How many piano tuners are in Chicago?
Tonight's idea
Enrico Fermi, who built the first nuclear reactor, liked to ask his students questions nobody could look up. The method he taught has three parts.
Break it into factors you can guess. You cannot guess the number of piano tuners, but you can guess how many people live in a city, roughly how many own a piano, how often a piano is tuned, and how many pianos one tuner can do in a year.
Round brutally. Round every number to 1, 3 or 10 times a power of ten: 7 becomes 10, 25 becomes 30, 400 becomes 300. Why 3? Because \(3 \times 3 \approx 10\), so 3 is the "middle" between 1 and 10 when you multiply.
Let the errors partly cancel. Each guess may be too high or too low. If the guesses are independent, the highs and lows tend to cancel: a factor of 2 up and a factor of 2 down multiply back to 1. That is good enough to tell 10 from 1,000 from 100,000. The danger is when every guess leans the same way, so push back on each one.
The same habit catches homework mistakes. Before you trust an answer, ask: is it roughly the right size? Are the units right? Should it be positive? A car that travels 3 km in a year or a classroom with 0.4 students has told you something went wrong.
Watch one: piano tuners
People in Chicago: about 3 million. People per household: about 3, so 1 million households. Households with a piano: maybe 1 in 20, so 50,000 pianos. Tunings per piano per year: about 1, so 50,000 tunings a year.
A tuner takes about 2 hours per piano including travel, so 4 a day. Working 250 days a year, that is 1,000 tunings per tuner per year.
\(50{,}000 \div 1{,}000 = 50\) tuners. Real listings for Chicago come to somewhere between 50 and 150. The estimate is in the right neighbourhood, and we never looked anything up.
Your turn
Write your chain of guesses, not just the answer. The chain is the point.
- How many seconds are in a year? Estimate first, then compute. Compare your number with \(\pi \times 10^7\).
- How many breaths will you take in your life? (How many per minute? Minutes in a day? Days in a long life?)
- How many words are in all seven Harry Potter books together? (Pages per book, words per page.)
- How many school buses would it take to move everyone in your town or suburb at once?
- How many pizzas does a city of one million people eat in a week?
- A homework answer says a bath holds 1,800 litres. Use a quick estimate (length × width × depth) to decide whether to believe it.
Stretch: hairs on a head
Estimate the number of hairs on a human head. You will need the area of a scalp (think of it as roughly half a sphere, or just a circle of some diameter), and how many hairs grow in one square centimetre (look at a small patch and guess). Then multiply. Biologists put the answer around 100,000. How close did you get, and which of your guesses was most uncertain?
Answers
Fermi answers are ranges. Anything within a factor of three of these is a good estimate.
- \(365 \times 24 \times 60 \times 60 = 31{,}536{,}000 \approx 3.15 \times 10^7\). And \(\pi \times 10^7 = 31{,}415{,}927\). It is a coincidence, but a handy one: physicists really do use "pi times ten to the seven seconds" as a year.
- About 15 breaths a minute, 1,440 minutes a day, so about 20,000 a day; times 365 is about 7 million a year; times 80 years is about \(6 \times 10^8\), six hundred million.
- About 4,000 pages in total, about 300 words a page, so about 1.2 million. The actual count is about 1.08 million.
- Depends on your town. A bus holds about 50. A town of 20,000 needs about 400 buses; a city of 2 million needs about 40,000.
- If one person in ten eats pizza on a given day and a pizza feeds two, that is 50,000 pizzas a day, so about 350,000 a week. Anything from 100,000 to a million is defensible.
- A bath is roughly 1.5 m long, 0.6 m wide, 0.4 m deep: \(1.5 \times 0.6 \times 0.4 = 0.36\) cubic metres, which is 360 litres. 1,800 litres is five times too big, so the answer is wrong (perhaps a unit slip).
Stretch. A scalp is about 600 square centimetres (a circle of diameter 28 cm gives about 615). Hair density is roughly 150 to 200 per square centimetre. \(600 \times 170 \approx 100{,}000\). The density guess is usually the shaky one; most people guess far too low until they actually look at a square centimetre.