Credit Card (CC) M offers a rebate of cash back of $a$ dollars, with NO annual fee.
CC S offers a rebate of cash back of $b$ dollars, but charges an annual fee of $f$.How much must I spend with CC S to breakeven with CC M, to ensure my regaining $f$ and the advantage in choosing CC S for its higher $b$?
Call total spending $s$. Then I need: Rebate Revenue from CC S = Rebate Revenue from CC M
$\iff bs - f = as \iff s = \dfrac{f}{b-a}$.
For the following quote, $a = 0.02, b = 0.04, f = 129$. Then my work yields: $s = \dfrac{f}{b-a} = \dfrac{129}{0.02} = 6450/$year $ \iff \color{green}{\$537.5}$/month.
But the quote beneath's breakeven spending is $\color{red}{\$470}$/month? Thus who is correct?
- Scotia Momentum Infinite Visa (link)
Right now, this is my favorite premium credit card out there due to the huge $4\%$ cash back that they offer on gas and grocery spending. In fact, if you spend more than $\color{red}{\$470}$/month in gas/groceries then this card will offer you more cash back than the free $\color{purple}{\text{[MBNA]}}$ Smart Cash card $\color{purple}{\text{[with 2% cash back on gas and groceries]}}$ (assuming \$129/year for $\color{purple}{\text{[Scotia]}}$ primary and secondary card).
In addition to the 4% cash back, they offer 2% on drug store and recurring payments, then 1% on everything else.
Seems you are correct. If we are just trying to determine the breakeven total spending, we can use this equation: $$ s - bs + f = s - as $$ Substituting the appropriate values in ($b = 0.04$, $f = 129$, and $a = 0.02$) gives us your result of $6450$/year or $537.5$/month.