Question : A vessel is filled with liquid, 5 parts of which are water and 11 parts syrup. What part of the mixture must be drawn off and replaced with water so that the mixture may be syrup and water in the ratio 3 : 2?
Option 1: $\frac{14}{45}$
Option 2: $\frac{27}{35}$
Option 3: $\frac{36}{65}$
Option 4: $\frac{7}{55}$
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Correct Answer: $\frac{7}{55}$
Solution :
5 parts water and 11 parts syrup (considering the unit volume of the mixture)
Water = $\frac{5}{16}$ and syrup = $\frac{11}{16}$
Let $x$ amount of liquid is replaced by water.
Water = $\frac{5x}{16}$ and syrup = $\frac{11x}{16}$
According to the question,
$\frac{\frac{11}{16}-\frac{11x}{16}}{\frac{5}{16}+x-\frac{5x}{16}}=\frac{3}{2}$
Or, $\frac{\frac{11}{16}-\frac{11x}{16}}{\frac{5}{16}+\frac{11x}{16}}=\frac{3}{2}$
Or, $\frac{15}{16}+\frac{33x}{16}=\frac{22}{16}-\frac{22x}{16}$
Or, $33x+22x=22-15$
Or, $x=\frac{7}{55}$
Hence, the correct answer is $\frac{7}{55}$.
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