One of my favorite mental exercises is something I like to call “ballparking.” The idea is to make a quick, rough estimate for some quantity using prior knowledge and some mental arithmetic. Here is an example I often use with my students:

Finding the radius of the Earth

It’s about 3000 miles from the west coast of the United States to the east coast.

There are 4 time zones in the continental U.S.

There are 24 time zones in the whole world, so the circumference of the Earth is about 18000 miles.

To get the radius, we just divide by 2\pi, or about 6, and we get about 3000 miles for the radius.

If you prefer the metric system, then there are about 2 km in a mile, so the radius of the Earth is about 6000 km. According to Wikipedia, the radius of the Earth is 6378 km, so my rough estimate was good to one significant figure.

You may have noticed I used the word “about” an awful lot in that calculation. This was by design. I do not know the exact distance from the west coast to the east coast, or the exact fraction of the 4 time zones taken up by that distance. I did not take into account the fact that we are not located at the Equator, but a little more than 30 degrees above it. I divided by 6 instead of 2\pi, mostly because it required less effort, and I was a bit off in my conversion from miles to kilometers. Despite all these imprecisions, I was still able to make an estimate that was “in the ballpark” and then get on with my life. If, for whatever reason, I needed a more accurate value for the radius of the Earth, I could go back and correct all these estimates, and get it. I could also use the ballpark estimate as a reference in case I make a mistake in the more laborious calculations needed to find the exact value.

One physicist who was famous for this type of calculation was Enrico Fermi. He was able to estimate the blast power of the atomic bomb Trinity by eyeballing how far it blew some pieces of paper miles from the blast site. Another classic question answered using this technique was “how many piano tuners work in the city of Chicago?” He was so good at this type of estimate, that most people refer to this technique as a “Fermi Calculation.” It was a method that helped scientists determine what a reasonable value for something would be before anyone had figured out what kind of math to use in order to determine the exact value.

Things for my readers to do:

What are some good problems for ballparking? I would like to use some of your suggestions for a class I am teaching this summer.


5 Responses to “Ballparking”

  1. “How many windows are within the city limits of San Jose?” One of my teachers would ask this during interviews to see what approach was used in order to answer an almost unanswerable question. I’ve been thinking about it ever since.

  2. A simple ballparking method for measuring a level length: two moderate paces (left-right) = five feet.

  3. But this isn’t what you are looking for, I think. I’ll ponder some more.

  4. Mike, that’s a good one. A variation on the classic piano tuner question. Here are a couple more:
    How many marbles can fit in a schoolbus? And a personal favorite: How many transistors are in this room?

  5. Andrea Pavellas Montano Slosarik Says:

    Alex, I want to remind you that not all can be explained or “ballparked”, some can only be referred to as “magic”: as one boy said to his mom after witnessing his sister give birth to an 11lb baby. He said “there is no mathmatical equation that can explain that size baby coming out of that size of an opening” (paraphrasing).

    I could go on about birth relatead numbers questions, such as: If the average size baby in the US is 7lbs 4oz why does San Jose have a Cesarean rate of 40%? But this is more political than mathmatical, so I shall refrain from spouting births stats that have nothing to do with Ballparking and all to do with fear, money and power.

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