Fenceposts and Gunshots

Key idea. Elapsed time is counted between shots, not per shot. So $n$ shots span $n-1$ gaps.

The six shots from the first to the last span $5$ equal intervals in $60$ seconds. Each interval is therefore

Firing $3$ shots, from the first to the third, spans $2$ such intervals:

The tempting answer of $30$ seconds comes from dividing $60$ by $6$ shots. It wrongly treats each shot as taking up its own block of time.