mpg frame

Adam switches from a gas-guzzler of 12 mpg to a slightly less voracious guzzler that runs at 14 mpg. The environmentally virtuous Beth switches from a 30 mpg car to one that runs at 40 mpg. Suppose both drivers travel equal distances over a year. Who will save more gas by switching?
[Thinking, Fast and Slow location 6263 by Daniel Kahneman]

Answer:

Spoiler: Show

Adam will save more gas. Think in terms of gallons per 100 miles instead of mpg. Adam goes from 8.33 to 7.14 saving 1.18 gallons. Beth goes from 3.33 to 2.5 saving just 0.83 gallons.

This is exactly why stating gas mileage in mpg is scandalous. It obfuscates the true efficiency of the vehicle.

Gallons per mile is a MUCH more intuitive result. Not only in the way this problem was solved.
It also represents a real, physical quantity.

Yeah, gpm is a way better ratio to use.

Surely mpg and gpm are reciprocals? Are they not just the same ratio expressed differently? There's nothing scandalous going on that I can see. Adam is still using a lot more gas. Am I right to assume that a car that gets 14 mpg will travel for 14 miles (on average, of course) on one gallon of gas? Because if that's not the case, then yes I can see a big problem with this measuement. But the same would be true of the reciprocal, no? If mpg is misleading, then why is gpm not misleading?

I'm confused why the two measurements are so vastly different in accuracy.

Yes, it's a reciprocal relationship.

Look at Eric's example.
Intuitively, Adam only got +2 mpg, while Beth got +10 mpg. So Beth is saving much more... 5 times more than Adam.
But for real, though... Eric's solution is correct.


How many people do you know who are good at doing reciprocals in their head?
How about doing 2 reciprocals and remembering the first while you calculate the 2nd, so that you can subtract them?
It's much easier to have the reciprocals stated and just subtract.

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Intuitively, Adam only got +2 mpg, while Beth got +10 mpg. So Beth is saving much more... 5 times more than Adam.

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No, intuitively, Beth got 5 times more extra distance than Adam did for a gallon. I can see how it confuses people, and yeah less people would be confused if we used gpm because then people would see how much fuel they save per mile, rather than how much more distance they travel per gallon. If people are assuming that Beth saves 5x the fuel that Adam does, then they're just bad at understanding how maths works. I guess that's a lot of people, I guess that's something I take for granted.

I should add that while I have a pretty solid understanding of reciprocals for an uneducated waster, I wouldn't be able to crunch these numbers in my head in any hurry. It doesn't surprise me that Adam "saves" more fuel, but the logic isn't obvious and I'd need to spend more time than I'm willing to right now to wrap my head around it properly.

If I can't do this quickly in my head, then there's no way most people will. So yeah, I guess I have to agree that gpm is a far superior means of expressing fuel to distance ratios.

Math alert!

Taking 1/X is pretty easy with a little bit of memorization. First memorize your inverses from 1 to 10

1/1 = 1.00
1/2 = .50
1/3 = .33
1/4 = .25
1/5 = .20
1/6 = .17
1/7 = .14
1/8 = .125
1/9 = .11
1/10 = .10

Most people already know at least a few of these.
Now the trick is that you work your way back up that list, except backwards and inverted while standing on your head*.

1/11 = .09
1/12.5 = .08
1/14 = 0.7
1/17 = .06
1/20 = .05
1/25 = .04
1/33 = .03
1/50 = .02
1/100 = .01

Obv. most of these should be "~=" but you get the point. You can get away with only memorizing the top half and the bottom half comes with a little trick. Multiplying by a factor of ten easily extends this range.
I.e. if 1/2 is .5 and 1/20 is .05, then 1/200 is .005

What's in the ballpark of .015?
1/7 is .14, so 1/70 is about .014, so maybe 1/68...?
(Calculator check shows 1/68 to be .0147. Pretty close for a 2 second estimate.)



*Truth be told, standing on your head is optional.

Everything MMJ said, plus...

Up to ~30 MPG you see better than expected results, after that you see less than expected results. That is the point where MPG ~= kMPG (thousand miles per gallon). The k puts the numbers in the same ballpark, and makes them more easy to understand vs our societal understanding.

The problem is that miles are the dependent variable. You do not fill up your tank, then decided how far you are going to drive. Most people drive a certain distance, and should think about how many gallons it takes to get there. We (the U.S. of A.) will adopt the metric system well before we change this "standard".

Yeah, we need the metric system and liters-per–100 kilometers. Until then using gallons-per-100 miles would save a lot of confusion.

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Originally Posted by swiggidy

Up to ~30 MPG you see better than expected results, after that you see less than expected results. That is the point where MPG ~= kMPG (thousand miles per gallon). The k puts the numbers in the same ballpark, and makes them more easy to understand vs our societal understanding.

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@underline: Must be a typo in there.
One of those MPGs should be a GPM.

Found it!
30 MPG ~= .03 GPM
multiply both sides by 1,000

30 kMPG ~= 30 GPM

Good catch, one more update for posterity. (I took me a minute to find it, had done confused myself.)

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30 kMPG ~= 30 GPM

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Should be

30 MPG ~= 30 GPkM

Thirty miles per gallon is approx. equivalent to thirty gallons per thousand miles.