In physics, there is a famous physicist named Enrico Fermi. You may remember that I have written about him before. One of the things I mentioned in that article that is named after him is the Fermi problem. This is a type of problem that physicists love. You may have heard them referred to before as a “back of the envelope” calculation (I have also heard it referred to as “street fighting math”). That essentially sums up this kind of problem. You use facts that you know and that are rounded to nice numbers that you can easily multiply together without the use of a calculator. Often, these kinds of estimates can provide questions where a more exact answer is impossible to find. These problems are used extensively through physics (I use them all the time in research) and other sciences as well. They provide a good starting point to look for a solution to a complicated problem. To better illustrate their power, I decided to sit down and work some out showing the power of the technique. I laid out some rules for myself in this challenge:
- I cannot look up any values. I can only use numbers I know or estimate.
- I cannot use a calculator to help with the math. I must be able to do it in my head.
Problem 1: What is the radius of the Earth?
To start out, it is always good to use as much background knowledge as you can that you know with some certainty. So, the first fact I will utilize is that I know there are 24 time zones around the earth corresponding to the 24 hours in the day. If I can estimate the width of each of these time zones, I can find the circumference of the earth and therefore the radius. So, with that goal in mind, I know that there are four of these time zones across the continental United States. Also, it is between 3,000 and 4,000 miles across that same distance. We can estimate that at 3,500 miles, but to be easily divisible by 4, I am going to estimate it as 3,600 miles. That means that in the US, the average time zone is about 900 miles across. However, we have to remember that the US is not at the equator where they will be bigger. So, let’s add 100 miles to the width of the average time zone, that makes the average time zone at the equator a nice, easy to work with 1,000 miles. Like I said before, there are 24 time zones meaning that the circumference of the earth is approximately 24,000 miles. To get the radius from the circumference, I need to divide by 2*Pi or about 6. That means my estimate for the radius of the earth is 4,000 miles. I purposely chose the first problem to be one where there is a measured value that is easy to look up to show how accurate these estimates can be. So, dear reader, the average radius of the earth is 3,959 miles! I was only off by 41 miles! That is an error of around 1%. I am going to have to ask you to trust me that I did this honestly because I was frankly stunned I got so close after I checked.
So, now that we know this technique can provide startlingly accurate answers, let’s move on to some problems where the answer is not easy or impossible to find. For the upcoming problems, there is no “correct” answer that I could find and I had to make quite a few approximations that I am not sure how accurate they are. So, if you have a better figure for me to use or a better way to approach the problem, let me know! I am not opposed to refining estimates.
Problem 2: How many Caucasian women in the USA make over $250,000 annually?
To start this problem, I know there are approximately 300 million people in the US. Of those, right around half are female. That means 150 million females in the US. Then, about 60% of the population is Caucasian (I think). That means I need to take 150 million, divide by ten to get 15 million and then multiply by 6 to get 90 million. This is the approximate number of Caucasian women in the US. Then, if Barack Obama is to believed, and a tax raise on people making over $250,000 a year would only effect 2% of the population, I need to take 90 million, divide by 100 to get 0.9 million and then multiply by 2 to get 1.8 million. So, my final estimate for the number of Caucasian women making over $250,000 a year is around 1.8 million in the US. This is about 1 in every 200 people. What do you think? Does this sound about right?
Problem 3: How many customers need to pass through a shop in Salem, MA in order for them to make a net profit?
First, a little background on this question. My girlfriend and I recently visited Salem, MA and were stunned by the number of shops that sell the same kind of merchandise (witch memorabilia). It made me wonder how all of those shops stay in business if they are in such direct competition with each other. So, I decided to use the experience to see if they would need a huge number of customers in order to stay afloat.
Now, let’s get estimating! I am going to estimate that the average shop employs around 6 people on an average day working for 8 hours a day. This accounts for the fact that the shop is open for more than 8 hours and therefore there is not always 6 people in the store. If they are making just above minimum wage, say $9 an hour, that puts their total pay at around $500 for the day. Now, I am going to say that the store needs to sell roughly that much in merchandise each day to justify that much in wage expenses. That brings the total expenses to $1,000. Then, let’s increase that number by 50% for other expenses such as rent, utilities, and benefits bringing the total daily expenses to about $1,500. Now, I am going to estimate that the average price of an item in the store is about $20 (say for a sweatshirt). Also, it takes about ten customers going through the shop to buy one item. That means for every ten customers, the store takes in $20. So, to break even, the store needs 75 purchases of $20, or 750 customers to enter the store. This number seems a bit high to me, but when we went to the shops, it was kind of late and they were getting ready to close, so I may have not seen them very busy (it was also a day with poor weather). If anybody has retail experience at a shop like this, is this close to the mark or did I completely miss? Let me know!
Problem 4: What is the total lost revenue for the federal government due to the current unemployment level?
This is a pretty quick and simple one, but has a lot of ramifications related to budget debates in Washington DC. Again, I will start this problem with the estimate that there are 300 million people in the US. The current unemployment (if you haven’t been following the news) is sitting at around 10%. That means 30 million people. If the average job in the US pays around $40,000 a year (I don’t know if this is a good estimate or not), then those 30 million people making $0 would instead be making $1.2 trillion dollars if they were not unemployed. Now, I am going to guess that the average person pays a total tax rate of about 10% (again, not sure if this is good or not). That means, of that $1.2 trillion, the US government would get $120 billion. So there you have it, if everybody in the US was employed, the federal government would be about $120 billion richer every year. That number may seem large, but compared to the massive size of the budget, it is only about four percent, just a little more than a drop in the bucket.
So, now that you have seem some examples of the Fermi problem, I encourage you to try it out yourself. It is fun to try to figure out answers to crazy questions. I know Google loves to ask these kinds of problems at interviews. One I know they have asked before is “How many golf balls would fit in a school bus?” Fermi himself once came up with an estimate for the number of piano tuners in the city of Chicago. I have also found that using facts like the width of the continental US in problems like this helps you remember them better so that you can impress your friends at a trivia competition.