An empty evaporating dish weighs $20\mathrm{.}000g.$ An unknown mixture containing $Fe,$NaCl, and $Si{O}_2$ is added to the dish and the dish and mixture together weigh $35\mathrm{.}000g.$ The mixture is then stirred thoroughly with a magnet which is then removed, and the dish and remaining mixture weigh $29\mathrm{.}550$ g$.$ After the extraction with several portions of water and drying the remaining solid, the dish and the remaining solid weigh $27\mathrm{.}680g.$ Calculate the mass and mass percent of $Fe,$NaCl, and $Si{O}_2$ in this sample. Show all your work.