
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
Malcolm.Swan@nottingham.ac.uk
Jim Ridgway
School of Education
University of Durham
Jim.Ridgway@durham.ac.uk
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.

 Breathing
How many days would it take you to breathe a million times?
Solution:
Assumptions
You breathe once every 4 seconds.

Reasoning
Calculation:
4,000,000 seconds is about 46 days.
Answer: About 50 days.

 Paper Clips
Suppose a chain is made from a million paper clips. How far will it stretch?
Choose suitable units for your answer.
Solution:
Assumptions
Each paper clip is about 4 cm in length.

Reasoning
Calculation:
4,000,000 cm = 40 km (or about 25 miles)
Answer: About 40 kilometres or 25 miles.

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


Solution:
Assumptions
A brick is about 3 inches high and 9 inches long.

Reasoning
Calculation:
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)

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


Solution:
Assumptions
A bucket holds about 8 liters.
A drip has a diameter of 2 mm

Reasoning
Calculation:
The volume of a drip is given by
^{4}/_{3} (1)^{3} ≈ 4 mm^{3}
A million drips will therefore have a volume of
4 x 10^{6} mm^{3} = 4 liters
Answer: About one half a bucket.

 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!


Solution:
Assumptions
You can write down 2 digits per second

Reasoning
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)
