Electron and positron densities for a number of positron binding atoms and ions are used to benchmark methods used to compute positron annihilation rates in solids. The electron and positron densities from ab initio calculations are multiplied by some commonly used enhancement factors and integrated to give the annihilation rate. These are then compared with the close to exact annihilation rate calculation using the fully correlated positronic atom wave functions. The results reveal deficiencies in the one-component local-density approximation (LDA) and the Boronski-Nieminen two-component LDA. A simple modification of the one-component LDA leads to an enhancement factor that predicts annihilation rates more reliably for some positronic systems with a minimal increase in complexity. An approach based on an existing version of the generalized gradient approximation tends to underestimate the annihilation rate with the valence electrons.