How much does a million dollars weigh?

Classical Mechanics Level 2

This is a Fermi Estimation problem that can be calculated with some approximations and assumptions

$How$ $much$ $does$ $a$ $million$ $dollars$ $weight?$

Details and Assumptions:

$You$ $only$ $have$ $one$ $dollar$ $notes$

Hint:

$Think$ $how$ $big$ $it$ $is$ $an$ x $pages$ $book$

1000kg 100kg 10kg 5000kg 500kg

2 solutions

Michael Mendrin
Aug 6, 2018

A dollar bill is about 0.1 mm x 66 mm x 156 mm, so that a million dollar bills fills a cube of just about 1 cubic meter. The weight of paper is about 1000 kg per cubic meter, or just over a ton. It's a terrible way to transport a million bucks.

Fabrizio Melges Ferro
Aug 7, 2018

A Fermi estimate is one done using back-of-the-envelope calculations and rough generalizations to estimate values which would require extensive analysis or experimentation to determine exactly.

When a problem like this doesn't provide enough informatiom, the best thing to do it's to make general assumptions and approximations.

In order to solve this I will break the big problem into appropriate sub-problems.

-1) What is the volume of a dollar note?

A dollar is about 15cm long and 5cm wide... now how thick is it? we don't know that, but money is made out of paper, and a book (it can be any book,but I am considering "Astrophysics for people in a hurry") is around 200 pages long, or 100 sheets of paper. and I think they are about 1.5cm thick => thick = 1,5cm/100 Now we can find the volume of a dollar:

15cm x 5cm x 1.5/100 = v => v = 1.125cm³

-2) What is the weigh of a dollar note?

We still don't know how much paper weighs though, But we know that a cm³ of water weighs a gram So we will say a cm³ of dollars weighs 0.9 gram

-3) Conclusion

One dollar weigh = 0.9 x 1.125 = 1.0125g

One million dollars weigh = 1.0125 x 1,000,000 = 1,012,500g = 1012.5kg

And that result it's close to the oficial one according to an website one million dollars weighs 1000kg

Physics is celebrated for its ability to make extremely accurate predictions about tough problems.. However, in FERMI ESTIMATE it is usually a good idea to relax demands and accept an approach that while imprecise, offers a prayer of moving forward.

I would like to thanks July Thomas, Azhaghu Roopesh M, Josh Silverman, and 3 others contributed that made an wiki about Fermi estimate:

https://brilliant.org/wiki/fermi-estimate/?subtopic=units-and-measurements&chapter=fermi-estimation

here it is the link of the website i mentioned:

http://1000000-euro.de/how-much-does-a-million-dollars-weigh/index.php#weigh-dollar

How much does a million dollars weigh?

2 solutions

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