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A box on my credit card bill says that I will pay off the balance in three years if I pay a certain amount. What does that mean? Do I have to pay that much? If I pay that much and make new purchases will I still owe nothing after three years?

Card issuers are required to let you know how long it will take you to pay off your current balance if you make no further charges and pay only the minimum amount due each month. They are also required to tell you on each statement, based on the balance as of the date of the statement, how much you need to pay each month to pay off your current balance in 36 months. 

These amounts are calculated based on your current balance and do not take into consideration any future purchases.

You do not have to pay any more than the minimum amount shown on your statement. However, the more you pay each month, the less interest you will pay over time. If you make only the minimum payment, it could take years to pay off your credit card.