During the Middle Ages, rats were responsible for the transfer of fleas that carried the deadly Black Plague. A bounty was placed upon each rat that was caught. Since it was not necessary to produce the entire rat as proof of capture, the rat's hind quarters were cut from the body - leaving only the tail and the "ass". The amount paid by the local governments were so low however, they were almost worthless. Hence the phrase - "not worth a rat's ass".