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Tools - Math 'Plausible Estimation' Estimating for Amazing Facts Tasks, Set #3 (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. High stack
    Suppose you have a very large sheet of paper. You tear it in half and put one half on top of the other. You now have a stack of two sheets.

    You now tear the whole stack in half and place one half on top of the other to make a new stack. You repeat this process, tearing 50 times. (Yes I know its impossible - just imagine you could).

    How high would the stack be? 50 feet? 100 feet? A mile? or more...? Make a sensible estimate, based on careful reasoning.

    Picture of a stack of paper.



  • A ream of paper (500 sheets) is about 2 inches thick


Each sheet is therefore approx 1/250th of an inch thick.

After 50 tears, the stack will contain 250 sheets of paper.

This is equal to approximately

1.1259 x 1015 sheets or 4.5 x 1012 inches = 71million miles.

Answer: 71 million miles!

  1. The swimming pool and the glass.
    How long would it take you to empty an olympic size swimming pool with a glass? Picture of a glass of water.



  • Dimensions of swimming pool = 50 meters x 20 meters
  • Depth of water = 6 feet (2 meters)
  • The glass holds half a pint.
  • You empty the pool at one glass per second.


This gives volume of pool as 2000 cubic meters or 2 x 106 liters.

One half pint glass has a volume of about 25 x 10-5 cubic meters.

Thus approximately 8 million glasses will empty the pool.

This would take 2,215 hours or about 90 days.

Answer: 92 days or about 3 months, working day and night!

  1. The briefcase of cents
    Suppose you filled a briefcase with one cent coins.

    How much would the money be worth?

    Picture of a briefcase.   Picture of the back of a penny.



  • A cent coin has a diameter of 20 mm (0.75 inches) and a thickness of 1.5 mm
  • The briefcase is approx 100mm x 350mm x 500mm


The volume of the coin is therefore given by

Greek letter pi.r2h = 3.14 x (20/2)2 x 1.5

≈ 470 mm3

The volume of the case is approx 500 x 100 x 350 = 17,500,000 mm3

Thus the number of coins that will fit in (assuming no gaps) will be about 17,500,000 ÷ 470 = 37,000 (approx).

There would be some gaps between coins, so

Answer: The money will be worth around $350.

(Note that the total weight of the case would be almost 100 kilograms - far too heavy!)

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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