In this article, we study multi-agent consensus algorithms with information reuse by intentionally introducing the outdated state information into the traditional consensus algorithms. In the continuous-time case, we first show that the outdated state information combined with the current state information does not necessarily jeopardise the stability of a single system, but may improve the convergence speed without increasing the maximal control effort. Then this idea is extended from the single-agent case to the multi-agent case. When the directed communication graph is fixed, the corresponding Laplacian matrix and the outdated state information satisfy certain conditions, we show that the consensus algorithm with both the current and outdated states can achieve a faster convergence speed than the standard one. We also consider the case of a switching directed communication graph and derive corresponding conditions. In the discrete-time case, we propose a discrete-time consensus algorithm with both the current and outdated states under an undirected fixed communication graph. We then derive conditions on the communication graph, the sampling period and the outdated state information such that the proposed algorithm can achieve a faster convergence speed than that using the standard one. In both the continuous-time and discrete-time settings, we show that the maximum control efforts for the proposed consensus algorithms are identical to those for the standard ones. Several simulation examples are presented as a proof of concept.