One painter can paint a room in 6 hours and another in 4 hours. How long do they take together?

Problem: One painter can paint a room in $6$ hours and another in $4$ hours. How long do they take together?

Steps

1. Find the rate of the first painter
   $$\frac{1}{6} \text{ of the room per hour}$$
2. Find the rate of the second painter
   $$\frac{1}{4} \text{ of the room per hour}$$
3. Add the rates together to find the combined rate
   $$\frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12} \text{ of the room per hour}$$
4. Take the reciprocal of the combined rate to find the total time
   $$t = \frac{12}{5} \text{ hours}$$