courses › Thinking

Fermi estimates

level 17 course

Chain rough numbers to size up anything.

Best in your head · no pen, no calculator why?

Opens at level 17.

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

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

How many workers, shops or vehicles a place needs.

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

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

A small steady rate over a long period, with a change of 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

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

Tips by skill

TipSmall rate, long time: Put the period in the rate's time unit and multiply. Then convert to the unit asked for, and round at the end.
TipEstimate from given facts: Divide total demand by what one worker or vehicle handles in the same period, then check the power of ten.

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.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

Small rate, long time par 25 s not tried yet
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
1. 20 × 365 = 7,300 minutes a year.
2. ÷ 60 ≈ 121.67 hours, about 122 hours.
Estimate from given facts par 45 s not tried yet
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?
1. 9
2. 9,000
3. 900
4. 90
Answer: 90
1. Visits a year: 200,000 × 2 = 400,000.
2. One dentist sees 20 × 220 = 4,400 patients a year.
3. 400,000 ÷ 4,400 ≈ 91, so about 90.
Real-world magnitudes coming soon not tried yet

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1 day
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60 days
mastered · every 90 days

your rounds

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