# Classic Fermi Questions

### with annotated solutions

Two "typical" Fermi questions are posed and solved in this section.
• How many piano tuners are there in New York City?
• How many jelly beans fill a one-liter jar?
WARNING: Fermi questions are not really typical and Fermi questions do not have single solutions. These two examples are provided simply for illustrative purposes.

## Example 1: HOW MANY PIANO TUNERS ARE IN NEW YORK CITY?

How might one figure out such a thing?? Surely the number of piano tuners in some way depends on the number of pianos. The number of pianos must connect in some way to the number of people in the area.

Approximately how many people are in New York City?
10,000,000

Does every individual own a piano?
No

Would it be reasonable to assert that "individuals don't tend to own pianos; families do?
Yes.

About how many families are there in a city of 10 million people?
Perhaps there are 2,000,000 families in NYC.

Does every family own a piano?
No.
Perhaps one out of every five does.
That would mean there are about 400,000 pianos in NYC.

How many piano tuners are needed for 400,000 pianos?
Some people never get around to tuning their piano; some people tune their piano every month. If we assume that "on the average" every piano gets tuned once a year, then there are 400,000 "piano tunings" every year.

How many piano tunings can one piano tuner do?
Let's assume that the average piano tuner can tune four pianos a day. Also assume that there are 200 working days per year. That means that every tuner can tune about 800 pianos per year.

How many piano tuners are needed in NYC?
The number of tuners is approximately 400,000/800 or 500 piano tuners.

Try it yourself.
Use different assumptions for various factors. It is unlikely that you can justify an answer greater than a factor of 10 or smaller than a factor of 10 from the number originally obtained; that is to say, there are probably not more than 5000 tuners and surely no less than 50. Thus the answer obtained is good to within an "order of magnitude".

## Example 2: HOW MANY JELLY BEANS FILL A ONE-LITER BOTTLE?

As with any Fermi question, there are multiple directions from which the problem can be approached. Solution 1 illustrates a more algorithmic approach; solution 2 is more intuitive. In both solutions, it is understood that one liter is equal to 1000 cubic centimeters.

### Solution 1

What is the approximate size a jelly bean?
An examination of a jelly bean reveals that is approximately the size of a small cylinder that measures about 2 cm long by about 1.5 cm in diameter.

Do jelly beans "completely fill the liter bottle"?
The irregular shape of jelly beans result in them not being tightly packet; approximately 80% of the volume of the bottle is filled

The number of jelly beans is the occupied volume of the jar divided by the volume of a single jelly bean

Number of beans = (Occupied Volume of Jar)/(Volume of 1 Bean)

The volume of one jelly bean is approximated by the volume of a small cylincer 2 cm long and 1.5 cm in diameter

Volume of 1 Jelly Bean = h(pi)(d/2)^2 = 2cm x 3 (1.5cm/2)^2 = 27/8 cubic centimers
Thus the approximate number of beans in the jar is
Number of beans = (.80 x 1000 cubic centimeters)/(27/8 cubic centimeters) = approx 240 jelly beans

Have your students try it out with jelly beans and a liter bottle or jar. If you don't have a liter jar, use a quart jar (.95 liter)

### Solution 2

Have your students construct or visualize a paper cube that measures 1 cubic inch.

How many jelly beans will fit in the cube?
Approximately 4

How many cubic inches are there in 1 liter?
1 inch = approx 2.54 centimeters. Therefore 1 cubic inch = approx. 16 cubic centimeters
1000 cubic centimeters/16 cubic centimeters = approx 62 cubic inches in one liter.

How many jelly beans are there in the one liter container?

62 x 4 = approximately 248 jellybeans