Question : From a cask filled with wine, 75 litres are first drawn and replaced with water. From this mixture, 60 litres are drawn and replaced with water. The ratio of wine to water in the cask is now 3 : 2. How many litres of wine did the cask initially hold?
Option 1: 225
Option 2: 250
Option 3: 375
Option 4: 300
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Correct Answer: 300
Solution :
Let the cask initially hold $x$ litres of wine.
After 75 litres are first drawn and replaced with water,
Now, wine = ($x$ – 75) litres and water = 75 litres
Ratio of wine and water = $(x–75) : 75$
Again, from this mixture, 60 litres are drawn and replaced with water.
So, wine drawn = $60×(\frac{x–75}{x})$ litres and water drawn = $60×(\frac{75}{x})$ litres
The wine's amount = $(x–75) – 60×(\frac{x–75}{x})$ litres
According to the question,
($x–75) – 60×(\frac{x–75}{x})=\frac{3x}{5}$
$⇒2x^2–675x+22500=0$
$⇒(x–300)(2x–75)=0$
$⇒x=300$ (As $x$ can not be $\frac{75}{2}$)
$\therefore$ the cask initially holds 300 litres of wine.
Hence, the correct answer is 300.
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