Can you carry a balance on Amex?

Balance should be paid in full ideally to avoid affecting credit score ratings. It's either you pay in full or don't pay in full, there's no ideal amount you need to pay every month, otherwise everyone would be hitting that "ideal amount" every month instead of paying in full and banks offering credit cards will definitely not want that.

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If you can't pay in full now, then just pay what you can to avoid late fees and charges. You should ideally be checking with your bank on these details.