Translation:Professor Rania's room is old and expensive.
I put the room of professor Rania not knowing how the expected answer would look. The translation in the answer is more natural and how I would have preferred to write it, though.
That is Duolingo's text (both the Arabic and the translation), not mine. I am commenting on it. It's an idafah construction so the the would go in front of room in my translation (The room of professor Rania) and in their translation there's no the at all because AL does not have to be translated as the so long as the object is definite (i.e., Rania or Professor Rania or Professor Rania's room are all already definite without having to add the).