We consider the coded caching problem with an additional privacy constraint that a user should not get any information about the demands of the other users. We first show that a demand-private scheme for files and users can be obtained from a non-private scheme that serves only a subset of the demands for the files and users problem. We further use this fact to construct a demand-private scheme for files and users from a particular known non-private scheme for files and users. It is then demonstrated that, the memory-rate pair , which is achievable for non-private schemes with uncoded transmissions, is also achievable under demand privacy. We further propose a scheme that improves on these ideas by removing some redundant transmissions. The memory-rate trade-off achieved using our schemes is shown to be within a multiplicative factor of 3 from the optimal when and of 8 when . Finally, we give the exact memory-rate trade-off for demand-private coded caching problems with .