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Tools - Math 'Plausible Estimation' Estimates for a Million Tasks, Set #1 (solutions)

Estimates for a Million, Set #1 (solutions)
Estimating for Amazing Facts: Set #2 (solutions) || Set #3 (solutions)
Estimates for the USA, Set #4 (solutions)

Malcolm Swan
Mathematics Education
University of Nottingham

Jim Ridgway
School of Education
University of Durham

The aim of this assessment is to provide the opportunity for you to:
  • develop a chain of reasoning that will enable you to estimate quantities to an appropriate degree of accuracy
  • choose suitable units for your estimate
  • communicate the assumptions upon which your estimate is based.

  1. Breathing
    How many days would it take you to breathe a million times?


You breathe once every 4 seconds.



4,000,000 seconds is about 46 days.

Answer: About 50 days.

  1. Paper Clips
    Suppose a chain is made from a million paper clips. How far will it stretch?
    Choose suitable units for your answer.

    A picture of 4 paper clips strung together


Each paper clip is about 4 cm in length.



4,000,000 cm = 40 km (or about 25 miles)

Answer: About 40 kilometres or 25 miles.

  1. The brick wall
    Suppose you use a million household bricks to build a wall four feet high. How long would the wall be? Picture of a man laying bricks to build a wall


A brick is about 3 inches high and 9 inches long.



The wall will be (4 feet ÷ 3 inches) = 16 bricks high

This means that the wall will be (1 million ÷ 16) 62,500 bricks in length

= 62,500 x 9 inches

= 8.9 miles.

Answer: Just under 9 miles.
(Maybe with mortar it would be just over 9 miles)

  1. The dripping faucet
    A faucet drips a million times. How many buckets will it fill? Picture of a dripping faucet


A bucket holds about 8 liters.

A drip has a diameter of 2 mm



The volume of a drip is given by

    4/3 Greek letter pi (1)3 ≈ 4 mm3

A million drips will therefore have a volume of
4 x 106 mm3 = 4 liters

Answer: About one half a bucket.

  1. Writing a million
    How long would it take you to write out all the numbers, from one to a million?

    Remember that some numbers have more digits than others!

    Picture of a pencil twisted into a question mark


You can write down 2 digits per second


Most ( 90%) of the numbers have 6 digits.

So we need to write down nearly six million digits.

This would take nearly 3 million seconds = 35 days (approx)

Answer: 35 days, working day and night.

Estimates for a Million, Set #1 (solutions)
Estimating for Amazing Facts: Set #2 (solutions) || Set #3 (solutions)
Estimates for the USA, Set #4 (solutions)

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