As a property manager, I have had my share of this. There are some really good things to do. One, don't put things like celery or potato skins down the disposal, as they are stringy and starchy, and gum things up. Two, run water before and after you use the disposal. Three, once a week, throw in a few ice cubes and coffee grounds to keep the blades sharp. Four, if the trap is clogged, it's just a couple of turns with a wrench or even your hand (PVC is easy) with a tub or big bowl under neath. For those "plumbing challenged" folks, that is the "U" shaped one under the sink. Clean it out, reattach and you are good to go.
When I moved into an apartment that had both a dishwasher and garbage disposal(which was the first time I'd had either) I had that problem, too. The maintenance mad said you should always run the disposal before running the dishwasher. The water from the dishwasher will still back up into the sink that has the disposal in it, but at least you won't get whatever you have dumped into the disposal. If anybody has more information on this it might be helpful.
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