It's usually the float mechanism or the rubber flapper that are the problem. They're cheap, but you need to shut off the water to replace them, so if you don't have a shut off valve by the toilet, you may have to have your landlord shut it off. Take the back off the tank and have a look. If the flapper is sitting properly but the water's still running, it will need replacing. If the flapper isn't seated correctly, you may be able to adjust the chain to make it seat properly. All the parts are cheap--the flappers cost a few bucks, the float costs a few bucks, not expensive at all, and the stuff is easy to replace. You can probably find a youtube video of how to replace the whole thing if you need help.
at 11:07 AM on Sep. 16, 2009