In what ratio should water and wine be mixed so that, when the mixture is sold at the cost price of pure wine, the seller makes a profit of 33.33% on the actual cost?

Introduction / Context:
This mixture and profit question explores how adding free water to wine affects overall profit when the mixture is sold at the original cost price of pure wine. It tests understanding of how to relate profit percentage to the ratio of free component (water) to the costly component (wine).

Given Data / Assumptions:

• Wine has some cost price per litre, say Rs. C.
• Water is free and has zero cost.
• The mixture is sold at the cost price of pure wine, i.e., selling price per litre of mixture = C.
• The desired profit percentage on the actual cost is 33.33%, which is approximately one-third.
• We must find the ratio water : wine in the mixture.

Concept / Approach:
Let the quantity of wine be W litres and water be A litres. Cost comes only from wine, so total cost = W * C. Total volume of mixture = W + A litres, and this is sold at C per litre. Thus, total revenue = (W + A) * C. Profit percentage is (Revenue - Cost) / Cost * 100. Setting this equal to 33.33% and simplifying leads to a relationship between A and W, from which the ratio water : wine can be found.

Step-by-Step Solution:
Step 1: Let wine quantity = W litres and water quantity = A litres.Step 2: Cost price of wine per litre = C, water is free.Step 3: Total cost = W * C.Step 4: Total mixture sold = W + A litres at price C per litre, so total revenue = (W + A) * C.Step 5: Profit = Revenue - Cost = (W + A) * C - W * C = A * C.Step 6: Profit percentage = (Profit / Cost) * 100 = (A * C) / (W * C) * 100 = (A / W) * 100.Step 7: Given profit percentage is 33.33% ≈ 1 / 3, so A / W = 1 / 3.Step 8: Therefore water : wine = A : W = 1 : 3.

Verification / Alternative check:
Assume W = 3 litres, A = 1 litre, and C = Rs. 10 per litre.Total cost = 3 * 10 = Rs. 30.Total mixture = 4 litres. Revenue at cost price C = 4 * 10 = Rs. 40.Profit = 40 - 30 = Rs. 10, profit percentage = 10 / 30 * 100 = 33.33%, confirming the mixture ratio.

Why Other Options Are Wrong:
Ratios such as 1 : 4, 2 : 3 or 3 : 4 yield different values of A / W and therefore produce profit percentages different from 33.33%. A ratio of 1 : 2 gives A / W = 1 / 2 or 50% profit. Only 1 : 3 corresponds to adding one-third as much water as wine, matching the required profit percentage.

Common Pitfalls:
Some learners invert the ratio and use wine : water instead of water : wine. Others mistakenly assume that 33.33% refers to the fraction of wine in the mixture rather than profit on cost. It is important to clearly distinguish between cost-bearing and free components and to base the profit calculation on cost.

Final Answer:
Water and wine should be mixed in the ratio 1 : 3 (water : wine).