Cash back rewards are calculated on spending, not on any interest charged, which means carrying a revolving balance specifically to 'earn more cash back' is a mathematical error: the interest charged on a carried balance -- commonly 20% APR or higher -- overwhelms any realistic cash back rate, which tops out well under 10% even in the most generous bonus categories.
For example, carrying a $1,000 balance for a year at 22% APR costs approximately $220 in interest, while even a generous 5% cash back rate on that same $1,000 in original spend only returns $50 -- meaning the interest cost is more than four times the reward earned.
This makes the standard, non-negotiable recommendation for anyone carrying a credit card balance clear: paying off the balance in full each month, or as quickly as possible if a balance already exists, takes priority over any cash back optimization strategy, since no reward rate available on the market compensates for typical interest rates.
Cash back optimization is a strategy for spending that would happen anyway and gets paid off in full -- it is not a strategy for increasing spending or delaying payoff in pursuit of rewards, since doing so virtually always produces a net financial loss once interest is accounted for.
For anyone currently carrying debt, a 0% introductory APR balance transfer card, focused purely on debt payoff, is typically a far more valuable tool than any cash back card, and cash back optimization is best revisited once existing balances are cleared.
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No -- typical cash back rates (under 10%) are far outweighed by typical credit card interest rates (20%+ APR) on any carried balance.
Debt payoff should always come first -- cash back optimization is most valuable for spending that's paid off in full each month.