'Plausible Estimation' Estimates 
for a Million Tasks - Set #1

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.

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1. 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. 

	

2. 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.

	

3. 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) 	

4. 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 p (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.	

5. 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.