A common gravity-flush toilet issue is a running toilet. These can usually be identified by the toilet constantly running water inside the tank or the toilet intermittently refilling on its own. Running toilets can waste up to 4.5 gallons per minute (nearly 300 gallons an hour or 6,480 gallons a day), which can cost over $60 a day.
If the water is constantly running and going into the overflow tube inside the tank, the fill valve likely needs adjustment or replacement. Test the float cup on the fill valve by lifting it and if the water stops, it likely just needs adjustment. Many fill valves have a spline with a screw on the side that can be turned counterclockwise to move the float down below the overflow tube. If this does not correct the running water, replacing the fill valve is the next step.
Once this is finished, take the new fill valve and install the rubber washer on the bottom of the fill valve stem and install the fill valve into the hole in the tank. Lastly, secure the valve with the retaining nut, reattach the supply line, reconnect the overflow tube to the overflow, and turn the water supply back on at the shut off valve.
Another possible cause for a running toilet is a flapper that is in need of replacement. This can usually by determined by the tank constantly filling, stopping, water level slowly going down, and tank refilling again. This can cause significant water waste and can often easily be fixed by replacing the flapper.
First, turn off the water at the supply line, remove the rubber flapper valve from the two studs on the sides of the flush valve, and remove the chain from the flush handle lever. Install the new flapper valve by connecting the holes on the sides of the flapper to the studs on the flush valve and connect the chain to the flush handle lever.
Below is a video detailing the replacement of both a fill valve and flapper.