This is part 3 of my series on Cash Back Credit Cards.
Almost everyone spends a non-trivial amount of money each month at gas stations, drugstores, grocery stores so it would be foolish to not have a card that gives you extra rewards in these categories. There are many people who have cards that give 3-5% cash back on g/d/g with no strings attached but I don’t think any of these cards are still available. If you have one of these cards, consider yourself lucky and for the love of god keep your card in good standing because it is probably better than any g/d/g card available at this time. If you don’t have one of these old cards, here are your choices.
Capital One No Hassle Cash Rewards Mastercard (link)
- 2% cash back for gas and groceries (x)
- 1% cash back on everything else (y)
This card is pretty straight forward, but there is no reason to get this when you can get the Fidelity Amex that gives you 2% cash back on everything. Next!
American Express Blue Cash (link)
- 1% cash back at supermarkets, gas stations, and drugstores
- 5% cash back at supermarkets, gas stations, and drugstores after spending $6,500 in one year
- 0.5% cash back everywhere else
- 1.25% cash back everywhere else after spending $6,500 in one year
This card sounds great until you read the fine print (which I conveniently bolded) that says the maximum cash back bonus doesn’t kick in until after you spend $6,500 in one year. On the first $6,500 you spend each year, this card gives a dismal 1% cash back on g/d/g and 0.5% on everything else, so how much do you need to spend to make the average cash back something reasonable?
The Fidelity Amex will give you 2% cash back on everything, so let’s calculate how much you would have to spend on g/d/g for the Blue Cash to average 2% cash back.
0.02 = (0.01*6500+0.05*(x-6500))/x
x = $8,666 per year or $722 per month
That means if you are spending less than $722 a month of g/d/g, the Blue Cash will probably give you less cash back then the Fidelity Amex would. I say “probably” because it actually depends on how much you spend in “other” categories. Here is the formula for how much cash back you would receive if X is the amount you spend on g/d/g and Y is the amount you spend on everything else. This equation assumes that X and Y are evenly spent throughout the year.
Let t = x+y
c = .01*6500*x/t + .005*6500*y/t + .05*(x-6500*x/t) + .0125*(y-6500*y/t)
Substitute for t and reduce to get
c = 0.05*x + 0.0125*y – (260*x + 48.75*y)/(x+y)
The formula for the Fidelity Amex is pretty simple
c = 0.02*x + 0.02*y
Using these formulas we can figure out which strategies maximize cash back given your spending.
If I had to choose one, which should I get?
To find the line where the cards are equal, just set the equations equal to each other and plot (the intersection of two surfaces is a line!)
0.02*x + 0.02*y = 0.05*x + 0.0125*y – (260*x + 48.75*y)/(x+y)
Simplified
0 = 3*x^2 + 2.25*x*y – 0.75*y^2 – 26000*x – 4875*y
Graph (using Gnuplot)

If you spend $X on g/d/g every year and $Y on everything else, then just find the point (x, y) on the graph. If you are to the left of the red line then a Fidelity Amex is better for you than a Blue Cash. If you are to the right of the red line then a Blue Cash is better for you than a Fidelity Amex.
If I already have a Blue Cash, should I get a Fidelity Amex?
Yes. If you aren’t reaching the $6500 tier every year then you a strictly better off with the Fidelity Amex. If you are reaching your $6500 tier then it doesn’t make sense to keep spending “other” categories on your Blue Cash at 1.25% cash back when you can be earning 2% on your Fidelity Amex.
If I already have a Fidelity Amex, should I get a Blue Cash?
There are two potential strategies if you have both cards.
- Use Blue Cash only for g/d/g and Fidelity Amex for everything else
- Use Blue Cash for everything until you hit the $6500 tier and then switch to strategy 1
Let’s compare, here is the formula for strategy 1
c = 0.01*6500 + 0.05*(x – 6500) + 0.02*y
Simplified
c = 0.05*x + 0.02*y – 260
And the formula for strategy 2. Let t = x+y
c = .01*6500*x/t + .005*6500*y/t + .05*(x-6500*x/t) + .02*(y-6500*y/t)
Simplified
c = 0.05*x + 0.02*y – (260*x + 97.5*y)/(x+y)
Set them equal to each other to find their intersection. Pay attention to the steps…
0.05*x + 0.02*y – 260 = 0.05*x + 0.02*y – (260*x + 97.5*y)/(x+y)
-260 = -(260*x + 97.5*y)/(x+y)
260*x + 260*y = 260*x + 97.5*y
260*y = 97.5*y
The strategies are equal if y = 0, but if y > 0 then working backwards we have
260*y > 97.5*y
260*x + 260*y > 260*x + 97.5*y
-260 < -(260*x + 97.5*y)/(x+y) [multiplied by a negative so flip inequality]
0.05*x + 0.02*y – 260 < 0.05*x + 0.02*y – (260*x + 97.5*y)/(x+y)
strategy 1 < strategy 2
So why is strategy 2 always better than strategy 1? As I calculated earlier, strategy 1 doesn’t make make any sense unless you spend at least $8,666 per year or $722 per month on g/d/g, otherwise you would be better off just putting everything on Fidelity Amex. But for the sake of argument, lets assume that you are using strategy 1 because you DO spend at least $8,666 per year on g/d/g. Now think about what would happen if you moved $1 of “other” spending from your Fidelity to your Blue Cash before you hit the $6500 tier on your Blue Cash. Well, you would lose the $0.02 cash back Fidelity would have paid you, but you would push $1 more of g/d/g spending to the $6500 tier which means Blue Cash would pay you $0.05 for that dollar instead of $0.01. That is a net gain of $0.05 – $0.01 – $0.02 = $0.02! The previous logic holds for any non negative amount so we just proved (twice) that strategy 1 is a dominated strategy and we can forget about it.
Now compare strategy 2 with using only a Fidelity Amex
0.02*x + 0.02*y = 0.05*x + 0.02*y – (260*x + 97.5*y)/(x+y)
Simplified
0 = 3*x^2 + 3*x*y + 26000*x – 9750*y
And the new graph

Once again, if you are to the left of the red line then you are better off without a Blue Cash.
Verdict
It depends, but luckily I just gave you all the graphs you need to figure out which is best for you! Here are some more general points.
- Amex is not accepted everywhere, so you should always have a backup Visa or Mastercard (I recommend Fidelity Visa with 1.5% cash back)
- Your Amex only gives you cash back once a year as a statement credit, so if something bad happens to your account you could lose it
Next: Cash Back Credit Cards – Other Categories