6 * x = 6x rupees. Water adds no cost.Step 3: Selling price of the 8 litres mixture = 8 * 2x = 16x rupees.Step 4: Profit amount = selling price - cost price = 16x - 6x = 10x rupees.Step 5: Profit percentage = (Profit / Cost price) * 100 = (10x / 6x) * 100 = (10 / 6) * 100 = (5 / 3) * 100 = 166.67% approximately.

Verification / Alternative check:
Assume x = 1 for simplicity. Then cost of 6 litres of milk is Rs. 6, and selling price of 8 litres of mixture is Rs. 16.Profit = 16 - 6 = Rs. 10. Profit percentage = 10 / 6 * 100 ≈ 166.67%.This matches the result from the algebraic method, confirming correctness.

Why Other Options Are Wrong:
60% and 100% are too low and would correspond to much smaller markups than doubling the price while adding free water. A profit of 116% ignores the extra 2 litres of water revenue. The value 133.33% comes from mixing ratios other than 6 litres of milk to 2 litres of water. Only 166.67% fits the given data.

Common Pitfalls:
Many learners mistakenly treat the total 8 litres as if all were purchased at cost, which underestimates profit. Others compute profit on the selling price instead of cost price, or they ignore that water is free. Carefully distinguishing between what is paid for and what is sold is critical in all mixture-based profit questions.

Final Answer:
The milkman's profit percentage on his cost is approximately 166.67%.