Reading here and there on the internet, I stumbled upon a text by Lawrence Weinstein and John A. Adam titled "More or Less How Much?", just to quote the source, which is very pleasant and concerns estimates for making approximations. It shows us that when faced with a doubt, for example when buying a pair of shoes, we can have three perplexities: it costs too much, it costs too little, or it is an adequate cost. Adequacy usually deviates from the maximum and minimum limits we set by a factor of 10. Continuing to read, I come to an example: "Piano Tuners". I will quote the text below...
"Now let's try a more difficult problem: how many piano tuners are there in a city like Rome?"
This example - similar to the original devised by Enrico Fermi - is used at the beginning of many physics courses because it forces one to use methods and reasoning common to all problems of this type, but without requiring specific knowledge of physics.
Solution: this problem is so complicated that you cannot simply estimate the answer: to solve it, we must break it down into more limited questions. We must estimate (1) how many pianos there are in Rome and (2) how many pianos each tuner can handle.
To estimate the number of pianos, one needs to know (a) the population of the city, (b) the fraction of the population that owns a piano, and (c) the number of schools, conservatories, and so on where pianos are located.
Then, to estimate how many pianos each tuner can handle, we will need to know (A) how often pianos are tuned, (B) the time required for the operation, and (C) how many hours each tuner works.
Overall, to solve the problem, we must estimate the following things:
1. The population of Rome
2. The number of pianos per inhabitant
3. The frequency with which a piano is tuned
4. The time required to tune it
5. The annual working hours of each tuner
Let's start from the top.
1. The population of Rome must be less than 10^7 (since the Italian population is 6 x 10^7) and greater than 10^5 (the inhabitants of a medium-sized city); we will estimate it at 10^6.
2. Piano owners are families and schools. About 10% of the population plays a musical instrument (certainly the fraction is more than 1% and less than 100%). Not all musicians play the piano or own one, so we estimate that 3% of musicians own a piano; this is equivalent to 3% x 10% = 3 x 10^-3 of the population (three per 1000 inhabitants).
The number of schools with a piano will be more or less one for every 500 students, i.e., about one for every 1000 inhabitants, which is 10^-3 pianos/inhabitant.
In total, there will therefore be about 4 x 10^-3 pianos per person, and the total number of pianos in Rome will be more or less 10^6 x 4 x 10^-3 = 4 x 10^3.
3. Pianos are probably tuned less than once a month and more than once every ten years; we will estimate the frequency to be once a year.
4. Tuning a piano (that is not too out of tune) must take more than 30 minutes and less than a day; we will estimate that 2 hours are needed. Another way to consider the matter is that the piano has 88 keys; dedicating one minute to each key would take 1.5 hours, while two minutes per key would take 3 hours.
5. A full-time job takes 8 hours a day, for 5 days a week and 50 weeks a year, which is equivalent to 8 x 5 x 50 = 2000 hours.
Since we have estimated two hours per piano, in one year a hard-working tuner manages to tune about 1000 pianos (wow!).
All this means that the 4 x 10^3 pianos in the city of Rome require 4 tuners.
How close did we get? Well, on the Yellow Pages of Rome under the entry > there are seven advertisements for companies that deal with tuning them. Probably each advertisement corresponds to a couple of tuners, who perhaps do not dedicate themselves to this activity full-time.
This means that our estimate is probably too low by a factor of two or three. And it's still much more accurate than the number we would have obtained by guessing. Remember that we are only trying to estimate the answer within a factor of ten.