Miguel is walking with his dog, Luna. When they reach the entrance to a park, Miguel throws a ball straight ahead and continues to walk at a steady pace. Luna sprints toward the ball, which stops by a tree. As soon as the dog reaches the ball, she brings it back to Miguel. Luna runs $5$ times faster than Miguel walks. What fraction of the distance between the entrance and the tree does Miguel cover by the time Luna brings him the ball?

Video solution:

Click to load, then click again to play

Written solution:

Let the entrance-to-tree distance be $D$, and let Miguel's speed be $1$. Luna's speed is $5$. Luna reaches the tree in time $D/5$, during which Miguel walks $D/5$. The remaining distance between Luna and Miguel is $4D/5$, and they close it at combined speed $6$, taking time $2D/15$. Miguel walks a total distance $D/5+2D/15=D/3$, which is $1/3$ of the entrance-to-tree distance.