How can two slightly different shades of color of a hardwood floor be made to match one another?

I just bought a house that had two large area rugs over the top of the hardwood floors in the living room and dining room. When I took up the rugs, I noticed that the shade of color of the wood under the rugs is slightly lighter than the wood that was not ever covered up.

Also, there was a lot of very fine grit that must have worked its way through the rugs over the years. Did this grit basically sand the floors under the rugs and that's the reason for the color difference?

Other than completely sanding the entire floor, re staining, refinishing, etc... Is there a way to blend the areas so that the difference is not so immediately noticeable (mostly at the areas where the borders of the rugs were)?

Has anybody ever had this problem? If so, how did you fix it?