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 $N$ files and $K$ users can be obtained from a non-private scheme that serves only a subset of the demands for the $N$ files and $NK$ users problem. We further use this fact to construct a demand-private scheme for $N$ files and $K$ users from a particular known non-private scheme for $N$ files and $NK-K+1$ users. It is then demonstrated that, the memory-rate pair $(M,\min \{N,K\}(1-M/N))$, 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 $K < N$ and of 8 when $N\leq K$. Finally, we give the exact memory-rate trade-off for demand-private coded caching problems with $N\geq K=2$.