Step 1 :Let's denote the volume of the $10 \%$ solution as $x$ and the volume of the $60 \%$ solution as $y$.
Step 2 :The total volume of the solution is 200 milliliters, so we have the equation \(x + y = 200\).
Step 3 :The total amount of salt in the solution is $30 \%$ of the total volume, so we have the equation \(0.10x + 0.60y = 0.30 * 200\).
Step 4 :We can solve these two equations simultaneously to find the values of $x$ and $y$.
Step 5 :By solving the equations, we find that $x = 120$ and $y = 80$.
Step 6 :Final Answer: Maria must mix \(\boxed{120}\) milliliters of $10 \%$ solution and \(\boxed{80}\) milliliters of $60 \%$ solution.