Enrico Fermi designed the first nuclear reactor to achieve a fission chain reaction. As a professor at the University of Chicago, Fermi would challenge his students to solve problems that seemed impossible due to lack of given information. Fermi wanted his students to develop their abilities to make assumptions and estimations and work out rough solutions on the back of an old envelope rather than making detailed (with calculators in your case) calculations. The typical Fermi problem was to **determine the number of piano tuners in Chicago given only the population of the city**.

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**Fermi’s solution: **

Given Chicago has a population of about 3 million people (in the 1940’s).

Assume that the average family has four members so that the number of families in Chicago must be about 750,000.

Assume one out of five families owns a piano, there will be 150,000 pianos in Chicago.

If the average piano tuner worked 5 days a week tuning 4 pianos each day and had a two week vacation during the summer….

In one year (50 weeks) he would tune 1,000 pianos. Therefore there must be about 150 piano tuners in Chicago.

**This method is not guaranteed correct but it is a valid first estimate that might be off by no more than a factor of 2 or 3 but will definitely not inaccurate by an order of magnitude (we should not expect 15 piano tuners, or 1,500 piano tuners). Notice how many assumptions were made to come up with the answer.**

**Estimation Strategies**

Don’t make numbers more precise than necessary

- Round to the nearest 0, 5 or 10
- Don’t worry about numbers like pi (call it 3 .. or even 2)

Guess numbers that you don’t know

- Try to use common sense to make good guesses
- Accurate guesses requires experience and common sense … this comes from education …. pay attention

Make complicated geometry simple

- Example “assume the cow is a sphere . . .

Extrapolate from what you do know

- Use ratios
- Use known similar values (for example you could reasonably assume the density of gasoline is 1 g/mL)

When estimating, keep things as simple as possible. You will use estimate in your lab work, but most often estimation can be used to see if a calculated answer is reasonable (in other words you did not make a calculation error).

**APPLY **- Try these

- How many jelly beans fill a one litre jar?

- How many times does the human heart beat in one week?

- How many tennis balls would cover the area of two tennis courts?

- How high would a pile of one trillion 1 dollar bills reach?

An amusing video showing how estimating can be used to solve improbable problems related to the Boston Marathon

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