Question below is followed by three statements I, II and III. You have to determine whether the data given in the statement is sufficient for answering the question. $y$ litres mixture of milk and water contains milk and water in the ratio $a : b$ respectively. The milkman sold $z$ litres of the mixture and added 10 litres of pure milk and 8 litres of water to the remaining mixture such that the ratio of milk and water in the final mixture became $5 : 4$ respectively. Find the respective ratio of amount of milk in the initial mixture and amount of milk in the final mixture. Statement I: A 54 litres mixture of syrup and water contains syrup and water in the ratio $a : b$ respectively. If 10 litres of syrup and 6 litres of water is added to the mixture, the ratio of syrup and water becomes $4 : 3$ respectively. Statement II: A 60 litres mixture of milk and water contains milk and water in the ratio $7 : 5$ respectively. If $z$ litres of the mixture is replaced with water, the ratio of milk and water in the mixture becomes $7 : 23$ respectively. Statement III: $y = 3z$

### Concept & Mixtures Data Sufficiency To find the initial to final milk ratio, we need the initial quantity of milk and the final quantity of milk. Initial Milk = $y \times \frac{a}{a+b}$. Final Milk = $(y - z) \times \frac{a}{a+b} + 10$. We need to determine the variables $a, b, y,$ and $z$. ### Step-by-Step Solution * **Evaluate Statement I:** 54L mixture in ratio $a:b$. Adding 10L syrup and 6L water gives $4:3$. Let initial syrup be $S$ and water be $W$. $S+W = 54$. $$ \frac{S + 10}{W + 6} = \frac{4}{3} \implies \frac{S + 10}{54 - S + 6} = \frac{4}{3} $$ $$ 3S + 30 = 240 - 4S \implies 7S = 210 \implies S = 30 $$ So, $W = 24$. The ratio $a:b = 30:24 = 5:4$. Statement I gives us $a$ and $b$, but not $y$ or $z$. * **Evaluate Statement II:** 60L mixture in $7:5$ ratio ($35L$ milk, $25L$ water). Replaced $z$ litres with water, new ratio is $7:23$. Milk left = $35 - \frac{7}{12}z$. Water = $25 - \frac{5}{12}z + z = 25 + \frac{7}{12}z$. $$ \frac{35 - \frac{7}{12}z}{25 + \frac{7}{12}z} = \frac{7}{23} $$ $23(35 - \frac{7}{12}z) = 7(25 + \frac{7}{12}z)$. Solving this yields $z = 36$. Statement II gives us $z$, but not $a, b,$ or $y$. * **Evaluate Statement III:** $y = 3z$. This only gives a relationship between $y$ and $z$. * **Combining Information:** Combining I, II, and III gives $a:b = 5:4$, $z = 36$, and $y = 3 \times 36 = 108$. With all variables known, we can find the exact required ratio. ### Exam Strategy & Shortcut Instead of fully solving Statements I and II, recognize what variables they allow you to isolate. Stmt I has only 1 unknown proportion set ($S$ out of 54), so it yields $a:b$. Stmt II has 1 unknown ($z$), so it yields $z$. Stmt III connects $y$ to $z$. To solve the main equation, all components are necessary. ### Common Pitfall Assuming that providing absolute values for a different liquid (syrup) in Statement I makes it invalid. The ratio $a:b$ is an independent constant linking the scenarios. ### Final Answer Therefore, the correct answer is **All I, II and III together**.