Two vessels contain equal quantities of $40\%$ alcohol. Sachin changed the concentration of the first vessel to $50\%$ by adding extra quantity of pure alcohol. Vivek changed the concentration of the second vessel to $50\%$ replacing a certain quantity of the solution with pure alcohol. By what percentage is the quantity of alcohol added by Sachin more/less than that replaced by Vivek?

### Concept & Strategy To adjust the concentration of a mixture, adding pure solute (Sachin's method) increases the total volume, while replacing part of the mixture with pure solute (Vivek's method) maintains a constant total volume. We can use basic ratios and the Alligation rule to find the exact quantities involved in both methods. ### Step-by-Step Solution * **Given:** * Initial concentration in both vessels = $40\%$ alcohol. * Final concentration in both vessels = $50\%$ alcohol. * Let the initial volume of both vessels be $100$ units. * **Sachin's Calculation (Addition):** * Initial alcohol = $40\%$ of $100 = 40$ units. * Let Sachin add $x$ units of pure ($100\%$) alcohol. * New total alcohol = $40 + x$. New total volume = $100 + x$. * Target concentration is $50\%$ (which is $\frac{1}{2}$): * $\frac{40 + x}{100 + x} = \frac{1}{2}$ * $80 + 2x = 100 + x \implies x = 20$. * Sachin added $20$ units. * **Vivek's Calculation (Replacement):** * Vivek replaces $y$ units of the mixture. This is equivalent to mixing the remaining $40\%$ solution with added pure $100\%$ alcohol to get a $50\%$ mixture. * Using Alligation: * Left side (remaining mix) = $40\%$. Right side (added pure) = $100\%$. Target (mean) = $50\%$. * Ratio of remaining volume to added volume = $(100 - 50) : (50 - 40) = 50 : 10 = 5 : 1$. * This means out of $6$ total parts ($5+1$), $1$ part was the replaced/added quantity. * Total volume = $100$. Quantity replaced $y = \frac{1}{6} \times 100 = \frac{50}{3}$ units. * **Comparison:** * Sachin's quantity = $20$. Vivek's quantity = $\frac{50}{3}$. * Difference = $20 - \frac{50}{3} = \frac{60 - 50}{3} = \frac{10}{3}$. * Percentage by which Sachin's quantity is more than Vivek's = $\left( \frac{\text{Difference}}{\text{Vivek's Quantity}} \right) \times 100$. * $\left( \frac{10/3}{50/3} \right) \times 100 = \frac{10}{50} \times 100 = 20\%$. ### Exam Strategy & Shortcut Assume initial volume is $60$ (the LCM of the alligation parts $6$ simplifies fractions). Sachin: $\frac{24+x}{60+x} = \frac{1}{2} \implies 48 + 2x = 60 + x \implies x = 12$. Vivek: Ratio $5:1$, so replaced is $\frac{1}{6}$ of $60 = 10$. Sachin added $12$, Vivek replaced $10$. Sachin is $2$ units more than Vivek ($10$). $\frac{2}{10} = 20\%$. Choosing a smart assumed value eliminates complex fractions! ### Common Pitfall Students frequently misread the final question and calculate the percentage based on Sachin's quantity as the denominator ($\frac{10/3}{20} = 16.66\%$), matching trap option (c). The phrasing "more/less than that replaced by Vivek" strictly dictates that Vivek's quantity must be the denominator. ### Final Answer **Therefore, the correct answer is 20% more.**