Question : From a cask of milk containing 64 litres, 8 litres are drawn out and the cask is filled up with water. If the same process is repeated a second, then a third time, what will be the proportion of milk to water in the resulting mixture?
Option 1: 343 : 179
Option 2: 333 : 169
Option 3: 443 : 179
Option 4: 343 : 169
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Correct Answer: 343 : 169
Solution : The amount replaced by water each time is 8 litres. The amount of milk = Original amount$\times(1- \frac{\text{Amount replaced by water}}{\text{Original Amount}})^n$, where $n$ = No. times process repeated So, the amount of milk $= 64\times (1-\frac{8}{64})^3$ $= 64\times (1-\frac{1}{8})^3$ $= 64\times (\frac{7}{8})^3$ $= 64\times \frac{7\times 7\times7}{8\times 8\times 8}$ $= \frac{343}{8}$ Amount of water $= 64 - \frac{343}{8}= \frac{169}{8}$ So, the required ratio $=\frac{343}{8}:\frac{169}{8} =343:169$ Hence, the correct answer is 343 : 169.
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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
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
Question : 80 litres of a mixture contains milk and water in a ratio of 27 : 5. How much more water is to be added to get a mixture containing milk and water in a ratio of 3 : 1?
Option 1: 5 litres
Option 2: 10 litres
Option 3: 15 litres
Option 4: 20 litres
Question : 150 litres mixture of milk and water contains 30 percent milk. How much more milk must be added to make the water percentage in the mixture as 40 percent?
Option 1: 120.5 litres
Option 2: 157.5 litres
Option 3: 132.5 litres
Option 4: 112.5 litres
Question : A 10-litre solution of milk and water contains 8 litres of milk. 2 litres of the solution is replaced by pure milk and mixed. The process is repeated two more times. How much milk (in litres) is present in the mixture so obtained?
Option 1: 8.976
Option 2: 8.597
Option 3: 8.796
Option 4: 8.679
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