Some banks have monthly fees for their accounts, some do not, and some have them but waive them if you meet certain criteria. So whether or not you pay your bank depends on your specific situation.
Some banks pay you interest on checking accounts, but most do not. Most banks pay you interest on a savings account, but that could be as little as 0.01% or as much as 4% right now. The bank is loaning out the money in your savings account to other people and charging them a higher interest rate than they pay you, so the bank is also making money on your account.
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