Math And Real World Applications

Recently the Freakonomics website posted an article on a topic that I've seen before: how the reporting of fuel efficiency for American cars in miles per gallon is misleading when one is considering ways to reduce dependency on oil. You can read about it at the preceding link, or you can continue on and see my explanation (or you could always do both!).

I wanted to explain this to Vicki (without sending her to the Freakonomics article) and figured that, as a Math tutor, I should really be able to do it. My first attempt died a death born of its own feebleness, but my second approach worked better. So that's what I'll use here.

Basically the problem is this: why is it that, when you consider improvements in fuel efficiency in terms of miles per gallon, you don't get an accurate picture of how fuel efficiency actually works? In the linked-to article, for example, there are 3 sample MPG values used:

1. 14 MPG
   
2. 24 MPG
3. 42 MPG
   

and then the question is asked: is it more of a savings to go from 14 MPG to 24 MPG, or to go from 24 MPG to 42 MPG? Common sense would seem to suggest the latter, because it's an "MPG increase" of 18 (42 minus 24), compared to one of only 10 (24 minus 14) in the first case. But in fact you'd save more gas going from 14 to 24 than you would going from 24 to 42. Why?

It's because the measurement of miles per gallon is considering the ratio in the wrong order (or, in fraction terms, inverting the fraction so that that numerator is swapped with the denominator). If you're trying to save gas/save money (/save the world!), then you want to reduce gas usage as much as possible. So in that light, you should be considering gallons per miles driven, not miles per gallons used. When you do that, you see the numbers in the context of the conversation (how much fuel is being used to go a certain distance for each of the 3 vehicles). In fact, if you take those same 3 MPG values in the preceding paragraph, and convert them to "gallons per 1000 miles" (in other words, how many gallons of gas does it take to drive each vehicle 1000 miles), then the fuel efficiencies of the 3 vehicles become:

1. 71.4 G/1000M
2. 41.7 G/1000M
3. 23.8 G/1000M
   

Now the greatest difference is between cars 1 and 2 (almost 30 gallons of gas saved), rather than between cars 2 and 3 (almost 18 gallons). In other words, it went from looking like you were doing almost twice as much improvement in that scenario (18 additional miles to a gallon compared to 10) to actually being 40% less of an improvement (only 18 gallons of gas saved compared to 30). And all because the MPG rating focuses on how miles you get to a gallon instead of how many gallons it takes to go a certain # of miles.

Since many other countries (including Canada and much/all of Europe) actually "do it right" and quote their fuel efficiencies as "L/100KM" (amount of fuel to go a set distance), I have to wonder why the MPG usage is still standard in the States. It certainly benefits the auto manufacturers there for people to think that it's not that big of a difference between 14 MPG and 24 MPG (and not to realize just how much gas they could be saving by buying a more fuel efficient car), but is there really anything conspiratorial at work in that standard?

A commenter on the Freakonomics site was the one who suggested that the U.S. go to a "Gallons per 1000 Mile" measurement, because 1000 miles is probably not too terribly far off from what a typical car owner travels in a month. Therefore, you could look at the various fuel efficiency ratings, multiply by the current price of a gallon of gas (which is what's posted at every gas station), and quickly estimate your gasoline budget for a month. For the cars in my example, using $3/gallon, you'd get:

1. $214.29
2. $125.00
3. $71.43

So if you're driving car # 1 with its 14 MPG efficiency, you could save almost $90/mth by getting a 24 MPG car, while upgrading from the 24 MPG car to # 3 with its 42 MPG rating would save you almost $54/mth. I wonder how many SUV gas guzzler owners realize that they could save almost $1100 each year in gas stops by increasing their MPG by 10? And would it be so hard to get higher fuel efficiency laws passed in the U.S. if people could so easily convert fuel efficiencies into actual dollars? (Do we do better at it here in Canada, for example?)