The bathroom door in my place had been scraping the floor since I moved in. Every time someone closed it, there was this horrible grinding sound, and a permanent arc wore into the floorboards. I lived with it for a year before finally fixing it.

It took one afternoon and cost me nothing, because the fix was not cutting the door. It was the hinges.

Try the hinge before the saw

A door that rubs usually is not too big. It has just sagged on its hinges over time, and the screws have loosened so the whole door hangs a little lower than it should.

1. Tighten every screw on all three hinges with a screwdriver.
2. If a screw just spins without tightening, the hole is stripped. Take it out and stuff the hole with a toothpick and wood glue, let it dry, then drive the screw back in.
3. Open and close the door to see if the rub is gone.

Most of the time, that is all it takes. My door lifted almost a quarter inch just from tightening the screws.

If it still rubs

If tightening does not fix it, the next step is sanding the bottom edge, not the floor. Mark the high spot with a pencil, then sand a little at a time with coarse sandpaper, checking the fit as you go. Go slow. You can always sand more, but you cannot put the door back.

I will warn you, I almost reached for the saw first out of impatience. Sanding a hair off the bottom is way safer than cutting, because it is reversible and you will not end up with a big gap under the door that lets in drafts.

The scrape is gone now, and the floor arc finally stopped growing.