A milkman pays Rs 6.40 per litre to buy pure milk. He then adds some amount of water, which is free, and sells the resulting mixture at Rs 8 per litre, thereby making a profit of 37.5% on his cost price. What is the ratio of water to milk in the mixture that the customers receive?

Introduction:
This question combines profit and loss with mixtures. The milkman buys milk at a certain cost price, mixes in free water, and sells the mixture at the original cost price of milk, yet earns a profit. We must compute the proportion of water and milk in the final mixture based on the percentage profit.

Given Data / Assumptions:

• Cost price of pure milk = Rs 6.40 per litre.
• Water is assumed to be free of cost.
• The mixture is sold at Rs 8 per litre.
• The milkman claims to sell at cost, but effectively earns a profit of 37.5%.
• We must find the ratio of water to milk in the mixture supplied to customers.

Concept / Approach:
Let m be the cost of pure milk per litre, here m = 6.40. Let the fraction of milk in 1 litre of the mixture be k, so the fraction of water is 1 - k. The cost price of 1 litre of mixture is then m * k, since only milk costs money. Selling price per litre of mixture is given as Rs 8. Profit percentage is 37.5%, that is 3 / 8 in fractional form. Use the relation:
SP = CP * (1 + profit rate). We equate SP to 8, CP to m * k, and profit rate to 3 / 8, and solve for k.

Step-by-Step Solution:
Step 1: Cost price of milk m = Rs 6.40 per litre. Step 2: Let fraction of milk in the mixture be k, so CP of 1 litre of mixture = 6.40 * k. Step 3: Profit percentage = 37.5% = 3 / 8. Step 4: Selling price SP = CP * (1 + 3 / 8) = CP * (11 / 8). Step 5: Given SP = Rs 8, so 8 = (6.40 * k) * 11 / 8. Step 6: Multiply both sides by 8: 64 = 6.40 * k * 11. Step 7: 6.40 = 32 / 5, so 64 = (32 / 5) * k * 11. Step 8: Simplify: 64 = (352 / 5) * k, so 64 * 5 = 352 * k. Step 9: 320 = 352 * k, hence k = 320 / 352 = 10 / 11. Step 10: Fraction of milk in mixture = 10 / 11, so fraction of water = 1 / 11. Step 11: Ratio of water to milk = (1 / 11) : (10 / 11) = 1 : 10.

Verification / Alternative check:
Assume 11 litres of mixture: milk = 10 litres, water = 1 litre. Cost of milk = 10 * 6.40 = Rs 64. Selling price at Rs 8 per litre for 11 litres = 11 * 8 = Rs 88. Profit = 88 - 64 = Rs 24. Profit percentage = 24 / 64 * 100 = 37.5%, matching the given profit. Thus the ratio 1 : 10 is correct for water to milk.

Why Other Options Are Wrong:
10 : 1 and 11 : 9 suggest more water than milk or comparable quantities, which would drastically change the cost and profit structure and not yield 37.5% profit.
9 : 11 and 1 : 4 do not match the calculated fractions obtained from the profit condition when you test them with assumed volumes and cost calculations.

Common Pitfalls:
Many learners mistakenly assume the selling price is equal to the cost price of pure milk, ignoring the given profit percentage. Others try to handle this as a simple alligation without incorporating profit, which leads to incorrect ratios. It is also easy to invert the ratio and give milk to water instead of water to milk, so careful reading is important.

Final Answer:
The ratio of water to milk in the mixture is 1 : 10.