Initially, the glass contains only milk. Let the initial amount of milk be \( M = 1 \). In the first step, \(\frac{2}{3}\) of the milk is removed and replaced with water.
The amount of milk remaining after step 1 is:
\[ M_1 = M - \frac{2}{3}M = \frac{1}{3}M \]Water added = \(\frac{2}{3}\).
In the second step, \(\frac{2}{3}\) of the current mixture (which contains both milk and water) is removed and replaced with water. This means \(\frac{2}{3}\) of the *milk* is removed.
Remaining milk after step 2:
\[ M_2 = \frac{1}{3} M_1 = \frac{1}{3} \times \frac{1}{3} M = \frac{1}{9} M \]Water added in step 2 = \(\frac{2}{3}\) of the total volume. Since the total volume is still 1, we can compute the amount of water added in step 2 as follows:
Water removed = \(\frac{2}{3}\) of the remaining milk, i.e., \(\frac{2}{3} \times M_1\). Therefore, the remaining milk is \(\frac{1}{3} M_1\). The amount of water added in step 2 is \(\frac{2}{3}\) of the total volume, which is 1.
For clarity, the remaining milk is reduced by a factor of \(\frac{1}{3}\) each time, leading to the equation for step 2:
\[ M_2 = \frac{1}{3} M_1 = \frac{1}{9} M \]In the third step, \(\frac{2}{3}\) of the current milk is again replaced with water:
Remaining milk after step 3:
\[ M_3 = \frac{1}{3} M_2 = \frac{1}{3} \times \frac{1}{9} M = \frac{1}{27} M \]The total amount of water in the glass is the initial volume (1) minus the final amount of milk (\(\frac{1}{27}\)).
Water added = \(1 - \frac{1}{27} = \frac{26}{27}\).
The final ratio of water to milk is:
\[ \frac{\frac{26}{27}}{\frac{1}{27}} = 26:1 \]