A set of phylogenetic trees with overlapping leaf sets is consistent if it can be merged without conflicts into a supertree. In this paper, we study the polynomial-time approximability of two related optimization problems called the maximum rooted triplets consistency problem (MAXRTC\textsc{MaxRTC}) and the minimum rooted triplets inconsistency problem (MINRTI\textsc{MinRTI}) in which the input is a set R\mathcal{R} of rooted triplets, and where the objectives are to find a largest cardinality subset of R\mathcal{R} which is consistent and a smallest cardinality subset of R\mathcal{R} whose removal from R\mathcal{R} results in a consistent set, respectively. We first show that a simple modification to Wu’s Best-Pair-Merge-First heuristic [25] results in a bottom-up-based 3-approximation for MAXRTC\textsc{MaxRTC}. We then demonstrate how any approximation algorithm for MINRTI\textsc{MinRTI} could be used to approximate MAXRTC\textsc{MaxRTC}, and thus obtain the first polynomial-time approximation algorithm for MAXRTC\textsc{MaxRTC} with approximation ratio smaller than 3. Next, we prove that for a set of rooted triplets generated under a uniform random model, the maximum fraction of triplets which can be consistent with any tree is approximately one third, and then provide a deterministic construction of a triplet set having a similar property which is subsequently used to prove that both MAXRTC\textsc{MaxRTC} and MINRTI\textsc{MinRTI} are NP-hard even if restricted to minimally dense instances. Finally, we prove that MINRTI\textsc{MinRTI} cannot be approximated within a ratio of Ω(logn) in polynomial time, unless P = NP.