A milkman claims to sell milk at its original cost price per litre, yet he still makes a profit of 30% because he mixes some amount of free water with the milk. Assuming he sells the adulterated mixture at the same price per litre that he paid for the pure milk, what percentage of the mixture is actually milk?

Introduction:
This question combines profit concept with mixture adulteration. The milkman adds free water to the milk but sells the mixture at the original cost price of pure milk. Despite this, he earns a 30% profit. From this information, we must determine what percentage of the mixture is actually milk.

Given Data / Assumptions:

• The milkman buys pure milk at some cost price per litre, say Rs M.
• He mixes water, which is free, with the milk.
• He sells the mixture at Rs M per litre, the same as the cost price of pure milk.
• He earns a profit of 30% on his cost price.
• We must find the percentage of milk in the mixture.

Concept / Approach:
Let the fraction of milk in 1 litre of the mixture be k, so the fraction of water is (1 - k). Because water is free, the cost price of 1 litre of the mixture is only due to the milk and equals M * k. The selling price per litre is M. Profit percentage is defined as:
profit% = (SP - CP) / CP * 100. Given the profit is 30%, SP = CP * 1.3. We equate the selling price M to 1.3 times the mixture cost M * k and solve for k.

Step-by-Step Solution:
Step 1: Let cost price of pure milk per litre be M rupees. Step 2: Let fraction of milk in the mixture be k, so CP of 1 litre of mixture = M * k. Step 3: Selling price of mixture per litre = M rupees. Step 4: Profit percentage = 30%, so SP = CP * (1 + 0.30) = CP * 1.3. Step 5: Thus M = 1.3 * (M * k). Step 6: Divide both sides by M (M is not zero): 1 = 1.3 * k. Step 7: So k = 1 / 1.3 = 10 / 13. Step 8: Fraction of milk in mixture is 10 / 13, so percentage of milk = (10 / 13) * 100. Step 9: (10 / 13) * 100 ≈ 76.92%.

Verification / Alternative check:
Assume the milkman sells 13 litres of mixture. Then milk in the mixture = 10 litres, water = 3 litres. Cost price of milk = 10 * M = 10M. Selling price of mixture = 13 * M = 13M. Profit = 13M - 10M = 3M. Profit percentage = 3M / 10M * 100 = 30%, which matches the condition given in the question and confirms that 10 / 13 or 76.92% of the mixture is milk.

Why Other Options Are Wrong:
71.02%, 63.22%, and 86.42% do not correspond to any simple fraction that gives exactly 30% profit given the structure of costs and selling price. Testing them by assuming a convenient volume of mixture would show that the profit percentage is not 30% for those compositions.
60% is too low and would yield a much higher profit percentage because the cost price per litre would be only 60% of M while the selling price remains M.

Common Pitfalls:
Many students confuse the direction of the profit calculation and set SP = CP / 1.3 instead of SP = CP * 1.3. Others wrongly treat M as a specific number instead of a variable, or mistakenly add water cost. Some learners also misinterpret the question and attempt to find the percentage of water rather than milk, leading to an inverted answer.

Final Answer:
The percentage of milk in the mixture is 76.92%.