I was thinking about how people travel to a distant gas station to get gas that's cheaper. I mostly thought of it as a waste of time, especially with the 10-gallon tank in the Civic. If I filled it all the way, a 5 cent difference in gas price would net me all of 50 cents. So if I'd be willing to cross the street, but not cross town — driving the distance isn't free.

So what's the break-even point? Well let's say the price of the cheapest fuel we can get without going out of our way is P0. Further, let's assume we fill up about the same amount — like if we run similar errands each week, we might need to buy half a tank each week. Let's call that volume of fuel V0, so we pay C0: P0 * V0. So now we see fuel at a cheaper price, P1, elsewhere. But it'll use more fuel to get there, so we'll have to now buy a volume of V1 instead, costing C1: P1 * V1. How far out of our way should we go to still save money?

The break-even point is when the two costs are the same:

C0 = C1

which is also

V0P0 = V1P1

but we want to talk about distance. The ratio of the the distance traveled to the volume of fuel used is your fuel efficiency or "gas mileage" in miles-per-gallon:

E = D/V

we can rearrange that so

V = D/E

and substitute above so

P0(D0/E0) = P1(D1/E1)

Now, if we make a broad assumption that the efficiency is about the same — that your gas mileage is the same whether you go to one place or another — then we get:

P0(D0/E) = P1(D1/E)

and we can eliminate the efficiency factor on both sides leaving

P0D0 = P1D1

So let's say that P1 is some percentage cheaper and that D1 is some distance further:

P1 = ( 1 – cheaper ) P0

D1 = ( 1 + further ) D0

And then substitute P1 and D1 to get:

P0D0 = ( 1 – cheaper ) P0 ( 1 + further ) D0

And now we can cancel out P0 and D0 from both sides:

1 = ( 1 – cheaper )( 1 + further )

But what we want to know is how much further we can go — for now as a percentage of how far we usually go before filling up. So we get:

1 + further = 1 / ( 1 – cheaper )

or, deriving some more:

further = 1 / ( 1 – cheaper ) – 1
further = 1 / ( 1 – cheaper ) – ( 1 – cheaper ) / ( 1 – cheaper )
further = ( 1 – ( 1 – cheaper ) ) / ( 1 – cheaper )
further = cheaper / ( 1 – cheaper )

So in other words, we'd need to travel less than (further) percent further to come out ahead which is the percentage cheaper divided by 100% – the percentage cheaper.

To bring this back to the real world, let's start with the fact that most cars today can go 300-600 miles per tank. So it's not unusual to get about 200 miles on a half of a tank. Now let's say we find gas that's 5 cents cheaper than other gas at $3.25 — about 1.5% cheaper. The percent difference is then (1.5%)/(1-1.5%) = 1.5%/98.5% = 1.52%. So if we filled the tank halfway, the cheaper gas would need to be 1.52% * 200 miles = 3 miles out of the way. That's "out of the way" so if it's a separate trip, it's only 1.5 miles each way.

And that's just to break even.

Alternatively, what if you could get gas at 10% off — something like $3.00 instead of $3.33. Then it's a distance difference of 10%/90% = 11%. If you had to get close to a full tank and you could get 400 miles, then you'd still save money up to 44 miles out of the way.

The trap, though, is that it's still not a lot of money. Like if you went 10 miles away, you'd have to get 420 miles of gas at $3.00 instead of 400 miles-worth at $3.33. If you were driving a truck that got 15 miles-per-gallon, that's 28 gallons at $3 or $84 versus 26.7 gallons at $3.33 or $88.91 so you'd save all of $4.91.

But at least now you have a way to figure it out.