I have a credit card that gives cash back on every purchase. I don’t remember the percentage off hand — so let’s call it X. I was wondering, if I always use my credit card to buy things, even the things i buy with the money I get back from them — what is the effective real cash back percentage?
For example, suppose that my credit card gives me 50% cash back on everything I purchase. I’m using an absurdly high number because it is easier to do the math — I know no such deal really exists!
Let’s start with ten dollars. I buy $10 with my credit card and get $5 back. If I spend that $5, I get $2.50 back. So far my $10 has bought $15 worth of merchandise, and I still have $2.50 left to spend! You can keep doing this forever, and if you do you’ll end up with exactly $20 worth of merchandise.
But how do we solve this problem for any cash back percentage? Well, to begin with, it is an infinite series, so we can never write the whole equation down. But we can start… Here I’ll use Y to represent the effective percentage rate, and also we’ll assume we have just one dollar.
In other words, the effective buying power of one dollar is one dollar plus one times the cash back amount, plus that amount times the cash back amount again and so forth. Each new term is how much cash back you got on the previous purchase.
To help get to the next step I’ll rearrange the terms just slightly:
Recall that X0 is equal to 1. Anything to the zero power is one.
One more rearrangement:
Notice that the term inside the parenthesis is equal to the right hand side of EQ. 1. Thus we can replace that infinite series with Y.
And now it is easy to solve for Y:
Ta dah! We can verify with our one known : When X is 50%, Y is 2. Remember that 50% is actually 0.5 in the above equation (% means divide by 100). Sure enough Y is 2 when you work that out. Go ahead and plug your cash back bonus amount to see what it is really worth.
4 replies on “My favorite kind of math”
this looks a helluva like algebra to me…….which is NOT my favorite kind of math
I guess I didn’t emphasize the cool part. The ability to transform the infinite series back into Y is really cool. That piece of insight turns an impossibly complex problem into a simple one. This is the kind of “ah hah” moment that I imagine really brilliant people like Einstein have when they conceive of something fantastic like Relativity.
Did Einstein carry Visa or Mastercard? Or is it all relative anyway?
Cool little calculating gizmo… but I’ll stick with my ‘free books from amazon’ card for now….