A dairyman pays ₹ 6.40 per litre of milk. He adds water and sells the mixture at ₹ 8 per litre, thereby making 37.5% profit. The proportion of water to milk received by the customers is

### Concept & Alligation with Free Additions When water (cost = 0) is mixed with pure milk to gain a profit by selling at a given price, alligation helps find the exact ratio. We must first find the true cost price of the mixture by removing the profit from the selling price. $$Cost\ Price\ (CP) = \frac{Selling\ Price\ (SP)}{1 + \frac{Profit\ \%}{100}}$$ ### Step-by-Step Solution * Selling Price (SP) of the mixture = ₹ 8 per litre. * Profit = 37.5% = $\frac{3}{8}$. * Cost Price (CP) of the mixture = $\frac{8}{1 + \frac{3}{8}} = \frac{8}{\frac{11}{8}} = \frac{64}{11}$ per litre. * Cost of pure milk = ₹ 6.40 (or $\frac{64}{10}$) per litre. Cost of water = ₹ 0 per litre. * Using the rule of alligation: * Ratio of Milk to Water = $(CP\ of\ mix - CP\ of\ water) : (CP\ of\ milk - CP\ of\ mix)$ * Ratio = $\left(\frac{64}{11} - 0\right) : \left(\frac{64}{10} - \frac{64}{11}\right)$ * Ratio = $\frac{64}{11} : \frac{64(11 - 10)}{110}$ * Ratio = $\frac{1}{11} : \frac{1}{110} = 10 : 1$. * The question asks for the proportion of water to milk, which is reversed: $1 : 10$. ### Exam Strategy & Shortcut If a mixture of milk and water is sold at a specific price, always convert to cost price first. A quick mental check: $37.5\%$ profit on $\frac{64}{11}$ is $\frac{64}{11} \times \frac{11}{8} = 8$. It works perfectly. Cross-multiply the denominators in alligation ($11 \times 6.4 = 70.4$ vs $64$) to see the ratio is $64 : (70.4 - 64) = 64 : 6.4 = 10 : 1$. ### Common Pitfall Students often calculate the milk-to-water ratio ($10:1$) correctly but hastily select an answer without reading carefully that the question specifically asks for the **water to milk** ratio ($1:10$). ### Final Answer Therefore, the correct answer is **1 : 10**.