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Q: "I have removed a sink and want to put in a new sink - except for the fact that I need to expand the hole another quarter-inch. How do you cut installed tile, or do I need to entirely redo the tilework?"
Before considering taking a sledgehammer to your tile sink, start small and work your way up. You may eventually find yourself with no other option but to demo the tile - but you'll have the satisfaction of having tried your level best. The solution is more a grab-bag of possibilities to try, rather than one "magic bullet" that will cure all ills:
Option: Hire a Tiler
If you have expensive tile and zero inclination to take on this tricky task, hire a tiler. It will be pricey to hire a tiler for the hour or less that this job will take, but you will have the satisfaction of getting the job done right.
Option: Hand-Held Wet Saw
If you have room to work, a circular hand-held wet saw will do a great job of cutting the tile and keeping the mess to a minimum.
Option: Regular Circular Saw with Tile Blade
Not keen on the idea of buying a specialized tool like the hand-held wet saw for a small project? Any circular saw equipped with a four-inch diamond blade can do the job - but it will get ugly. There is a reason why tilers use wet saws. Tile cutting is dusty and dangerous, with the potential for flying shards of sharp tile.
Option: Tile Nipper
A simple tool found at home improvement stores, the tile nipper is normally used prior to installing tiles to cut shapes that are non-linear. If your tile configuration allows, you can cut small parts off with the tile nipper. Caution: it is easy to break tile with the nipper, and it is tedious to use for large expanses of tile.
Option: Coping Saw
A coping saw equipped with a carbide blade can cut tile, but like all other manual methods of cutting tile, it is very labor intensive.
Option: Replace Only the Affected Tiles
Pull out the tiles that need recutting. You can do this by using a small angle grinder, a Roto Zip or Dremel tool, or elbow grease, in order to chip out mortar and replace the tiles.
Option: Replace All the Tile
If the tile wasn't very expensive or valuable in the first place, you may save more time and reduce aggravation simply by demolishing the existing tile and replacing all of it.