do banks accept torn paper money?

Paper money that has been mutilated or partially destroyed may in some cases be redeemable at full face value. Any badly soiled, defaced, torn, or worn-out currency that is clearly more than half of the original note can be exchanged at a commercial bank, which processes the note through a Federal Reserve Bank. More seriously damaged notes—those with clearly less than half of the original surface or those requiring special examination to determine their value—must be sent to the Department of the Treasury for redemption.

The redemption value of mutilated coins depends on their type, denomination, and the extent of their mutilation. Redemption of mutilated coins is handled by the U.S. Mint in Philadelphia. Coins that are merely bent or worn slick through natural wear are not considered mutilated and are exchangeable at full face value.

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RE:

do banks accept torn paper money?

Yes they do. Banks collect mutilated currency and send it back to the Federal Reserve bank. However, the serial numbers on both sides must match. If a bill is torn in half, you must present both pieces and match the complete serial number. If the tear goes through the serial number, they will not accept it. If just a corner is ripped it's no problem. But for them to accept it, you must have both complete serial numbers

I had two torn tens I tried to exchange at a bank and they said they couldn't because the money reader kicked them back out,

All the ones I've ever used did. There is a natural life cycle for paper money anyway. They will probably just put it in the pile to be destroyed, but it's still valid money. Attached is an interesting webpage about U.S. money

Mutilated Currency

The banks will NOT exchange torn money. They don't do that anymore!

Here is a link to a web site explaining the redemption policy. Basically any bill that is clearly more than half of the original (and whose value is easily determined) can be redeemed at your local bank. More seriously damaged bills need to be shipped to the Bureau of Engraving and Printing to determine the value. The address is on the link.

http://usgovinfo.about.com/library/weekly/aa070701...

Mutilated Money

Yes, most of it must be intact. Banks are given the provision of replacing damaged monetary notes by the Federal Reserve so you don't need to feel like scum trying to swap them.