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. 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 "arse". The amount paid by the local governments was approximately equivalent to a penny a dozen. The rat’s arses would be presented as a bundle and exchanged for money. However there became so many rats the price dropped incredibly low and they were almost worthless.
Hence the phrase -"not worth a rat's ass" – “don’t give a rat’s arse”.
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