This paper proposes new designs for Reduced Precision Redundancy (RPR) systems using Approximation (RPAs). Reduced redundancy is accomplished by utilizing approximate modules, hence requiring substantially different designs for the decision hardware for generating an output. Some of proposed schemes deal with a single erroneous data word generated by a module (in the presence of single and multiple bit errors) using three modules as inputs to the decision hardware of the RPA (denoted as 2E1A and 1E2A where E (A) stands for exact (approximate)). The remaining RPA schemes operate under single and multiple bit errors in at most two modules out of four input modules (denoted as 2E2A). Different from RPRs found in the technical literature, the proposed RPAs operate using only logic operations (so no arithmetic unit is involved as decision hardware). The probability of providing an exact data word at the output of the RPA under the above error conditions is analytically found; simulation results are also provided. It is shown that the difference between simulated and analytical probabilities is at most 5%. Circuit based metrics (such as delay, power dissipation, and area) of the proposed designs are simulated and compared with RPR; the proposed designs outperform RPR in all metrics, except the area of the four-input RPA (2E2A) due to the larger number of modules. The application of the proposed RPAs to image processing is also presented; the results show that the proposed schemes are very efficient for such application.