Answer: 20 minutes.
Explanation:
Think of the water stream as being still (perhaps like a solid pavement), from the point of view of objects floating on the surface. (of course, from the POV of anyone standing on the banks, the water flows downstream). Think of the boat as a bike riding on the pavement. The boat just turned upstream and the hat fell into the water. In this POV, the hat lies on the surface and the boat/bike is riding away. For 20 minutes. Then (Katya realizes she lost the hat and turns back), the boat/bike needs exactly the same time, i.e., 20 minutes to come back to the hat, which rests at the same spot on the surface!
What makes the question initially confusing is the fact that the river flows with respect to the banks and one's mind is tempted to view the situation from there, which makes it much more complicated.
In physics this is referred to as choosing the suitable "frame of reference."
