Step 1 :Let's denote the amount of the $25\%$ solution as $x$, the amount of the $45\%$ solution as $y$, and the amount of the $90\%$ solution as $z$.
Step 2 :We know that the total amount of solution is 100 liters, so we have the equation \(x + y + z = 100\).
Step 3 :We also know that the chemist uses 2 times as much of the $90\%$ solution as the $45\%$ solution, so we have the equation \(z = 2y\).
Step 4 :Finally, we know that the final solution contains $40\%$ acid, so we have the equation \(0.25x + 0.45y + 0.9z = 0.4 * 100\).
Step 5 :We can solve this system of equations to find the values of $x$, $y$, and $z$.
Step 6 :The solution to the system of equations is \(x = 70\), \(y = 10\), and \(z = 20\).
Step 7 :Final Answer: The chemist should use \(\boxed{70}\) liters of $25 \%$ solution, \(\boxed{10}\) liters of $45 \%$ solution, and \(\boxed{20}\) liters of $90 \%$ solution.