Question : A cask having a capacity of 600 litres is initially filled with wine. From this cask $y$ litres of wine are first drawn and replaced with water. From this mixture, 120 litres are drawn and replaced with water. From this further diluted mixture. 120 litres are drawn again and replaced with water. The ratio of wine to water in the cask is now 12 : 13. What is the value of $y$?
Option 1: 125
Option 2: 150
Option 3: 200
Option 4: 100
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Correct Answer: 150
Solution :
Let the initial amount of water in the cask be $y$ litres.
The initial amount of wine in the cask = $600-y$ litres
The ratio of wine to water = 12 : 13
So, Ratio of wine is to mixture = 12 : 25
We know, Final quantity = Initial Quantity(1 – $\frac{\text{Amount of mixture replaces}}{\text{Total amount of mixture}}$)
After 2 operations, the quantity of wine when 120 litres is taken out and replaced = $600(1-\frac{y}{600})(1-\frac{120}{600})^2$
According to the question,
$(1-\frac{y}{600})(1-\frac{120}{600})^2=\frac{12}{25}$
⇒ $(1-\frac{y}{600})=\frac{12}{16}$
⇒ $\frac{y}{600}=\frac{1}{4}$
⇒ $y=150$ litres
Hence, the correct answer is 150 litres.
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