Fermi estimates
Chain rough numbers to size up anything.
Best in your head · no pen, no calculator why?
Opens at level 17. You're level 1. You can read and practice here now.
the idea
A Fermi estimate sizes up something you can't look up by chaining numbers you can. How many barbers does a city need? People times haircuts a year gives the demand. One barber's haircuts a year gives the supply. Divide one by the other.
Write the unit on every number so you can see them cancel: minutes a day × days gives minutes. Convert to the unit asked for at the end, with the conversions you are given. Often you need only the right power of ten, not the last digit. The classic slip is losing a factor of ten somewhere along the chain.
techniques
Demand over capacity
- Demand: start from the population, narrow it to who needs the service, and multiply by how often.
- Capacity: what one worker or vehicle handles in the same period.
- Divide demand by capacity.
- Count the zeros to check the power of ten before you choose.
worked example
A town has 100,000 people. Each gets a haircut every 10 weeks (52 weeks a year). A barber does 13 haircuts a day, 200 days a year. How many barbers does the town need?
- Haircuts a year: 100,000 × 52 ÷ 10 = 520,000.
- One barber: 13 × 200 = 2,600 a year.
- 520,000 ÷ 2,600 = 200 barbers.
Answer: 200
Rate × time, then units
- Put the period in the time unit of the rate: seconds for a drip every few seconds.
- Multiply the rate by the period. Minutes a day × days gives minutes.
- Then convert to the unit asked for: ÷ 60 for minutes to hours, ÷ 1,000 for millilitres to litres.
- Round only at the end, as the question asks.
worked example
You spend 40 minutes a day waiting in line. How many hours is that in a year of 365 days? Round to the nearest hour.
- 40 × 365 = 14,600 minutes a year.
- 14,600 ÷ 60 ≈ 243.3 hours.
- Rounded: 243 hours.
Answer: 243
watch out for
- Losing a factor of ten when dividing millions by thousands. Count the zeros on both sides before you choose.
- Stopping in the wrong unit: millilitres instead of litres, or minutes instead of hours.
- Reading "a haircut every 6 weeks" as 6 a year. It is 52 ÷ 6, about 8.7 a year.
practice
Small rate, long time
worked example
You spend 20 minutes a day looking for misplaced things. How many hours is that in a year of 365 days? Round to the nearest hour.
Answer: 122 hours
- 20 × 365 = 7,300 minutes a year.
- ÷ 60 ≈ 121.67 hours, about 122 hours.
Estimate from given facts
worked example
A city has 200,000 people, and each person visits a dentist twice a year. A dentist sees 20 patients a day, 220 days a year. About how many dentists does the city need?
- 9
- 9,000
- 900
- 90
Answer: 90
- Visits a year: 200,000 × 2 = 400,000.
- One dentist sees 20 × 220 = 4,400 patients a year.
- 400,000 ÷ 4,400 ≈ 91, so about 90.