## 20 June 2008

### Gallons per mile?

Experts find key to saving fuel: say gallons per mile.

I'll summarize: d/dx 1/x = -1/x2.

Research has been done that shows that people believe that an improvement of 1 mpg will always save them the same amount of gas. But this is obviously false, as some simple arithmetic shows. Let's say I drive 12,000 miles a year. (Why 12,000? Because everything that follows will work out to be integers.) If I improve from 15 mpg to 16 mpg, I go from using 800 gallons of gas a year to 750, a 50-gallon reduction. But if I improve from 24 mpg to 25 mpg, I go from using 500 gallons of gas a year to 480, a 20-gallon reduction.

A driver driving m miles per year in a car getting x miles per gallon will of course use m/x gallons of gas; the derivative of this is -m/x2. So if you get x miles per gallon already, improving by one mile per gallon saves m/x2 gallons. (I'm assuming here that 1 is small compared to x.)

The article claims that this means people wanting to get better gas mileage, if they have multiple vehicles, should always target the least efficient vehicle -- but that's going too far. It might be cheaper to get a 1-mpg improvement for less fuel-efficient cars. There's no reason that the cost of a car should be linear in the number of miles per gallon it gets, all else being held constant.

Note that I'm also not saying the cost of a car should be linear in the number of gallons per mile it gets! In fact this would be impossible, because it would predict that cars that get zero gallons per mile could be made for a finite amount of money.

"Gallons per mile" is kind of an annoying unit, though, because all cars get less than 1. Perhaps "gallons per 100 miles" would be a good way to go, with most cars measuring between perhaps 3 and 6 on this scale. And people can picture driving 100 miles. (For example, if they have a ten-mile commute each way, it's five round-trips to work.) But on the other hand, there's a temptation to not have to deal with decimals, and there's a big difference between 4 gallons per 100 miles and 5 gallons per 100 miles. Perhaps "gallons per 1000 miles" works nicely; typical values are now 2-digit integers, and rounding to the nearest integer gives roughly the same precision as the current system.

(Readers from other countries: please spare me the "in my country we measure fuel economy in liters per 100 km" comments. I know this.)

thecooper said...

Maybe gallons per 300 miles, or 500 miles? I think 1000 miles might be hard for people to think in terms of.

Isabel Lugo said...

But then it feels like you're just picking arbitrary numbers. And if people want to know "how much gas am I going to use in a year?", which seems like a reasonable thing to be calculating, it's easier to start from the gallons per 1000 miles.

Come to think of it, most people probably figure their budgets monthly, not yearly. And I've often heard 12,000 miles per year as an "average" figure for mileage put on cars, which is of course 1,000 per month.

SlimAndSlam said...

Perhaps we should go one step further, and put it in terms that people would really understand: dollars per mile (or dollars per 1000 miles, if you prefer).

Of course, this involves some amount of speculation about what will have happened to gasoline prices by the time you drive the 1000th of those 1000 miles, but one could use current prices as a rough estimate.

Let's see... at \$4.00 per gallon, an improvement from 15mpg to 16mpg translates into an improvement from \$266.67 per 1000 miles to \$250.00 per 1000 miles, or a savings of \$16.67 per 1000 miles.

From my personal experience, flipping this around to "miles per dollar" also keeps things concrete. "If I put ten dollars' worth of gas in the tank, how far could I then travel?" is a real question, and it would often be quite beneficial to know the answer-- or else you might discover that you don't have enough money to get home...

misha said...

It looks like gallons per mile is more convenient because people more often ask "how much it would cost to do X" than "what I could do if I had Y\$," and multiplication is easier than division.

CarlBrannen said...

This reminds me of a problem on the practice prelim exams in physics at U. Cal., Irvine, what is the approximate acceleration of earth's gravity at the surfance in units of furlongs per square fortnight?

Pseudonym said...

Just for reference, in Australia we use litres per 100km, and the numbers are in a pretty reasonable range.

Approximate Identity said...

I've thought about this before and thinks it makes sense for the reasons you mentioned. It's also psychologically closer to how a spendthrift would decide to measure it and considering the recent increase in prices, more could use some thrifty thinking.

In our current system, we might have 10 gallons and wonder how long it can take us. I.e. we're planning on using it up before really know what we need it for.

In this other system we might say well if I drive those 80 miles how much gas will it cost me?

I just think that the latter thinking leads to a better point-of-view if you're trying to save gas, since it leads to thinking out the costs first (at least in theory...anyone who really cared could still figure it out, but it's the ease of calculation I'm talking about).

Anonymous said...

Harald Hanche-Olsen said...

Despite the final exhortation in Isabel's post, I cannot resist the temptation to point out that whether you use gallons per mile or liters per 100 km, we're talking about volume divided by length, in other words area. So we metric users could express our gas economy in square millimeters. Unfortunately, that produces numbers around 0.01, while the next step, square micrometers, produces uncomfortably large numbers. Here you users of imperial units have the advantage: If I am not mistaken, a pico-acre would be just the right sort of unit.

Harald Hanche-Olsen said...

Just to follow up on my own comment: In metric units we have ares of course, and a reasonably economical small car will use about half a nano-are (na) of fuel. If you prefer integers, maybe better express in in pico-ares (pa). Hmm, as the more commonly used unit hecto-are is shortened to hectare, should we do so with other prefixes to, creating units of nanares (na) and picares (pa)?

kickingbottom said...

71276508266.6402 furlongs per square fortnight. a rather easy question.

It is a tad easier outside the US to work in litres per 100km as that gives a much larger figure than gallons per 100 miles.

Of course why not just do it in gallons per average year. or dollars per average year. (average year being 12000 miles) I am sure that would help most people and stop them having to work out the complex maths.

Joe said...

kickingbottom has an interesting point. I'd love to see fuel efficiency in terms of gallons per year (average use). Don't they already do things like that for refrigerators? I think they have signs indicating the operating cost for a given price of energy. (At least I think they do this. I've never had to buy a refrigerator.)

Cost per year might be a little difficult given the wild fluctuations in gas prices.

Joe said...

Also, I'd really like to get away from the miles per gallon metric, even if it's just so I don't have to hear another commercial talking over and over again about "better emm-pee-gee." I don't think the advertisers realize that it's just a unit. (Computer manufacturers don't say their product gives you better GHz.)

Isabel Lugo said...

I'm pretty sure they do that for refrigerators, although like you I've never bought one.

I think there's a problem with using "average use" -- the mileage people put on cars probably varies a lot more than the amount of electricity they use to power their refrigerators. (But I don't know for sure; I'm not a refrigeratorologist.)

Isabel Lugo said...

Joe,

I never thought of that! Now it's going to bother me when I hear it.

Another one: cars advertised as having "a GPS" included. Really, the entire Global Positioning System comes with the car?

David said...

I vote for gallons per hundred. With most cars, it'll work out to something convenient and comfortable [1.0-ish,6.0-ish]

Ian said...

Just to take the opposite view for a moment (though I will say I agree with the idea in general): Perhaps listing this would lead to worse buying behaviors?

I'm thinking of someone who currently has a Ford Excessive or whatever, getting ~11"mpg". They are told the benefits of moving up even 1 unit via the new Gallons-per-mile metric, and even understand on cars with far better efficiency, the step up in one conventional "mpg" doesn't net as much improvement. "Well," they say, "I'm doing great even if just get something that has ~15'mpg', so lets go look at the Ford Epoch with massive wheels in case I need to run through jungle on the way to get groceries..."

Point being, the push should be towards people driving less overall and driving far more efficient cars when they can. Just to be the doubting Thomas, I'd worry that people would feel inappropriately good about only minor changes in fuel efficiency.

But that's just a weird economic point. Overall, I agree that MPG is a goofy measure.

Anonymous said...