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A Fermi problem asks for a rough number when you seem to have almost no data: how many piano tuners work in a city, or how much energy a blast released. The name comes from the physicist Enrico Fermi, who won the 1938 Nobel Prize in Physics for demonstrating new radioactive elements produced by neutron irradiation and for discovering nuclear reactions brought about by slow neutrons. The skill is useful in exams too. A quick estimate tells you whether an answer is sensible before you mark it.
Fermi's two famous examples
The pieces of paper at Trinity. On 16 July 1945, Fermi watched the first nuclear test from about ten miles away. In his own written account, from a document held in the US National Archives, he describes dropping small pieces of paper from about six feet before, during and after the blast wave reached him, about 40 seconds after the explosion. The paper moved about 2½ metres, which he estimated corresponded to the blast from ten thousand tons of TNT. The US Department of Energy's history of the test gives the yield as finally calculated as 21,000 tons, more than twice Fermi's figure. That is impressive for a handful of paper and a few seconds of thought.
The piano tuners of Chicago. A NASA Glenn Research Center page retells a problem Fermi used to set his classes. Its reasoning goes:
Chicago has about 3 million people.
With about four people per family, that is about 750,000 families.
If one family in five owns a piano, there are about 150,000 pianos.
A tuner servicing four pianos a day, five days a week, for 50 weeks, tunes about 1,000 pianos a year.
So Chicago needs about 150,000 ÷ 1,000 = 150 piano tuners.
The page adds that this kind of estimate may be off by a factor of two or three, but it rules out 15 or 1,500.
How to do a Fermi estimate
Say exactly what you are estimating, with units.
Break it into factors you can guess: people, rates, times, sizes.
Round each factor, often to the nearest power of ten.
Multiply, keeping track of units.
Sanity-check against anything you know.
Rounding to powers of ten is the idea NCERT calls order of magnitude. NCERT's example: Earth's diameter is 1.28 × 10⁷ m, of order 10⁷ m, and a hydrogen atom's is 1.06 × 10⁻¹⁰ m, of order 10⁻¹⁰ m. So Earth is 17 orders of magnitude larger.
A biology example
How many times does your heart beat in a day? NCERT's circulation chapter gives 72 beats a minute. A day has 60 × 24 = 1,440 minutes. So 72 × 1,440 is about 100,000, or 10⁵ beats a day. If a calculation gives 10⁸, something is wrong.
Using estimates to check exam answers
You won't be asked to estimate piano tuners in JEE, NEET or boards. But a 20-second estimate can catch a slipped power of ten or a wrong unit before you move on.
Free fall. A stone dropped from 20 m: with g ≈ 10 m/s², v = √(2gh) = √400 = 20 m/s. If your answer is 200 m/s, recheck.
Moles. NCERT's first chemistry chapter gives the molar mass of water as 18.02 g mol⁻¹ and one mole as 6.02214076 × 10²³ particles. So a few grams of water holds about 10²³ molecules. If you get 10³⁰, a power of ten has slipped.
Circuits. A 2 Ω resistor across a 6 V supply carries 3 A. If you get 3 mA, check the units of each value.
Before each estimate, ask: what size should this be? A lab beaker can't hold a tonne of water, and a cricket ball can't weigh 50 kg.
How to practise this week
Pick one estimate a day and write it in five lines, as above. Try: the number of breaths you take in a day, the mass of air in your classroom, or the number of steps from your home to school. Then do your normal problem set with one change: write an expected size next to each question before you solve it.
When your estimate and your answer disagree and you can't find why, compare your steps with a full solution. The Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer, so you can see where the power of ten slipped.
For another 30-second check, read dimensional analysis for physics answers. For a fuller problem-solving method, see Polya's How to Solve It on a real physics problem. And for exam-day checking, use how to check your answers in the last 10 minutes.
Try the heartbeat estimate now, without a calculator.
