A vessel contains a mixture of Grape, Pineapple and Banana juices in the respective ratio of $4 : 6 : 5$. $15$ litres of this mixture is taken out and $8$ litres of grape juice and $2$ litres of pineapple juice is added to the vessel. If the resultant quantity of grape juice is $10$ litres less than the resultant quantity of grape juice is $10$ litres less than the resultant quantity of pineapple juice. What was the initial quantity of mixture in the vessel? (in litres) [IBPS—Bank PO/MT (Pre.) Exam, 2015]

### Concept & Ratio Equations When a specific amount of mixture is removed, it is drawn out in the exact same ratio as the original mixture. Tracking the algebraic changes to each component allows us to solve for the original total. *(Note: The question contains a redundant phrase error, but the mathematical logic relies on: "grape juice is $10$ litres less than... pineapple juice")*. ### Step-by-Step Solution * **Given:** Initial ratio $G : P : B = 4 : 6 : 5$. Total initial parts = $15$. Let total initial quantity be $15x$. * $15$ L of mixture is removed. It leaves in the ratio $4:6:5$. * Grape removed = $\frac{4}{15} \times 15 = 4$ L. * Pineapple removed = $\frac{6}{15} \times 15 = 6$ L. * Initial component quantities: Grape = $4x$, Pineapple = $6x$. * Quantities after removal: * Grape = $4x - 4$ * Pineapple = $6x - 6$ * Quantities after addition ($8$L Grape, $2$L Pineapple): * Final Grape = $4x - 4 + 8 = 4x + 4$ * Final Pineapple = $6x - 6 + 2 = 6x - 4$ * The problem states Final Grape is $10$ L less than Final Pineapple: * $(4x + 4) = (6x - 4) - 10$ * $4x + 4 = 6x - 14$ * $18 = 2x \Rightarrow x = 9$ * Initial quantity = $15x = 15 \times 9 = 135$ L. ### Exam Strategy & Shortcut You can bypass calculating the removed fractions by treating the remaining mixture as the baseline. Let the remaining mixture components be $4k$, $6k$, $5k$. Final Grape = $4k + 8$. Final Pineapple = $6k + 2$. $4k + 8 = (6k + 2) - 10 \Rightarrow 4k + 8 = 6k - 8 \Rightarrow 2k = 16 \Rightarrow k = 8$. Total remaining mixture = $15k = 15 \times 8 = 120$ L. Initial total = Remaining + Removed = $120 + 15 = 135$ L. This avoids fractional subtraction early on! ### Common Pitfall Getting confused by the typo in the original text ("resultant quantity of grape juice is $10$ litres less than the resultant quantity of grape juice is $10$ litres less than..."). Ignore the repetition and extract the core mathematical equation: $G = P - 10$. ### Final Answer Therefore, the correct answer is **$135$**.