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.