Josie deposited $450 in a bank account at the beginning of the month. A payment of $35.20 is automatically deducted each month for her gym membership.
At the beginning of the month, Josie's bank account balance is $450.
Her gym membership payment of $35.20 is automatically deducted from her account each month.
Therefore, her bank account balance at the end of the month is $450 - $35.20 = $<<450-35.20=414.80>>414.80. Answer: \boxed{414.80}.