Assess the damage.
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Light scratches require a different repair approach from deep scratches.
For light scratches, lightly rub fine steel wool over the scratch.
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Be sure to rub the steel wool along the grain of the wood or you’ll have more scratches to repair.
For deeper scratches, use lightweight sandpaper on the scratch.
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Be sure to sand along the grain of the wood.
Rub mineral spirits over the sanded area.
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Rub mineral spirits over the area to smooth it out and to pick up the fine dust from the sanding.
Fill the scratch with premixed wood filler.
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Use a plastic putty knife to fill the scratch with wood filler that matches the color of the floor. Let the filler dry thoroughly.
Be sure to use a plastic putty knife to apply the filler (you don’t want to create any new scratches).
Sand the filler and wipe up all the dust.
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Once the filler has dried, use lightweight sandpaper (use 180 grit). Be sure to only sand the excess filler.
Seal the sanded area.
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Refinish the board with varnish, polyurethane, or a similar product to match the rest of your floor.
Warning: Don’t use a foam roller or brush to apply the polyurethane — you’ll end up with air bubbles in the finish. Go with a lambswool applicator or natural bristle brush, instead.