When the plane flies with the wind its speed is p+10 where p is the plane's speed in still air. When it flies against the wind its speed is p-10. s=vt, where s=distance (miles), v is speed (mph), t is time (hours).
t=s/v, so, since the times are the same for the two distances, 390/(p+10)=325/(p-10).
Cross-multiplying: 390(p-10)=325(p+10), 390p-3900=325p+3250, 65p=7150, p=7150/65=110mph.
So the plane's speed in still air is 110mph, as you thought. (Time is 3¼ hours.)