In a certain medical group, Dr. Schwartz schedules appointments to begin $30$ minutes apart, Dr. Ramirez schedules appointments to begin $25$ minutes apart, and Dr. Wu schedules appointments to begin $50$ minutes apart. All three doctors schedule their first appointments to begin at $8{:}00$ in the morning, which are followed by their successive appointments throughout the day without breaks. Other than at $8{:}00$ in the morning, at what times before $1{:}30$ in the afternoon do all three doctors schedule their appointments to begin at the same time?
I'm not sure how to approach this question. They all begin at the same time at $8{:}00$. Also doesn't the overlapping of appointments depend on how long their respective appointments take?
I think that for the matter of the problem they don't want you to consider the time that those appointments may take... it seems to be a problem that can be solve with LCM (least common multiple), if you do LCM(30, 25, 50) you get 150 minutes = 2hr 30min.
So you do: 8:00 + 2:30 = 10:30
then
10:30 + 2:30 = 13:00
and you should stop there... It may help to sketch a straight line and represent those minutes with cm or inches and see when the three doctors will met again on that time-line