If Evelyn were to paint her living room alone, it would take 4 hours. Her sister Rachel could do the job in 5 hours. How many hours would it take them working together? Express your answer as a fraction reduced to lowest terms, if needed.

Answer
Keypad
Keyboard Shortcuts

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.