Fermi Estimation (AQA Level 3 Mathematical Studies (Core Maths))

Fermi Estimation

What is Fermi estimation?

• Fermi estimation involves making fast, rough estimates of quantities which may be difficult or impossible to measure directly

• For example:

  • How many slices of pizza are eaten in the world every day?

  • How many leaves are on a typical tree?

  • How many bricks were used to build the Great Wall of China?

• To do this, factors which affect the answer are identified, estimated, and combined

• Fermi estimation is an example of a mathematical model

How do I perform a Fermi estimate?

• Consider the scope of the question and make sure it is well-defined

  • E.g. "How much pizza is eaten in the world?" could become "How many tonnes of pizza is eaten each day globally?"

• Consider which factors, that can be estimated, would affect the answer

  • These factors will be greatly simplified

  • For example

    • How many people are on Earth?

      • Around 8 billion

    • On average, how many pizzas does a person eat per week/month?

      • Around 1 per week

      • Answers could vary a lot for this, but Fermi estimation should be quick and approximate

    • What proportion of people eat pizza at all?

      • Perhaps half the planet, 50%

      • Similarly, this proportion could be researched more, but focus on a rough estimate

      • It is definitely not 5%, and definitely not 95%

    • How much does an average pizza weigh?

      • Around 0.5 kg

• Perform a calculation, or multiple calculations using the estimated factors

  • When calculating, be rough with estimates

    • The level of accuracy of the approximated factors is already low

  • 8 billion people × 0.5 × 1 = 4 billion people eat pizza each week

  • 4 billion ÷ 7 = approximately 600 million pizzas eaten globally each day

  • 600 million × 0.5 kg = 300 million kg

  • 300 million kg ÷ 1000 = 300 000 tonnes

    • As 1000 kg = 1 tonne

• Check your answer seems reasonable

  • This can obviously be tricky given the nature of the question

  • Focus on the order of magnitude of your answer

    • Consider if your answer be in the thousands, millions, billions, etc.

    • E.g. If the answer was calculated to be 8 billion kg of pizza per day, this would imply everyone on the planet ate 1 kg of pizza per day, which is obviously incorrect

What are some useful facts to know to help with Fermi estimates?

• To help make estimates, having some knowledge about the rough sizes of quantities really helps

• Time

  • 24 hours per day, 7 days per week, approximately 30 days per month, 365 days per year, and 12 months per year

  • Human life expectancy is 73 (globally)

• Population

  • Globe: around 8 billion (8 000 000 000)

  • United Kingdom: around 67 million

  • Europe: around 750 million

  • A city: hundreds of thousands, to millions

  • A town: thousands, or tens of thousands

  • A village: hundreds

• Mass

  • Bag of sugar: 1 kg

  • Human: around 60-80 kg

  • Car: 1 to 2 tonnes

  • 1000 kg = 1 tonne

• Length/Distance

  • Height of a person: around 1.6-1.8 m

  • House: 8-10 m tall

  • Across a town: 5-10 km

  • Length of Britain: around 1000 km (600 miles)

  • Circumference of earth: 40 000 km

  • 1000 m = 1 km

• Area

  • Table: 1 m²

  • Football field: 10 000 m²

  • A city: 200 km²

  • United Kingdom: 200 000 km²

  • 1 000 000 m² = 1 km²

• Volume

  • Cup: 250 ml

  • Bath: 150 litres

  • Olympic Swimming pool: 2500 m³

  • 1000 ml = 1 litre

  • 1000 litres = 1 m³

• Speed

  • Walking: 5 km per hr

  • Car: 100 km per hr

  • Plane: 800 km per hr