From a can full of milk, $10$ litres of milk is taken out and replaced with water and this operation is repeated twice. If the capacity of the can is $50$ litres, what is the ratio of water and milk in the can after this operation?

### Concept & Successive Replacement When a fixed quantity of a mixture is repeatedly removed and replaced with water, the amount of the original pure liquid remaining can be calculated using the successive dilution formula. $$ \text{Final Quantity} = \text{Initial Quantity} \times \left(1 - \frac{\text{Replacement Volume}}{\text{Total Capacity}}\right)^n $$ Where $n$ is the *total* number of times the operation is performed. ### Step-by-Step Solution * **Given:** - Initial quantity of milk ($I$) = $50$ liters - Replacement volume ($R$) = $10$ liters - The operation is done once, and then "repeated twice", meaning the total number of operations ($n$) is $1 + 2 = 3$. * **Calculation:** 1. Calculate the final volume of milk left after $3$ operations: $\text{Final Milk} = 50 \times \left(1 - \frac{10}{50}\right)^3$ $\text{Final Milk} = 50 \times \left(1 - \frac{1}{5}\right)^3$ $\text{Final Milk} = 50 \times \left(\frac{4}{5}\right)^3$ $\text{Final Milk} = 50 \times \frac{64}{125} = 2 \times \frac{64}{5} = \frac{128}{5} = 25.6 \text{ liters}$ 2. Calculate the final volume of water in the $50$-liter can: $\text{Final Water} = \text{Total Capacity} - \text{Final Milk}$ $\text{Final Water} = 50 - 25.6 = 24.4 \text{ liters}$ 3. Find the ratio of water to milk: Ratio = $24.4 : 25.6 = 244 : 256$ 4. Simplify the ratio by dividing by $4$: $61 : 64$ 5. The ratio $61 : 64$ is not present in the given options ($2:3$, $9:16$, $16:9$). ### Exam Strategy & Shortcut Recognize the linguistic traps in dilution problems. Calculate the multiplier $(1 - \frac{10}{50}) = \frac{4}{5}$. If the operation is done $3$ times total, the final milk fraction is $(\frac{4}{5})^3 = \frac{64}{125}$. Milk = $64$ parts, Water = $(125 - 64) = 61$ parts. Ratio of Water to Milk = $61 : 64$. None of the options match. ### Common Pitfall A massive trap is misinterpreting the phrase "and this operation is repeated twice" as meaning the operation was performed a *total* of $2$ times. If $n=2$, the final milk fraction is $(\frac{4}{5})^2 = \frac{16}{25}$. The ratio of milk to total is $16:25$, meaning the ratio of Water to Milk is $9:16$, which perfectly matches option (b). However, strict English interpretation of "repeated twice" means $3$ total operations. ### Final Answer Therefore, the correct answer is **None of these**.