From $5$ litres of a $20\%$ solution of alcohol in water, $2$ litres of solution is taken out and $2$ litres of water is added to it. Find the strength of alcohol in the new solution.

### Concept & Logic This is a **Replacement Problem**. Removing a portion of the fully mixed solution removes proportional amounts of alcohol and water, but does *not* change the concentration of the remainder. Adding pure water afterward acts as a dilution. ### Step-by-Step Solution * **Given:** * Initial total volume = $5$ litres * Initial alcohol concentration = $20\%$ * Volume removed = $2$ litres * Pure water added = $2$ litres * **Calculation:** * Step 1: Handle the removal. When $2$ litres are removed, the remaining volume is $5 - 2 = 3$ litres. * This remaining $3$ litres still has the original concentration of $20\%$ alcohol. * Calculate absolute alcohol remaining: $20\%$ of $3$ litres $= 0.20 \times 3 = 0.6$ litres. * Step 2: Handle the addition. $2$ litres of pure water is added back, bringing the total volume back to $5$ litres ($3 + 2 = 5$). * The total absolute alcohol remains strictly $0.6$ litres, as only water was added. * Calculate the final strength: $\left( \frac{\text{Remaining Alcohol}}{\text{New Total Volume}} \right) \times 100$. * $\left( \frac{0.6}{5} \right) \times 100 = 0.6 \times 20 = 12\%$. ### Exam Strategy & Shortcut **Fractional Retention Method:** Understand that removing $2$ litres out of $5$ means keeping $3$ litres out of $5$. You are retaining exactly $\frac{3}{5}$ of the original active solute (alcohol). Because the total volume is perfectly restored to $5$ litres with neutral water, the final concentration simply scales by the retained fraction. New Concentration = Original Concentration $\times$ Retained Fraction. New Concentration = $20\% \times \frac{3}{5} = \frac{60}{5} = 12\%$. Solved mentally in 5 seconds! ### Common Pitfall A common error is calculating $20\%$ of the initial $5$ litres ($1$ litre of alcohol), then mistakenly thinking the $2$ litres removed is pure alcohol, leading to negative or impossible values. Always remember that removing *solution* removes the components proportionally, leaving the remaining concentration temporarily unchanged. ### Final Answer **Therefore, the correct answer is 12%.**