A dishonest milkman buys pure milk at Rs. 24 per litre and then adds water equal to one-third of the quantity of milk. He sells the resulting milk–water mixture at Rs. 32 per litre. What is his percentage gain on the whole transaction?

Introduction / Context:
This question tests the concept of profit percentage when a dishonest milkman mixes water (which is free) with milk and then sells the mixture at a higher rate. Such aptitude questions combine simple percentage profit with the idea of adulteration, where the cost price is based only on the pure component while the selling price is based on the full mixture volume.

Given Data / Assumptions:

• Cost price (CP) of pure milk = Rs. 24 per litre.
• Water is added equal to one-third of the quantity of milk.
• Water is considered free and has no cost.
• Selling price (SP) of the mixture = Rs. 32 per litre.
• We need to find the overall percentage gain on the transaction.

Concept / Approach:
The main idea is that the milkman pays only for milk but earns money on the combined quantity of milk plus water. We first assume a convenient quantity of milk, calculate the total mixture volume after adding water, then compute the total cost and total selling price. Finally, we use the standard profit percentage formula: profit% = (profit / cost price) * 100.

Step-by-Step Solution:
Assume the milkman buys 3 litres of milk (any convenient quantity works). Cost price of 3 litres of milk = 3 * 24 = Rs. 72. He adds water equal to one-third of milk: water = (1 / 3) * 3 = 1 litre. Total mixture volume = 3 litres milk + 1 litre water = 4 litres. Selling price per litre of mixture = Rs. 32. Total selling price = 4 * 32 = Rs. 128. Profit = Total selling price - Total cost price = 128 - 72 = Rs. 56. Profit percentage = (Profit / Cost price) * 100. Profit percentage = (56 / 72) * 100. Simplify 56 / 72 = 7 / 9 ≈ 0.7778. Therefore, profit percentage ≈ 0.7778 * 100 = 77.78%.

Verification / Alternative check:
We could also assume 1 litre of milk. Then water added = 1 / 3 litre, mixture = 4 / 3 litres. Cost price = 24 * 1 = Rs. 24. Selling price = mixture volume * 32 = (4 / 3) * 32 = 128 / 3 ≈ Rs. 42.67. Profit ≈ 42.67 - 24 = 18.67. Profit percentage ≈ (18.67 / 24) * 100 ≈ 77.78%, which confirms the earlier calculation.

Why Other Options Are Wrong:
33.33%: This would arise if one incorrectly assumed gain only on the extra water without considering the full mixture price. It underestimates the profit.
50%: This assumes incorrect proportional reasoning between 24 and 32 without including the free water effect.
66.67%: This could come from partial calculation where some steps are approximated or misapplied.
60%: This is a random plausible value but does not match the exact ratio of profit to cost price.

Common Pitfalls:
Students often forget that water has zero cost and treat the entire mixture as if the cost is based on the mixture volume. Another mistake is to assume the milkman adds one-third of the total mixture instead of one-third of the milk, which gives a different and incorrect ratio. Some also directly take (32 - 24) as profit over 24 to get 33.33% and ignore the role of extra volume. Careful attention to the wording “adds one-third of water to it” is crucial.

Final Answer:
The dishonest milkman makes a gain of 77.78% on his cost price.