A cache-aided K-user Gaussian broadcast channel (BC) is studied. The transmitter has a library of N files, from which each user requests one. The users are equipped with caches of different sizes, which are filled without the knowledge of the user requests in a centralized manner. Differently from the literature, it is assumed that each file can be delivered to different users at different rates, which may correspond to different quality representations of the underlying content, e.g., scalable coded video segments. Accordingly, instead of a single achievable rate, the system performance is characterized by a rate tuple, which corresponds to the vector of rates users' requests can be delivered at. The goal is to characterize the set of all achievable rate tuples for a given total cache capacity by designing joint cache and channel coding schemes together with cache allocation across users. Assuming that the users are ordered in increasing channel quality, each file is coded into K layers, and only the first k layers of the requested file are delivered to user k, k=1, ..., K. Three different coding schemes are proposed, which differ in the way they deliver the coded contents over the BC; in particular, time-division, superposition, and dirty paper coding schemes are studied. Corresponding achievable rate regions are characterized, and compared with a novel outer bound. To the best of our knowledge, this is the first work studying the delivery of files at different rates over a cache-aided noisy BC.