Step 1 :Let's denote the rate of work done by Evelyn as \(E\) and by Rachel as \(R\).
Step 2 :Given that Evelyn can paint the room in 4 hours, her rate of work is \(E = \frac{1}{4}\) of the room per hour.
Step 3 :Similarly, Rachel can paint the room in 5 hours, so her rate of work is \(R = \frac{1}{5}\) of the room per hour.
Step 4 :When they work together, their rates of work are added. So, the combined rate of work is \(E + R = \frac{1}{4} + \frac{1}{5} = \frac{5}{20} + \frac{4}{20} = \frac{9}{20}\) of the room per hour.
Step 5 :Therefore, the time it would take for them to paint the room together is the reciprocal of their combined rate of work, which is \(\frac{20}{9}\) hours.
Step 6 :So, it would take them \(\boxed{\frac{20}{9}}\) hours to paint the room together.