A dishonest milkman professes to sell his milk at cost price but he mixes it with water and thereby gains 25%. The percentage of water in the mixture is

### Concept & Profit from Free Ingredient When a mixture containing a free ingredient (water) and a pure ingredient (milk) is sold at the cost price of the pure ingredient, the profit percentage is equivalent to the ratio of the free ingredient to the pure ingredient. $$ \text{Ratio of Water : Milk} = \text{Profit Percentage} : 100 $$ ### Step-by-Step Solution 1. Understand the source of profit: - The $25\%$ gain is entirely due to the added water, since the milk is sold at its original cost price. 2. Determine the ratio of Water to Milk: - Water : Milk = $25 : 100 = 1 : 4$ 3. Calculate the total parts in the mixture: - Total mixture = $1 \text{ (water)} + 4 \text{ (milk)} = 5 \text{ parts}$ 4. Find the percentage of water in the total mixture: - Percentage of water = $\left(\frac{1}{5}\right) \times 100\%$ - Percentage of water = $20\%$ ### Exam Strategy & Shortcut If gain is $P\%$, the proportion of water in the mixture is $\frac{P}{100 + P}$. Here, $\frac{25}{125} = \frac{1}{5} = 20\%$. This bypasses the ratio step entirely. ### Common Pitfall A very common mistake is confusing the "percentage of water to milk" ($25\%$) with the "percentage of water in the mixture" ($20\%$). Always check what the denominator should be. ### Final Answer Therefore, the correct answer is **20%**.