Fermi Questions: Back of the Envelope Calculations

The Art of Estimation
When I was in high school, in the days before electronic calculators were available, we learned how to use mechanical slide rules (see ChemMatters, April 2004, p.4) for our calculations. While they were great for getting a good answer to math problems, they didn’t keep track of the decimal place. That had to be done by keeping track of an order of magnitude estimate of the answer.

Fermi Questions
Physicist Enrico Fermi, creator of the first nuclear reactor was a genius in this kind of estimation. He was able to derive answers to problems by making reasonable estimates of values that were either too difficult to measure or were imprecise. People still refer to Fermi questions as challenges set up for doing this kind of estimates. Several science competitions have events based on student answers to Fermi questions.

Back of the Envelope Calculations
This kind of informal calculation is also referred to as ‘back of the envelope calculations‘. The goal is to quickly derive a ballpark estimate of the true value. Just like in a real baseball park, where the ball has to land somewhere within the confines of the field, the hope is that back of the envelope calculation is within a reasonably close range to the true answer.

But why would anyone want to do a back of the envelope calculation, rather than a precise calculation using a computer or calculator? Here is an example. Suppose you are in charge ordering pizza for a ChemClub meeting. How many pizzas do you order? You know there are 25 students who regularly attend. Each student could probably eat three slices. Although some may eat more, others might eat less. If 25 students each eat three pieces it means you need 75 pieces of pizza. If each pizza is cut into 8 pieces you need about 10 pizzas. (You can round 75 slices up to 80 and then divide by 8 to get 10 total).

25 students x (3 slices / 1 student) = 75 pizzas

round 75 pizzas up to 80 pizzas

80 pizzas x (1 pizza / 8 slices) = ~ 10 pizzas

While we don’t know if this is the precise amount we need, it is a good estimate for making sure everyone has enough pizza!

(Find out about the activity, including a prize, as part of the ChemClub Virtual Book Club onThe Martian.)

Why are Estimates Useful?
But why would a rough approximation ever be useful? The biggest reason is to find out what the reasonable approximation of an answer should be. Many times students put blind trust in the answers their calculators give them. I’ve had students tell me that the average mass of a molecule of carbon dioxide is 1.34 x1024 grams! That is as much as the Earth weighs! Clearly something went wrong as the student was entering their numbers into their calculator, but worse yet, they did not do a quick back of the envelope estimate to see if their answer was reasonable. Maybe it’s time to bring back slide rules?

A New Club Activity
Providing Fermi questions for your ChemClub could be a fun club activity. There are lots of sample problems on the internet, but here are a few for you to consider.

  1. How many times does a person’s heart beat in their lifetime?
  2. What fraction of volume does one molecule of water have in a glass of water?
  3. How much does the total student population in your school weigh, in grams?
  4. What is the dollar value of gold in a solid gold tooth?
  5. How much would the gasoline cost to drive from coast to coast across the USA?

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