Question

A container contains only milk. 2/3 of the mixture is removed and replaced with water. This process is done once and then repeated another 3 times. What is the final ratio of milk and water?

Hint:

Assume a convenient starting volume for the milk, a number divisible by 3 several times over, such as 81 litres. This keeps every round of removal and replacement in whole numbers, making it easy to track the milk volume step by step instead of working with fractions throughout.

Answer 1

Step 1: Start with milk at \(100\%\) of the container. Each replacement keeps \(\tfrac13\) (about \(33.33\%\)) of whatever milk is currently present, since \(\tfrac23\) is removed each time.
Step 2: Track the milk percentage round by round: after round 1, \(33.33\%\); after round 2, \(11.11\%\); after round 3, \(3.70\%\); after round 4, \(1.2346\%\) (each is the previous value divided by \(3\)).
Step 3: As a clean fraction, \(1.2346\%=\tfrac{1}{81}\) of the original volume is milk, so water makes up the remaining \(\tfrac{80}{81}\).
Final Answer: milk : water \[ =\frac1{81}:\frac{80}{81}=\boxed{1:80} \]