Let the total mixture volume be \( V \). Initially, the water-to-milk ratio is \( 6 : 10 \). The water and milk amounts are:
\[ \text{Water} = \frac{6}{16}V \quad \text{and} \quad \text{Milk} = \frac{10}{16}V \]
Remove \( x \) volume of the mixture and replace it with water. Removing \( x \) decreases water by \( \frac{6}{16}x \) and milk by \( \frac{10}{16}x \). Adding \( x \) of water, the water amount becomes:
\[ \frac{6}{16}V - \frac{6}{16}x + x \]
For a final mixture of half water and half milk, set the water and milk amounts equal:
\[ \frac{6}{16}V - \frac{6}{16}x + x = \frac{10}{16}V - \frac{10}{16}x \]
Simplifying the equation:
\[ \frac{6}{16}V - \frac{6}{16}x + x = \frac{10}{16}V - \frac{10}{16}x \]
\[ \frac{6}{16}V + \left(1 - \frac{6}{16}\right)x = \frac{10}{16}V - \frac{10}{16}x \]
\[ \frac{6}{16}V + \frac{10}{16}x = \frac{10}{16}V \]
Solve for \( x \):
\[ \frac{10}{16}x = \frac{10}{16}V - \frac{6}{16}V \]
\[ \frac{10}{16}x = \frac{4}{16}V \]
\[ x = \frac{4}{16}V \times \frac{16}{10} = \frac{V}{5} \]
Therefore, remove and replace \( \frac{1}{5} \) of the total volume \( V \).