Question : A bottle is filled with liquid, of which 4 parts are water and 5 parts are fruit extract. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half fruit extract?
Option 1: $\frac{1}{10}$
Option 2: $\frac{9}{10}$
Option 3: $\frac{4}{9}$
Option 4: $\frac{5}{9}$
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Correct Answer: $\frac{1}{10}$
Solution :
Let $x$ be a part of the mixture that has drawn off and $v$ be the total volume.
Given: Initial ratio = Water : fruit extract = 4 : 5
Final ratio = water : fruit extract = 1 : 1
Taking the volume of fruit extract, we get:
Final volume = initial volume × $(1-\frac{x}{v})^n$
⇒ $\frac{1}{2}$ = $\frac{5}{9}(1-\frac{x}{v})^1$
⇒ $\frac{9}{10}$ = $1-\frac{x}{v}$
⇒ $\frac{x}{v}$ = $\frac{1}{10}$
⇒ $x$ = $\frac{1}{10}v$
Hence, the correct answer is $\frac{1}{10}$.
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