Mixtures: Repeated Remove & Replace

This aptitude practice question is a mixtures and alligation question. Concept tested: Finding quantity of milk in a cask after repeated removal of milk that is subsequently replaced with water.

Question

From a cask of milk containing 30 litres, 6 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 number of litres of milk left in the cask?

1. 0.512 liters
2. 12 liters
3. 14.38 liters
4. 15.36 liters

  Correct Answer      Choice (4). 15.36 liters

Explanatory Answer

The problem can be solved by brute force of removing and replacing 6 litres each time and finding the quantity of milk and water after each iteration. Finally, after step 3, we will get to the answer. But it is cumbersome to do that.
The question can be solved more effectively. Let us use that method to solve this question.

Let us find the fraction of milk left after n^th operation.
Fraction of milk in the cask after the nth iteration = \(\frac{x - y}{x}{)}^{n}).
'x' is initial quantity of milk in the cask 'y' is the quantity of milk drawn out in each iteration and 'n' is the number of iterations.
Fraction of milk left after the 3^rd iteration = \(\frac{30 - 6}{30}{)}^{3}) = \(\frac{24}{30}{)}^{3}) = \(\frac{5}{6}{)}^{3}) = \\frac{125}{216})
∴ the quantity of milk in the cask after the 3rd iteration = \\frac{125}{216}) * 30 = 15.36 liters.

  The correct answer is Choice (4).