How many piano tuners are there in Chicago? How can we find an answer as close as possible to reality without opening a local phone book? Enrico Fermi was an accomplished Italian-American physicist, winner of the Nobel prize in 1938 and an integral scientist in the American Nuclear programme during world war two. He developed methods to solve this problem and others like it using estimations. Watch the video and discover how many piano tuners are there in Chicago and how it can be solved.


Video: Michael Mitchell, animation: Mark Phillips. The lesson was produced for the TED-ED project.

The video also presents the question “how many M&Ms are there in the jar?” Want to try and solve it?
We will present here the solution to “how many jelly beans are in a one-liter jar?”

Image: Shutterstock
Image: Shutterstock

One liter is 1,000 cubic centimeters, meaning in a jar the size of one liter we can insert 1,000 cubes the size of one centimeter on each side.

We will now try and estimate the size of one jelly bean. A jelly bean is similar in shape to a small cylinder with the height of roughly two centimeters and the diameter of roughly 1.5 centimeter.

Do the jelly beans completely fill the jar? As you can see from the image, due to their disorganized shape the jelly beans will have gaps between them. Therefore we can estimate that the jar will only be 80% full.

The number of jellybeans that fit in the jar would be:

The volume of a cylinder is the surface of the circular base times its height, in our case:

$$\pi \cdot (\frac{1.5}{2})^2 \cdot 2$$

We can round $\pi$ down to 3 and solve:

$$3 \cdot (\frac{3}{4})^2 \cdot 2 =\frac{(3 \cdot 9 \cdot 2)}{16} = \frac{27}{8} $$

Meaning the volume of one jelly bean is $\frac{27}{8}$ cubic centimeters.

Therefore, the estimated number of jelly beans in a one liter jar would be:

$$\frac{(80\% \cdot 1,000)}{\frac{27}{8}} = \frac{(800 \cdot 8)}{27} = \frac{6,400}{27} = 237$$

And so we get 240 jelly beans. Now all that is left is to fill the jar with jelly beans, count them and examine how close we were to the correct answer.

What about the jar in the image above? How many jelly beans do you estimate are in there?