Arthur and betty start walking toward each other when they are 100 m apart. arthur has a speed of 3.0 m/s and betty has a speed of 2.0 m/s. their dog, spot, starts by arthur's side at the same time and runs back and forth between them at 5.0 m/s. by the time arthur and betty meet, what distance has spot run?

Let t be the time they both meet.  When they meet they both have covered the distance between them.

Arthur with 3 m/s would have covered 3t m
Betty with 2 m/s would have covered 2t m

3t + 2t = 100
5t = 100
t = 100/5
t = 20 seconds.

Spot who is helplessly runing back and forth at 5 m/s would have done this for 20 seconds.

This distance covered by spot =  5m/s * 20s = 100m

Some type of thing that was invisible