A milkman sells a mixture at the cost price of pure milk. In what ratio should water be mixed with milk to earn a gain of 16 2/3% (i.e., one-sixth) on selling the mixture at the cost price of milk?

Introduction / Context:
When a seller mixes free water with milk and sells the mixture at the cost price of milk, the observed profit arises from the water content. We must determine the water-to-milk ratio that yields a 16 2/3% (one-sixth) gain.

Given Data / Assumptions:

• Selling price per liter of mixture = cost price per liter of pure milk.
• Profit% on cost = 16 2/3% = 1/6.
• Water is free; milk costs 1 (normalized) per liter for ease of proportion.

Concept / Approach:
If milk fraction in the mixture is m, then revenue per liter = 1, cost per liter = m*1 + (1 − m)*0 = m. Profit% = (revenue − cost)/cost * 100 = (1 − m)/m * 100. Set this to 16 2/3%.

Step-by-Step Solution:
(1 − m)/m = 1/61/m − 1 = 1/6 ⇒ 1/m = 7/6 ⇒ m = 6/7Water fraction = 1 − m = 1/7Hence water : milk = (1/7) : (6/7) = 1 : 6

Verification / Alternative check:
With milk 6 liters and water 1 liter (7 liters total), revenue at “1 per liter” is 7, cost is 6, profit = 1, profit% = 1/6 = 16 2/3%.

Why Other Options Are Wrong:
Ratios like 4 : 3 or 6 : 1 would produce very different profits; only 1 : 6 gives exactly one-sixth profit at cost price.

Common Pitfalls:
Confusing water : milk with milk : water, or using profit on selling price instead of profit on cost price.

Final Answer:
1 : 6