Guesstimate and Fermi questions, with worked answers

Ask first

• The city of Chicago only, or the whole metro area?
• Full-time tuners, or anyone who tunes pianos part of the time?

One way in: people in Chicago ÷ people per piano (homes, schools and churches together) × tunings per piano per year ÷ tunings one full-time tuner does in a year (4 a day, 250 days)

From the people: 2.7 million people in Chicago ÷ 50 people per piano (homes, schools and churches together) × 1 tunings per piano per year ÷ 1,000 tunings one full-time tuner does in a year (4 a day, 250 days) = 54

From the households: 1.1 million households in Chicago × 4% share with a piano + 8,000 pianos in schools, churches, studios and venues × 0.9 tunings per piano per year (some never, some twice) ÷ 1,100 tunings per tuner per year (5 a day, 220 days) = 43

The routes are 27% apart, so give a range: 43 to 54 full-time piano tuners. Order of magnitude: about 10².

Tip: This is the question Enrico Fermi is said to have asked his students. The skill is the chain: people, pianos, tunings, tuners. Say each link before you put a number on it.

Full walkthrough with the follow-up questions

Ask first

• Is the bus empty, with the seats taken out, or do we pack around the seats?
• A full-size school bus, or a short one?

One way in: feet of usable interior length × feet of interior width × feet of height inside × cubic inches in a cubic foot × share of the space balls fill when poured in ÷ cubic inches per golf ball (1.68-inch diameter)

By volume: 35 feet of usable interior length × 7.5 feet of interior width × 6 feet of height inside × 1,728 cubic inches in a cubic foot × 64% share of the space balls fill when poured in ÷ 2.48 cubic inches per golf ball (1.68-inch diameter) = 702,348

By rows: 250 balls in a row along 420 inches (420 ÷ 1.68) × 54 rows across 90 inches (90 ÷ 1.68) × 43 layers up 72 inches (72 ÷ 1.68) = 580,500

The routes are 21% apart, so give a range: 580,000 to 700,000 golf balls. Order of magnitude: about 10⁶.

Tip: Volume questions are three lengths multiplied. Then say the packing out loud: round balls never fill a box, and a neat grid holds fewer than balls that settle into the gaps.

Ask first

• Gasoline only, or diesel and jet fuel too?
• Personal vehicles only, or delivery vans and trucks as well?

One way in: light vehicles on US roads (cars, SUVs, pickups) × miles driven per vehicle per year ÷ miles per gallon, on average

From the vehicles: 250 million light vehicles on US roads (cars, SUVs, pickups) × 11,500 miles driven per vehicle per year ÷ 23 miles per gallon, on average = 125 billion

From the gas stations: 145,000 gas stations in the US × 2,500 gallons sold per station per day × 365 days per year = 132 billion

Say about 130 billion gallons of gasoline per year. Order of magnitude: about 10¹¹.

Per American, that is about 380 gallons of gasoline per year. That is over 7 gallons a week for every man, woman and child, which fits most adults driving and filling up about once a week.

Real figure: The US Energy Information Administration puts US gasoline use at about 136 billion gallons in 2023. EIA: How much gasoline does the United States consume?

Tip: Two routes that share no inputs, vehicles on one side and stations on the other, are the strongest check you can give.

Ask first

• All aircraft, including private and cargo planes, or airline flights only?
• A busy weekday afternoon, or any moment including the middle of the night?

One way in: flights the FAA serves on a typical day (it says more than 44,000) × hours per flight, on average ÷ hours in the busy part of the day, when most flights happen

From the flights: 44,000 flights the FAA serves on a typical day (it says more than 44,000) × 2 hours per flight, on average ÷ 16 hours in the busy part of the day, when most flights happen = 5,500

From the fleet: 7,000 US airline passenger and cargo jets × 60% share flying at a busy moment + 1,500 private, training and other small planes flying at the same time = 5,700

Say about 5,600 aircraft in the air. Order of magnitude: about 10⁴.

Tip: Snapshot questions (how many at one moment) are a daily total times the time each one lasts, divided by the hours in the day. The same trick sizes a queue or a parking lot.

Ask first

• An average American, living to the average life expectancy?
• Do we allow for exercise and sleep, or use one resting rate?

One way in: beats per minute at rest, on average × minutes in a year (60 × 24 × 365) × years in a lifetime (US life expectancy at birth is about 79)

One average rate: 70 beats per minute at rest, on average × 525,600 minutes in a year (60 × 24 × 365) × 79 years in a lifetime (US life expectancy at birth is about 79) = 2.91 billion

Awake and asleep: 72,000 beats in 16 waking hours at about 75 a minute (75 × 60 × 16) + 28,800 beats in 8 hours of sleep at about 60 a minute × 28,850 days in 79 years (79 × 365, plus leap days) = 2.91 billion

Say about 2.9 billion heartbeats. Order of magnitude: about 10⁹.

Tip: Know 525,600 minutes in a year and about 31.5 million seconds. Time-based Fermi questions often need one of them.

Ask first

• Spoken out loud only, not typed or texted?
• A typical workday, or any day of the week?

One way in: hours a day spent talking × minutes per hour × words per minute in normal conversation

From talking time: 2 hours a day spent talking × 60 minutes per hour × 130 words per minute in normal conversation = 15,600

From conversations: 25 conversations in a day, short and long × 8 minutes per conversation × 50% share of the time you are the one talking × 130 words per minute = 13,000

The routes are 20% apart, so give a range: 13,000 to 16,000 words a day. Order of magnitude: about 10⁴.

Real figure: A study published in Science in 2007, which recorded 396 people for several days, found that women and men both spoke about 16,000 words a day. Mehl and others (2007), Science: Are women really more talkative than men?

Tip: When a question is about one person's day, build it from time: hours doing the thing, times the rate while doing it.

Ask first

• Along roads, not in a straight line?
• An average adult's stride, walking rather than running?

One way in: miles of walking route, New York to Los Angeles × feet per mile ÷ feet per step (a typical adult step length)

By distance: 2,800 miles of walking route, New York to Los Angeles × 5,280 feet per mile ÷ 2.5 feet per step (a typical adult step length) = 5.91 million

By time: 933 hours of walking at 3 miles an hour (2,800 ÷ 3) × 6,000 steps per hour (about 100 a minute) = 5.6 million

Say about 5.8 million steps. Order of magnitude: about 10⁷.

Tip: Distance and time give two routes, but both start from the 2,800 miles, so check the rates against each other: 3 miles an hour at 100 steps a minute is a 2.6-foot step, close to the 2.5 feet in the first route.

Ask first

• Live births in the 50 states and DC only?
• An average day, not a particular day of the week?

One way in: people in the US × births per person per year (about 11 per 1,000) ÷ days per year

From the population: 340 million people in the US × 1.1% births per person per year (about 11 per 1,000) ÷ 365 days per year = 10,247

From the mothers: 64 million US women aged 15 to 44 × 5.5% share who give birth in a year ÷ 365 days per year = 9,644

Say about 9,900 births a day. Order of magnitude: about 10⁴.

Real figure: The CDC counted 3,628,934 US births in 2024, about 9,900 a day. CDC National Center for Health Statistics: Births and Natality

Tip: Rates per 1,000 are easy to misplace by a factor of 10. Say the yearly total out loud before you divide by 365.

Ask first

• All five boroughs?
• Pet dogs only, not working dogs or dogs in shelters?

One way in: households in New York City × share that own a dog (lower than the suburbs) × dogs per dog-owning household

From the households: 3.2 million households in New York City × 16% share that own a dog (lower than the suburbs) × 1.2 dogs per dog-owning household = 614,400

From the people: 8.3 million people in New York City ÷ 15 people per dog in a dense city (small apartments, few yards) = 553,333

Say about 580,000 pet dogs. Order of magnitude: about 10⁶.

Tip: Adjust national habits for the place. A dense city with small apartments has fewer dogs per household than the country as a whole, so say why your share is lower.

Ask first

• Yellow school buses for K to 12 only, not transit or charter buses?
• Public and private schools, or public only?

One way in: public school students, K to 12 × share who ride a school bus ÷ riders per bus per day (about 36 a run, 1.5 runs)

From the riders: 49 million public school students, K to 12 × 50% share who ride a school bus ÷ 54 riders per bus per day (about 36 a run, 1.5 runs) = 453,704

From the schools: 99,000 public schools × 5 buses per school, on average = 495,000

Say about 470,000 school buses. Order of magnitude: about 10⁶.

Tip: Buses run more than one route a morning. Forget the runs per bus and your count comes out far too high.

Ask first

• End zones included?
• A large pizza is about 14 inches across?
• Laid flat with no overlap, so small gaps between them are fine?

One way in: feet long, end zones included × feet wide ÷ square feet per 14-inch pizza (π × 7² = 154 square inches) × share of the field circles cover when packed tightly

By area: 360 feet long, end zones included × 160 feet wide ÷ 1.07 square feet per 14-inch pizza (π × 7² = 154 square inches) × 90% share of the field circles cover when packed tightly = 48,449

By rows: 308 pizzas along the length (4,320 inches ÷ 14) × 137 pizzas across the width (1,920 inches ÷ 14) = 42,196

Say about 45,000 large pizzas. Order of magnitude: about 10⁵.

Tip: Circles in a square grid cover about 79% of the space. Staggered like a honeycomb they cover about 91%. Name the gap, and your range explains itself.

Ask first

• Emails written by people, not automated notices or marketing?
• Sent, not received?

One way in: US workers × share who use email at work × emails each sends per workday

From the senders: 160 million US workers × 60% share who use email at work × 40 emails each sends per workday = 3.84 billion

From desk hours: 160 million US workers × 45% share in office or desk jobs × 7 hours at a desk per day × 7 emails sent per desk hour = 3.53 billion

Say about 3.7 billion work emails a day. Order of magnitude: about 10¹⁰.

Tip: Segment early. A nurse and an analyst both work, but only one sends 40 emails a day.

Ask first

• An adult with a full head of hair, not counting the face or body?
• Scalp hair only?

One way in: square inches of scalp with hair (a hand is about 20) × hairs per square inch

In square inches: 120 square inches of scalp with hair (a hand is about 20) × 1,000 hairs per square inch = 120,000

In square centimeters: 600 square centimeters of scalp with hair × 170 hairs per square centimeter (about one every millimeter) = 102,000

The routes are 18% apart, so give a range: 100,000 to 120,000 hairs. Order of magnitude: about 10⁵.

Tip: Measure with your own body. Your palm is about 20 square inches, and a scalp is a few palms wide and a few long.

Ask first

• Article text only, not talk pages, images or references?
• Reading at a normal pace, 8 hours every day of the year?

One way in: English Wikipedia articles × words per article, on average ÷ words read per minute ÷ minutes per hour ÷ reading hours per year (8 a day × 365)

From the words: 7 million English Wikipedia articles × 650 words per article, on average ÷ 250 words read per minute ÷ 60 minutes per hour ÷ 2,920 reading hours per year (8 a day × 365) = 104

From the articles: 7 million English Wikipedia articles × 3 minutes to read one article (most are short) ÷ 60 minutes per hour ÷ 2,920 reading hours per year (8 a day × 365) = 120

The routes are 15% apart, so give a range: 100 to 120 years of reading. Order of magnitude: about 10².

Tip: Long chains of divisions are where answers slip by a factor of 60. Say the unit after every step: words, minutes, hours, years.

Ask first

• One independent shop, open every day of the week?
• Drinks and food together?

One way in: customers per day × average ticket × days open per year

From customers: 500 customers per day × $7 average ticket × 360 days open per year = $1.26 million

From the hours: 280 orders in the 4 morning rush hours (70 an hour) + 220 orders in the other 10 open hours (22 an hour) × $6.5 average ticket × 360 days open per year = $1.17 million

Say about $1.2 million in sales per year. Order of magnitude: about 10⁶.

Tip: For any one shop, the bottleneck is the rush. Seventy orders an hour is a little over one a minute, which is about what one register and a fast bar can do.

Ask first

• A station with a convenience store, open every day?
• Revenue, not profit?

One way in: gallons of fuel sold per day × price per gallon + in-store sales per day × days per year

Fuel plus store: 2,500 gallons of fuel sold per day × $3.3 price per gallon + $2,500 in-store sales per day × 365 days per year = $3.92 million

From visits: 300 visits per day (fuel, store or both) × $40 average spend per visit, fuel plus snacks × 365 days per year = $4.38 million

Say about $4.2 million in sales per year. Order of magnitude: about 10⁷.

Tip: Most of a gas station's revenue is fuel, but most of its profit is the store. Say that out loud and you have your follow-up answered.

Ask first

• Tickets and the snack bar together?
• A suburban multiplex, not a downtown flagship?
• Is it the only multiplex within a 15-minute drive?

One way in: screens × showings per screen per day × seats per auditorium × share of seats filled, on average × days per year × spend per ticket: about $11 ticket plus $6 at the snack bar

From the seats: 12 screens × 5 showings per screen per day × 200 seats per auditorium × 12% share of seats filled, on average × 365 days per year × $17 spend per ticket: about $11 ticket plus $6 at the snack bar = $8.94 million

From the moviegoers: 250,000 people living within a 15-minute drive × 2.4 movie visits per person per year × 75% share of those visits that come to this theater × $17 spend per ticket, ticket plus snacks = $7.65 million

The routes are 17% apart, so give a range: $7.7 million to $8.9 million in sales per year. Order of magnitude: about 10⁷.

Tip: Capacity questions need a utilization number. Theaters sell far fewer seats than they have, and saying so shows you understand the business.

Full walkthrough with the follow-up questions

Ask first

• One automated tunnel, open most days of the year?
• Do monthly memberships count, or single washes only?

One way in: cars washed per day × average price per wash × days open per year

From cars a day: 300 cars washed per day × $13 average price per wash × 350 days open per year = $1.36 million

Members plus singles: 2,500 monthly members × $30 membership price per month × 12 months per year + $420,000 single-wash sales per year (100 a day at $12, 350 days) = $1.32 million

Say about $1.3 million in sales per year. Order of magnitude: about 10⁶.

Tip: When a business sells subscriptions, size the members and the walk-ins separately. They behave differently and an interviewer will ask about both.

Ask first

• One restaurant with a drive-thru?
• Burgers only, not chicken sandwiches?

One way in: yearly sales of one fast-food burger restaurant ÷ days open per year ÷ average order × burgers per order (many orders are chicken, drinks or breakfast)

From yearly sales: $1.8 million yearly sales of one fast-food burger restaurant ÷ 360 days open per year ÷ $10 average order × 0.8 burgers per order (many orders are chicken, drinks or breakfast) = 400

From the meal rushes: 200 orders at lunch (2 hours at 100 an hour) + 160 orders at dinner (2 hours at 80 an hour) + 150 orders in the other 10 open hours (15 an hour) × 0.8 burgers per order = 408

Say about 400 burgers a day. Order of magnitude: about 10³.

Tip: Working back from yearly sales is a good second route whenever you can guess the revenue of one location.

Ask first

• Services only, or hair products sold at the desk too?
• Open six days a week?

One way in: stylists × clients per stylist per day × average ticket, cuts and color × days open per year

From the chairs: 6 stylists × 6 clients per stylist per day × $60 average ticket, cuts and color × 290 days open per year = $626,400

From the client book: 1,500 regular clients × 6.5 visits per client per year × $60 average ticket = $585,000

Say about $610,000 in sales per year. Order of magnitude: about 10⁶.

Tip: Supply (chairs and hours) and demand (the client book) should meet. If they don't, the salon is either turning people away or sitting idle.

Ask first

• An office building with about 300 people, open on workdays only?
• Sales, before the building's commission?

One way in: items sold per day × average price × workdays per year

From sales a day: 40 items sold per day × $2 average price × 250 workdays per year = $20,000

From the people: 300 people working in the building × 10% share who buy something on a given workday × 1.1 items per buyer × $2 average price × 250 workdays per year = $16,500

The routes are 21% apart, so give a range: $17,000 to $20,000 in sales per year. Order of magnitude: about 10⁴.

Tip: Small numbers are guesstimates too. Saying '$17,000 to $20,000, so a vending route needs dozens of machines' is the kind of so-what interviewers look for.

Ask first

• Membership dues only, or personal training too?
• A mid-priced gym, not a $10 budget chain or a boutique studio?

One way in: members × monthly dues × months per year

From the members: 2,500 members × $40 monthly dues × 12 months per year = $1.2 million

From the floor space: 20,000 square feet of gym × $55 yearly revenue per square foot = $1.1 million

Say about $1.2 million in sales per year. Order of magnitude: about 10⁶.

Tip: Revenue per square foot is how retailers and landlords compare stores. It makes a quick second route for any business with a floor.

Ask first

• A regular suburban supermarket, not a warehouse club?
• An average day across the week?

One way in: checkout transactions per day × average basket

From the checkout: 1,700 checkout transactions per day × $40 average basket = $68,000

From the neighborhood: 25,000 households within a 10-minute drive × 12% share that use this store as their main store × $150 grocery spend per household per week ÷ 7 days per week = $64,286

Say about $66,000 in sales a day. Order of magnitude: about 10⁵.

Tip: A store-level route (checkouts) and a customer-level route (households) are independent, which is what makes the second number worth having.

Ask first

• Revenue, not profit?
• Monthly permits and daily parkers together?

One way in: spaces × share filled on an average day × average daily take per filled space (permits work out to about $12 a day, daily tickets more) × full-revenue days per year (weekends run quieter)

From the spaces: 500 spaces × 72% share filled on an average day × $14 average daily take per filled space (permits work out to about $12 a day, daily tickets more) × 300 full-revenue days per year (weekends run quieter) = $1.51 million

Permits plus daily parkers: 300 monthly permit holders × $250 permit price per month × 12 months per year + $540,000 daily parkers per year at $25 (60 a day, 360 days) = $1.44 million

Say about $1.5 million in revenue per year. Order of magnitude: about 10⁶.

Tip: Check the two routes against each other: 300 permits plus 60 daily cars is 360 filled spaces, the same 72% the first route assumes. Mixed customers (permits and daily) are a hint to split the route. The split also tells you where the growth would come from.