Discussion
A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?
Answer: C
Take the vessel as 8 units: 3 of water and 5 of syrup. Drawing off a fraction \( f \) removes both in proportion, and the water poured back keeps the total at 8.
\[ \text{water after} = 3(1 - f) + 8f = 3 + 5f \]
Half water means 4 of the 8 units:
\[ 3 + 5f = 4 \quad\Longrightarrow\quad f = \frac{1}{5} \]
The replacement adds water twice over - it removes syrup and puts water in its place - which is why only a fifth needs drawing off to move the water from \( \tfrac{3}{8} \) to \( \tfrac{1}{2} \).
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