85% and 92% alcoholic solutions are mixed to get 35 litres of an 89% alcoholic solution. How many litres of each solution are there in the new mixture?

### Concept & Strategy When mixing two solutions of different concentrations to form a target concentration, the **Rule of Alligation** provides the exact ratio of the quantities required. It avoids complex linear equations by mapping the distances between the percentages. $$ \frac{\text{Quantity of Cheaper}}{\text{Quantity of Dearer}} = \frac{\text{Dearer Price} - \text{Mean Price}}{\text{Mean Price} - \text{Cheaper Price}} $$ ### Step-by-Step Solution * **Given:** * Solution 1 (Lower) = $85\%$ * Solution 2 (Higher) = $92\%$ * Target Mixture (Mean) = $89\%$ * Total Volume = $35$ litres * **Calculation / Deduction:** * Set up the Alligation cross: * Place $85\%$ on the top left and $92\%$ on the top right. * Place $89\%$ in the middle. * Calculate the diagonal differences: * Right diagonal (Difference between $92\%$ and $89\%$) $= 3$. This goes under the $85\%$ side. * Left diagonal (Difference between $89\%$ and $85\%$) $= 4$. This goes under the $92\%$ side. * The required ratio of Solution 1 to Solution 2 is $3 : 4$. * The total number of ratio parts is $3 + 4 = 7$ parts. * We know the total volume is $35$ litres. So, $7$ parts $= 35$ litres. * $1$ part $= 5$ litres. * Volume of Solution 1 ($3$ parts) $= 3 \times 5 = 15$ litres. * Volume of Solution 2 ($4$ parts) $= 4 \times 5 = 20$ litres. ### Exam Strategy & Shortcut The Alligation rule itself is the shortcut. However, to verify your answer even faster via Option Elimination: The target mixture ($89\%$) is closer to $92\%$ than it is to $85\%$. Therefore, the mixture must contain a larger quantity of the $92\%$ solution. Looking at the options, only (a) and (c) have more of the second solution. A quick mental check of the $3:4$ ratio confirms (c) is correct. ### Common Pitfall Students often place the calculated ratio components under the wrong initial solutions (swapping the $3$ and $4$). Remember: the difference calculated from the right side value dictates the proportion of the left side substance, and vice versa. ### Final Answer **Therefore, the correct answer is 15 of the first and 20 of the second.**