An intriguing gas discount

Mike emails me about a curious promotion and asks me to analyze it:

I noticed this interesting discount where a grocery store teamed up with Sunoco to give rewards to frequent shoppers. Here is the link with the details

Basically, for each $50 spent at the grocery store, you can redeem for a one-time $0.10 discount per gallon on up to 20 gallons of gas, a potential $2.00 savings. If you spend $200 at the store, you can take $0.40 off per gallon.

The question they asked me, and I haven’t been able to come up with a definitive answer: is it better to use the discount each time they fill up, or save for larger per gallon discounts? My first inclination is that it does not matter, but I can’t prove it mathematically. Any help here?

Can you come up with a proof? Give it a try before reading my answer.

Hint: the distributive law

I think the problem is about the distributive law, one of those useful but often forgotten tidbits from algebra.

As a refresher, the distributive law states:

a(b + c) = ab + ac

In other words, you get the same thing if you multiply the sum of two numbers or if you multiply each number and then sum it up.

The distributive law comes up in personal finance in a lot of different places. An example is that restaurants can safely split up a bill because applying tax to individual orders is the same as applying tax to the whole bill.

The answer to the gas discount

Let’s calculate a formula for savings from the gas promotion.

Assume you fill a constant number of gallons G at each fill-up. The savings is then G times the discount per gallon.

Now it’s easy to see why saving up discounts is unnecessary. Here’s the formula showing the equivalence of saving up for a $0.20 per gallon discount versus filling up two separate $0.10 discounts:

G(0.10 + 0.10) = G(0.10) + G(0.10)

This is nothing other than a specific application of the distributive law. The conclusion it doesn’t matter if you save up discounts, as long as you fill the same amount each time.

Clarification 1: The only other consideration is the promotion’s 20-gallon limit . If you have an SUV or van, it would be wise to fill up 20 gallons at a time rather than filling the whole tank.

Clarification 2: In the comments Sailesh points out an example that is not distributive. The issue is not filling up to the allowed 20-gallon limit. To avoid this,  we should add the constraint G = min(20, full tank)