Thursday, June 19, 2008

Learning to Think

We are all concerned about rising gas prices. I enjoyed listening to NPR today (Car Gas Mileage May Not Be All It Seems) as they talked about making the correct choices to reduce the amount of gasoline we use. Are we making the right decisions as we choose the vehicles we drive? Here's a math puzzle presented on the show.

A family has two vehicles both driven the same number of miles per year.

One is a mini-van which gets 18 miles per gallon. It could be traded for a station wagon which gets 28 mpg, gaining 10 miles per gallon of gasoline purchased.

The other vehicle is a sedan. It gets 30 mpg but could be traded for a hybrid car that gets 50 mpg, gaining 20 miles per gallon of gas.

Which vehicle should be traded in to minimize gasoline consumption?

(Answer to be shared soon. Please leave your answer and supporting theorems in the comments section. Anna solved this problem in under a half a minute, while I sat and figured a more, shall we say, roundabout way to get to the answer.)

The problem should be more complicated though. You have three vehicles, two of which are paid off but pitiful. These two are driven at sporadic times, sometimes on the highway, sometimes through town. You don't know how many miles you drive - it depends. While pulling horses, increase gas usage to only 10 mpg. You don't know how many miles you pull horses, only that you have to and this vehicle can't be traded because new trucks cost more than houses.

The minivan is paid off, but about 50,000 miles over the expected life of vans of this model. It has a name, "Eugene". You can't possibly trade it because (did I mention it was paid off?) it is like part of the family, even if it does have a peculiar odor to it. How could we do that to Eugene?

Trading in vehicles isn't a math problem, you see. It is a logistics problem - how many people you need to cart where and with what equipment or animals. It is a cleanliness problem - Eugene isn't picky and in a new car, well, spills just wouldn't be taken quite so calmly. Trading cars just simply can't be reduced to a math problem.

Anyway, take a pot shot at the above unrealistically simple problem about saving gas.


pita-woman said...

Oh, well, if you traded the mini-van for a station wagon, that would probably be a waste, as most people don't want to be seen riding in a station wagon... there's just some sort of stigmatism about them, so I'll have to go with trading in the sedan for the hybrid, as right now hybrids seem to be the "cool thing" to have.

pita-woman said...

we have our own "Eugene", so I completely understand.
My husband's suv looks like hell as the pain is worn away on the hood & roof, has some tempermental issues with the radio antenae and the rear hatch, it too smells funny and the stuffing is coming out of the drivers seat. But it's paid for and it still gets him from point A to B, so we're keeping it.

chuckmcc1 said...

I'd trade in the minivan. The minivan is probably used for many more miles each year, thus would get more gallons of gas saved. Also the incremental price of the hybrid is too much to allow the savings in gas to pay off the additional price in a reasonable time.

Flock Fold Kids said...


Flock Fold Kids said...

Sorry--I was testing.

According to my calculations, it would be better to trade in the van for the station wagon.

Assume each vehicle is driven 10K miles per year. With mpg of 18, 28, 30, and 50, we have gallons per mile of .056, .036, .033, and .020, respectively. Multiplying miles per year times gallons per mile yields gallons per year. For the four vehicles, these values are 555.55, 357.14, 333.33, and 200, respectively. Now, assuming gas at $4 per gallon, this yields annual fuel expenditures of $2222.22 (van), $1428.57 (sw), $1333.33 (sedan), and $800 (hybrid). The difference in annual fuel costs is then $2222.22 - $1428.57 = $793.65 for the van/sw comparison and $1333.33 - $800 = $533.33 for the sedan/hybrid comparison. Advantage: station wagon (bye bye, Eugene).

At least, that's what I got. :)


Ingrid said...

Right now, your average mpg is (18+30)/2=24. If you trade the minivan, your average is (28+30)/2=29. If you trade the sedan, your average is (18+50)/2=34.

You could also just use the difference in half (because that car is responsible for half your miles). So, trading the minivan improves by 10/2=5 mpg, and trading the sedan improves by 20/2=10 mpg.

Either of these approaches would easily work if one car was used more as well. If the minivan gets driven twice as much, for example, the minivan is responsible for 2/3 of the total miles (because 2/3 is twice 1/3, and they add to 1). So, trading the minivan results in improvement of 10*2/3=6 2/3, and trading the sedan results in improvement of 20*1/3=6 2/3. Equal in this case, although I didn't plan it.

Linda Clement said...

I think it is more complex than this: it depends a lot on what you're dragging around in the vehicles. If you are using the mini-van to full-capacity, there are going to be times when you're driving both vehicles at the same time, and that means you're effectively adding to the consumption, not taking away from it.

Personally, it makes the most (non-linear) sense to me to have 2 vehicles: one that gets used rarely (may even be only insured rarely) that is kept for 'big capacity' use only: full van, dragging horses, going camping or whatever, and; one 'smallest normal capacity'.

So if you almost never go out with the whole family of 8, but usually with a driver and 2 passengers, a small 4-seater (like the hybrids) that can squeeze in 5 rarely when necessary would be the general use/commuter car.

If one car is being driven my one person alone most of the time, it should be a smart car or something with equivalent passenger space and mileage. Motorcycle, scooter, bike, running shoes ;D

If, on the other hand, you always need the capacity for all the kids, horses or whatever, it makes the most sense to sell both vehicles, buy one that suits your actual capacity needs and has a new, well-tuned engine.

Oh, and it matters almost as much how well an engine has been maintained as it does what kind of car it is to begin with.

And let's give a shout-out for the idiots who are making cars that will save you gas... but require the most expensive premium fuel to do so (effectively costing the same amount as the less-efficient vehicle did). Whoo-hoo.

Ingrid said...

Oops. My analysis only works if you paid a fixed cost per mile, which is of course, the whole point. I hadn't followed the link before trying it myself. If you convert to gallons/mile first, I think the rest of what I said is right (and gives you the opposite answer). Thanks for making me think. It's obviously something I need a little practice with :)

Richard said...

You need to get rid of the biggest gas guzzler. That will save you money faster than one that already gets pretty good gas mileage.
If you take a hypothetial 1000 miles driven at $4/gallon for gas,
the mini-van will cost 1000/18mpg x $4 = $220; the station wagon will cost 1000/28 x $4 = $142. That is a savings of $78.
The sedan will cost 1000/30 x $4 = $133. The hybrid will cost 1000/50 x $4 = $80. That is only a savings of $53.
So, the mini-van needs to be exchanged for the station wagon.


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