[email protected] wrote:A museum offers four video programs that run continuously throughout the
day, each program starting anew as soon as it is finished. The first program
runs every 15 minutes, the second every 30 minutes, the third every 45
minutes, and the fourth every 40 minutes; the first show of each program
starts at 10:00 A.M. and the last showing of each program ends at 4:00 P.M.
If a tour group can watch the programs in any order, but needs at least ten
minutes between programs to regroup, what is the least amount of time
the group can take to watch all four programs?

To minimize the total time, as many breaks as possible must last only 10 minutes each.
List the starting times of each show and create a schedule from the available times.

The 15-minute video begins at:
10:00, 10:15, 10:30, 10:45
11:00, 11:15, 11:30, 11:45
etc.

The 30-minute video begins at:
10:00, 10:30
11:00, 11:30
12:00, 12:30
etc.

The 40-minute video begins at:
10:00, 10:40, 11:20
12:00, 12:40, 1:20
etc.

The 45-minute video begins at:
10:00, 10:45, 11:30, 12:15
1, 1:45, 2:30, 3:15
etc.

Note that no video prior to 2pm starts 10 minutes after the 45-minute video.
Prior to 2pm, the 45-minute video must be followed by a break of at least 15 minutes.
(While the 1:45 showing of the 45-minute video ends at 2:30 and thus could be followed by the 2:40 showing of the 40-minute video, the resulting schedule would not minimize the total amount of time needed to view all 4 videos.)

Schedule:
10:00-10:30 = 30-minute video.
10:30-10:40 = 10-minute break.

10:40-11:20 = 40-minute video.
11:20-11:30 = 10-minute break.

11:30-12:15 = 45-minute video.
12:15-12:30 = 15-minute break.

12:30-12:45 = 15-minute video.

Total time = 2 hours, 45 minutes.

Since the number of 10-minute breaks has been maximized, there is no way to minimize the time further.