Is it necessary to reply to a solution email by a thank-you email? If I asked via email someone, say a member of the graduate admission staff of a school I am applying to, some question, and if the staff member replied to my email with a solution to my problem, does the staff member expect to receive a thank-you email from me? 

At the present stage I still am not good at evaluating if it is suitable to reply to such an email by a thank-you note.
It is appropriate? Sure, what is the harm? It is always nice to see one's work recognised and appreciated. Just keep it short, polite, and to the point.

Is it expected? Probably no. The staff is busy, and if your question wasn't anything out of the ordinary, they would probably forget you asked soon.

Edit: mbert adds in the comments some interesting inside information:


  I am an assistant at my university and answer a lot of emails through a ticketing system. Our signature says something similar about replying. This is because when I answer an email, the ticket gets closed and marked done. If you answer to this email the ticket gets reopened and somebody needs to close it again. So I would suggest to answer with a short sentence but only if you are not emailing with a team that shares an email ( like staff@... or support@... )


(Emphasis mine).