MathBench > Measurement

Serial Dilution... or,
how to count to a million

Scaling Up

In this module, we're going to consider some practical implications of counting bacteria. Primarily, we're going to deal with 2 questions:

  1. how do I count something that runs into the millions or billions?
  2. what are the pitfalls?

We'll talk about a specific method (serial dilution, plating, counting, and scaling up), but I want to make it clear at the outset that the problem of counting huge numbers of things is not at all unique to counting bacteria. Many of the same issues (scaling up, error propagation) apply to other things that people want to count, like insects, birds, contaminants, pollutants, human population ...


So, we're going to start by talking about scaling up.

Richard Feynman, one of the most celebrated physicists in modern times, was famous for beginning his first-year honors physics class at Caltech by asking the students to estimate how many barbers work in Chicago. Presumably he was not interested in getting a haircut. Instead, he was (in part) asking his students to "scale up" an estimate based on small sample.

So, for example, you might estimate that in one square urban mile there is on average one barber shop (just barbers, not hairdressers) with 2 barbers. And the area of Chicago is about 2100 square miles (according to Wikipedia). So,

if there are 2 barbers per square mile, and

2100 square miles in Chicago

then how many barbers are there in Chicago?

The answer of course is 2100*2 = approximately 4200. And in fact you would be approximately right. According to one website, there are 4140 barbers in Chicago in early 2006 -- and according to that same website, this is a question sometimes used in job interviews to determine how well candidates think on their feet. Talk about far-reaching applications of mathematical concepts!

So let's try a few more:

plastic bag

How many plastic shopping bags are used per year in the U.S.?

Clearly this is not something you want to go out and count. But without doing a few calculations, its impossible to make even an informed guess. Is it billions? trillions? more?

Instead, try making an informed guess, based on your own use of plastic bags, and a US population of 300 million people...

(To make this problem interactive, turn on javascript!)

I think I have the answer: your daily use * one hundred billion


In fact, according to, the number is about 4.6 * 10^11, or 460 billion.

Notice that I am being a little cavalier with the numbers here. I say I use "about" 4 bags a day, and multiply by "about one hundred million". Not only is it OK to be a little vague, its absolutely necessary, otherwise you're claiming more knowledge than you actually have. Saying:

Americans use about 4.6 hundred billion bags/yr

is a reasonable statement. But saying

Americans use 304,339,593 * 365 * 4 = 444,335,805,780 bags/yr

is ridiculous, and it makes you sound like a pompous know-it-all to boot. For one thing, the U.S. population has changed since I wrote the above equation. For another thing, "4 bags a day" was only a rough guess. (And technically years have 365.25 days).

So I really don't know how many bags were used, but if I did enough research, I hope to be able to say "between 4 and 5 hundred billion", or even "in the hundred-billion range". In other words, not trillions, not millions, but hundreds of billions.