A milk vendor mixes water with milk in the ratio 1 : 4. He then measures 800 ml instead of a litre and sells the milk at a nominal profit of 20% over the cost price. What is his actual profit percentage?

### Concept & Faulty Measurement with Adulteration To calculate the true profit percentage when a seller uses false weights and adulterates goods, compare the total revenue collected for the claimed quantity against the actual cost incurred for the pure substance delivered. $$ \text{Actual Profit \%} = \left( \frac{\text{Selling Price of Claimed Quantity} - \text{Actual Cost Incurred}}{\text{Actual Cost Incurred}} \right) \times 100 $$ ### Step-by-Step Solution * Let the cost price (CP) of pure milk be ₹ 1 per ml. Thus, 1 litre (1000 ml) of pure milk costs ₹ 1000. * The vendor mixes water and milk in the ratio $1 : 4$. In any volume of this mixture, $\frac{4}{5}$ is milk and $\frac{1}{5}$ is water. * He sells 800 ml of this mixture when the customer asks for a litre. * The vendor charges for 1 litre at a $20\%$ nominal profit over the cost price. * Selling Price (SP) = CP of 1000 ml pure milk + $20\%$ = $1000 + 200 = 1200$. * The actual volume given to the customer is 800 ml of the mixture. * The amount of pure milk in this 800 ml mixture = $800 \times \frac{4}{5} = 640$ ml. * The actual cost incurred by the vendor is only for the pure milk he gives (since water is free). Actual CP = ₹ 640. * Actual Profit = SP - Actual CP = $1200 - 640 = 560$. * Actual Profit \% = $\left(\frac{560}{640}\right) \times 100 = \left(\frac{7}{8}\right) \times 100 = 87.5\%$. ### Exam Strategy & Shortcut Use proportional multipliers. The profit multiplier is $1.2$ (from $20\%$ nominal profit). The quantity manipulation multiplier is $\frac{1000}{800} = \frac{5}{4}$ (he gets paid for 5 parts while giving 4). The adulteration multiplier is $\frac{5}{4}$ (for every 4 parts of milk he pays for, he sells 5 parts of mixture). Overall multiplier = $1.2 \times \frac{5}{4} \times \frac{5}{4} = 1.2 \times \frac{25}{16} = \frac{30}{16} = \frac{15}{8}$. Profit = $\frac{15 - 8}{8} = \frac{7}{8} = 87.5\%$. ### Common Pitfall Calculating the actual cost based on 800 ml of pure milk, forgetting that the 800 ml delivered is a mixture that contains even less actual milk (640 ml). ### Final Answer Therefore, the correct answer is **87.5%**.