Fermi Questions Practice Tests

order-of-magnitude estimation, dimensional analysis, and logical reasoning.

Population & demographics

• US population: ~330 million (3.3×10⁸)
• World population: ~8 billion (8×10⁹)
• US households: ~130 million
• US land area: ~3.8 million mi² = ~10 million km²
• World land area: ~150 million km² (57 million mi²)
• Average US household: ~2.5 people
• Average human lifespan: ~80 years
• Human heartbeats: ~100,000/day, ~1/second at rest
• Human breaths: ~20,000/day
• Births per year (world): ~140 million
• Deaths per year (world): ~60 million

Physical constants

• Speed of light: c = 3×10⁸ m/s
• Speed of sound (air): ~340 m/s = ~1,100 ft/s
• Gravitational acceleration: g = 10 m/s² (for estimation)
• Earth radius: ~6,400 km = ~4,000 miles
• Earth circumference: ~40,000 km = ~25,000 miles
• Earth-Sun distance: ~150 million km = 1 AU = 8 light-minutes
• Earth-Moon distance: ~400,000 km = ~240,000 miles
• Moon radius: ~1,700 km (about 1/4 Earth)
• Sun radius: ~700,000 km (about 100× Earth)

Densities & masses

• Water: 1 g/cm³ = 1 kg/L = 1000 kg/m³
• Air (sea level): ~1.2 kg/m³
• Iron/Steel: ~8 g/cm³
• Wood: ~0.5-1 g/cm³
• Human body: ~1 g/cm³ (we barely float)
• Earth mass: ~6×10²⁴ kg
• Sun mass: ~2×10³⁰ kg

Time conversions

• 1 minute = 60 seconds
• 1 hour = 3,600 seconds = 60 minutes
• 1 day = 86,400 seconds ≈ 10⁵ seconds
• 1 year ≈ π×10⁷ seconds ≈ 3.15×10⁷ seconds
• 1 year = 365.25 days = 8,766 hours = 525,960 minutes

Length conversions

• 1 inch = 2.54 cm
• 1 foot = 30 cm = 0.3 m
• 1 yard = 0.9 m ≈ 1 m
• 1 mile = 1.6 km = 5,280 feet
• 1 meter = 3.3 feet
• Human height: ~1.7 m = ~5.5 feet
• Human arm span ≈ height

Volume conversions

• 1 gallon = 3.8 L ≈ 4 L
• 1 liter = 1000 cm³ = 10⁻³ m³
• 1 cup = ~250 mL
• Olympic swimming pool: ~2.5 million liters

Everyday objects

• Car: ~2,000 kg = ~4,000 lbs
• Car length: ~4-5 m
• School bus: ~10 m long
• Football field: 100 yards ≈ 100 m
• Classroom: ~10m × 10m = 100 m²
• Sheet of paper: ~0.1 mm thick
• Human hair: ~0.1 mm diameter
• Dollar bill: ~15 cm × 6 cm

Estimation techniques

1. Break into smaller parts (divide and conquer)

2. Round to powers of 10

3. Use dimensional analysis (check units cancel)

4. Geometric mean for ranges: √(low × high)

5. Sanity check your answer

6. Work from what you know to what you don't

Example problem

"How many piano tuners in Chicago?"

• Chicago population: ~3 million
• People per household: ~2.5 → ~1.2 million households
• Households with piano: ~5% → ~60,000 pianos
• Tunings per year per piano: ~2 → ~120,000 tunings/year
• Tunings per tuner per day: ~4 → ~1,000/year per tuner
• Piano tuners needed: ~120,000 / 1,000 = ~120 tuners

Common mistakes

• Forgetting unit conversions
• Not breaking complex problems into simpler parts
• Using wrong order of magnitude
• Not checking if answer is reasonable
• Forgetting factors of π or 2 that matter at order of magnitude