I am glad to report that the problem has been fixed. First of all, I would like to thank all of you who told me that a chain needs replacing once it is "stretched". I had no idea that it ever needed replaced. The chain had stretched almost 1/2 inch. I assumed the chainring on the bike would be too worn to replace with a new chain, so I replaced the chain on the good bike and put that slightly stretched chain on the old bike. Both bikes now are good as new.

And to anybody who has not changed their chain recently, I highly recommend it. Now that I have put a new chain on the good bike, I cannot believe how smooth it now rides...no noise what so ever. I never thought riding to work could be so peaceful.