two buses leave the terminal at 12 noon. bus #86 takes 45 minutes to complete its route. bus #79 takes 60 minutes when is the next time the two buses will arrive together at the terminal

3 hours from noon, or 3 pm.

Step-by-step explanation:

As of 12 noon, the frequencies are 45 and 60 minutes.

To find out when they leave at the same time again, we need the LCM (least common multiple) of the two frequencies, i.e.

LCM (45, 60) = common time from {45,90,135,180,225,270,315,360} and{60,120,180,240,300,360,...}

We see that they leave at the same time again at 180 and 360, ... minutes.  Since 180 minutes is the least common multiple, 180 is the next time they leave together again, or 3 hours after noon, i.e. 3pm.

Happy New Year, ... en quelques minutes!  :)