From Enrico Fermi to Bill Bennett

How big? How many? Quickly getting an approximate idea of the magnitude of a number and of its relation to other numbers is often helpful when doing physics.

To further this skill, the Nobel Prize-winning physicist Enrico Fermi often asked his students to estimate various bizarre quantities. Answering these questions required that they think critically about the quantity, make reasonable assumptions, ascertain basic facts, and then perform the required calculations.

A classic Fermi question is "How much tea is there in China?" Let me sketch the process before discussing its usefulness when reading the daily newspaper or Web site.

Tea in China

So how much tea is in China? We assume the following:

That there are about 1.2 billion people in China. That each person drinks an average of two cups of tea per day. That about four grams of tea leaves (post drying & processing) make one cup of tea. And, that there may be three or four months of leaves stockpiled at any one time.

With these (debatable) assumptions, the amount of tea in China is 1.2 billion people x 2 cups per person per day x 4 grams of tea leaves per cup x 100 days, which equals about 1 billion kilograms (or 1 million tons) of tea leaves in China.

By making our assumptions explicit, we can revise one or more of them if we find them to be way off.

However obtained (and there are other approaches), the answer certainly won't be exact. But if we're careful, it will probably be correct to within a power of three. For many purposes that's close enough.

If you want to try a Fermi problem, here's one (answer below): If the entire land surface of the Earth were to be divided into parcels of equal area, one for each human on the planet, how big would these parcels be?

Bill Bennett's Gambling

The appeal of these problems is that often without much more than arithmetic and a few facts we can obtain the approximate magnitude of numbers important to, but not explicitly mentioned in the news stories that we read. One example:

Recent articles have reported that conservative commentator Bill Bennet lost up to $8 million over the last decade playing slot machines and video poker at casinos. These might cause us to wonder how much time he spent in casinos over this period.

If we assume the payout on the machines was 96 percent (over time, one loses 4 percent of what one bets), then the $8 million loss is 4 percent of the total amount bet over this time. The latter would thus be about $200 million, of which he'd get back $192 million, the total of much smaller amounts recycled through the machines and recounted many times.

So how long does it take to bet $200 million?

Let's assume that Bennett pulled the lever on the $500-a-pull machines he preferred a rabbit-fast five times per minute on average (factoring in breaks). That's $2,500 per minute and $150,000 per hour. If we divide $200 million by $150,000 per hour, we get more than 1,300 hours of pulling machine levers, which is almost eight months worth of full 40-hour weeks spent in casinos over the decade!

The calculation suggests that the $8 million estimate of his losses may be high or that the machines had lower payouts or that he was even more enamored of gambling than previously thought. Given the games he played and only his documented losses, however, Bennett's claim that he gambled occasionally and more or less broke even is laughably innumerate. His Book of Virtues should have contained a section on arithmetic rectitude.

In summary, doing a few back-of-the-envelope calculations can sometimes be quite revealing, not just in physics, but in journalism as well.

Answer: We need to find out how much land area there is on Earth and what the population is. The Earth is a sphere whose radius is about 4,000 miles. Since the formula for the surface area of a sphere is 4pi*r^2, we find that the earth has a surface area of about 200 million square miles. There are 5,280 feet per mile so there are 5280^2 square feet per square mile, which means that 200 million square miles contains about 5.6 x 10^15 square feet. Multiplying this by .3 (since only 30 percent of the Earth's surface is land) yields about 1.7 x 10^15 of land surface area. Dividing this by approximately 6 billion humans gives each of us about 280,000 square feet to call our own. Taking the square root of this figure reveals that we each would have a square a bit more than 500 feet on a side. Of course, most of these parcels would be inhospitable.

Professor of mathematics at Temple University and adjunct professor of journalism at Columbia University, John Allen Paulos is the author of several best-selling books, including Innumeracy, and the just released A Mathematician Plays the Stock Market. His Who’s Counting? column on appears the first weekend of every month.