The optical properties of symmetric split-ring/ring dimer (SRRD) nanostructures composed of a small nanoring surrounded by an Ag splitting nanoring with a larger diameter are calculated theoretically. The apparent asymmetric Fano line shape in the spectra is related to fast switching of the bonding modes between the split-ring plasmon and ring dipole. The influence of the dimensions of the SRRD nanostructures on the spectral positions and intensity of Fano resonance is studied, and the asymmetric Fano line shape can be flexibly adjusted by varying the geometric parameters. In addition, relatively simple SRRD nanostructures have the same overall sensing figures of merit as conventional nanoparticles, thus rendering them suitable for high-performance optical sensors.