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Estimation and Fermi Questions

Date: 04/10/2002 at 13:10:05
From: Kenneth S. Kwan
Subject: Very general estimation question

Dear Dr. Math,

I'm currently learning about Estimation techniques similar to those 
used by the famous scientist Enrico Fermi, who proposed the question, 
"How many piano tuners are there in Chicago?"

The question I invented was the following:

If there are 12,000 students who attend a certain college, how many 
professors are employed by the college?

I came up with the following estimation. Can you tell me if my 
reasoning is reasonable? Thanks very much!

1) The average professor teaches about 1 hour a day, so in one week 
   (Monday - Friday) he teaches 5 hours. 

2) Each class takes about an hour, so in one week he teaches 5 

3) But he doesn't teach 5 different classes in one week; the same 
   classes are held 2-3 times a week (either Monday-Wednesday-Friday 
   classes or Tuesday-Thursday classes). To make it simpler, let's say 
   a professor teaches each class 2 times a week (assume only Tuesday-
   Thursday classes exist). 

   Therefore, he sees the same class 2 times a week, meaning every 
   half week he sees the same students, but that also means every week 
   he only sees the same students because the classes repeat. 

4) If a professor teaches for 2.5 hours per half week (5 hours per 
   week), where each class takes about an hour, and assuming that a 
   typical class consists of 50 students, then he sees 2.5 classes x 
   50 students = 125 students per week. 

5) Since there are 12,000 students for all professors to lecture, 
   then there are probably 12,000 / 125 = close to 100 professors on 

From a scale of 1-10, how would you rate this estimation? Any 
suggestions or comments are greatly appreciated.

Kenneth S. Kwan

Date: 04/10/2002 at 17:03:38
From: Doctor Peterson
Subject: Re: Very general estimation question

Hi, Kenneth.

It seems to me that you have left out one important factor: each 
student attends more than one class.

This can be tricky to describe clearly: in your model each class has 
50 students, and each student has, say, 5 classes. So is it 50 
students per class, or 5 classes per student?

One way to avoid this trouble is to think of a specific name for the 
relation of a student to a class. One I've thought of is "seat." Each 
class has 50 seats (students in the class); each student has 5 seats 
(in different classes). You can diagram this:

    Student ------> Seat <------ Class <------ Prof
            1     5      50    1       2.5   1

Looking for more on Fermi questions, I ran across these pages which 
may be of interest:

   Classic Fermi Questions with annotated solutions - Sheila Talamo 

   How To Solve Fermi Questions - Norman Rothery 

But I didn't find your question among these.

- Doctor Peterson, The Math Forum 

Date: 04/10/2002 at 19:56:01
From: Kenneth S. Kwan
Subject: Very general estimation question

Thanks very much Doctor Peterson for answering my "Fermi" question, 
but now I have a question about your answer:

The diagram you gave me was:

    Student ------> Seat <------ Class <------ Prof
            1     5      50    1       2.5   1

I'll understand that as:

Each student has 5 seats. There are 50 seats in one class. But does 
that mean each class holds only 10 students? I'm not sure I understand 
this concept. And I thought that the total number of students (12,000) 
attending the college would matter in the estimation. Can you explain 
this further? 

Thanks very much!


Date: 04/10/2002 at 22:28:44
From: Doctor Peterson
Subject: Re: Very general estimation question

Hi, Kenneth.

Maybe you can get a better understanding if you try to tell me what 
role the number of classes each student takes should play in your 
estimate. It takes a bit of wrestling on your own before you can quite 
pin this idea down.

Obviously I didn't say that each class has only 10 students; but if 
each student took only one class of ten students, you would need the 
same number of professors, so in a sense it is equivalent to that. Or, 
instead of 12,000 students taking 5 courses each, you could have 
60,000 students taking one course each; that would require the same 
number of professors, which is five times as many as you estimated. 
Does that help?

And I didn't say the number of students doesn't matter; my diagram 
(a variety of Entity Relationhip Diagram) only shows relative numbers, 
not absolute numbers. It says that for each student there will be 
5 seats (in different classes), each of which is 1/50 of a class, each 
of which needs 1/2.5 of a professor, so 12,000 students will need

                    5 seats    1 class    1 professor
 12,000 students * --------- * -------- * ----------- = 480 professors
                   1 student   50 seats   2.5 classes

There's a lot of useful math and logic in this question!

- Doctor Peterson, The Math Forum 

Date: 04/11/2002 at 14:43:16
From: Kenneth S. Kwan
Subject: Very general estimation question

Dear Dr. Peterson,

Thanks very much, I understand your solution. 

One last general question on how I can sort of think the way you do, 
so to speak. What you said about how I missed a link between the 
students and the class, where we're not sure if each student has 5 
classes or if each class has 50 students, makes me ponder about how 
you realized and picked that up. Have you worked with problems such as 
these repeatedly, so you know all the nitty gritty details? Or is it 
because I'm illogical or have less ability to reason that I totally 
neglected that fact. If so, is there a particular math course I can 
take to help open up my mind to incorporate more reason and logic into 
my mathematical thinking? I feel incompetent when I see other students 
being able to answer such questions while I'm struggling. Can you give 
some advice?

Thank you once again, Dr. Peterson!


Date: 04/11/2002 at 16:17:42
From: Doctor Peterson
Subject: Re: Very general estimation question

An interesting question!

I suspect a lot comes from experience - and that experience is 
probably what Fermi was trying to develop with his questions. Logical 
ability is not just something you are born with (though it may be that 
some of us naturally gravitate toward it, and therefore develop the 
skills without having to be forced into it); I think everyone has to 
develop it by practice.

In this case, as with the piano tuner problem, a lot of the thinking 
needed depends on specific knowledge of the subject matter. You have 
to picture a university and know that there are classes and professors 
and so on, or picture a piano tuner's job and see that he will do more 
than one piano a day, and they will be in different homes, and so on. 
In this case, I just thought about what factors would play a role, saw 
that each student would be in several classes, and expected to see 
that somewhere in your analysis. When I didn't, a red flag went up. No 
logic, just visualization. But I probably would have noticed it 
anyway, by going through your presentation in order  and falling off 
the end when you didn't mention what each student does. Maybe you can 
call that "follow-through" - you can't stop your logic just because 
you've made contact with the goal, but have to keep thinking until you 
can't think any further. You did fine until then; you just didn't take 
it all the way.

Assuming that "problem-domain" knowledge, you have to be able to think 
through the problem, both in a straight line (each professor teaches N 
classes; in each class there are N students; ...), and also sometimes 
coming at it from all sides, just brainstorming to think of all the 
relevant factors. The former requires the ability to stay focused and 
think in an orderly way; the latter requires defocusing and letting 
wild ideas come in. Both have their place, and some of us are probably 
better at one than the other. In this case, to find my own answer, I 
just went through it sequentially, starting at the student (since I 
knew the hard part was at that end).

Now I happen to have an advantage over you in doing a Fermi problem, 
one that I haven't seen mentioned in discussions of them. I am a 
computer programmer, and part of my work involves designing relational 
databases, where you might have one table listing all the students, 
another listing all the classes, and so on. That's where my Entity 
Relationship Diagram came from - it's a tool used to see how these 
tables relate to one another, and design additional tables that 
capture the information needed for these relations. While I was 
thinking about how to explain the "5 classes per student and 50 
students per class" problem, that method popped into my mind - I 
suppose that is an example of the non-linear type of thinking, pulling 
a tool out of my toolbox because the kind of thinking I was doing 
reminded me of it, not because it was the next thing to think of. 
That's not essential for this kind of problem, but it can always be 

Speaking of databases, if I had actually been trying to design one, I 
would have based my thinking on the paperwork involved in a 
university. Each of the 12,000 students has a course schedule, listing 
his or her 5 subjects. Each of those 60,000 subject lines corresponds 
to one seat in one class; 50 of them together form one of 1200 
classes. Each professor has a schedule listing 2 or 3 courses, among 
which the 1200 classes will be found. In the database, you would add a 
table containing the information from all the students' schedules, 
which relates students to classes. (That's why it's called a 
"relational" database.) This image of actual pieces of paper can often 
make the abstract ideas of numerical ratios more concrete, and help in 
logical thinking.

Where can you build these skills? Many math classes will incorporate 
them implicitly, in geometric proofs or word problems, for example. 
Other fields need it too, from Fermi's physics to law school. There 
are all sorts of puzzles you can find in books or elsewhere.

I went back to the Web to look for articles on Fermi problems that go 
a little deeper in talking about the kind of thinking involved. This 
is a long discussion that seems very helpful:

   Fermi Problems - Discovery Learning Project 

This looks good, too:

   'Back-of-the-Envelope' Calculations - Paul Francis 

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Logic

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