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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