The Other 60%: Sleight of Number

What is innumeracy?

One of the classic jokes in the workplace is the fact that 40% of sick days are taken on Mondays and Fridays. The comic strip Dilbert used this to launch a series in which the boss sets out to solve this problem. Of course, calmer heads will also notice that 40% of the work gets done on Mondays and Fridays, since those two days account for 40% of the work week.

This is one example of innumeracy, or sleight of number. By stating that "40% of sick days are being taken on Mondays and Fridays" we create the impression that this is somehow abnormal, and it is easy to avoid checking the facts before reacting.

Thinking about numbers before reacting. Not falling prey to scams based on numbers. Realizing when somebody is abusing numbers just to get a reaction. That's what these pages are about. Here we will look at some ways that numbers are abused in everyday life, and some ways that we can do 'quick checks' to see if somebody's numbers make sense. It's amazing how often they don't! From this page you can find out about:

What is the mass of the Earth?

Have you ever wondered what the mass of the Earth is ? I didn't think so. Even so, let's see how we can quickly figure it out. Let's start with two facts: the circumference of the Earth is about 25,000 miles, and the density of water is 1.

From the circumference, we can quickly get the diameter of Earth as 8000 miles, or about 13000 km, or about 13 million meters. The volume of a sphere is about half the volume of the cube, so we'll just take the volume of the cube as close enough. The volume, then, is about 13*13*13 million million million cubic meters. We can forget the extra digits, so 13 * 13 * 13 is about 10 * 10 * 10, or about a thousand. So, adding up all those groups of zeroes, we get the volume of Earth as about a thousand million million million cubic meters, or 1 with (3+6+6+6) zeroes. That's 21 zeroes!

Now, a cubic meter of water is a million cubic centimeters, and a cubic centimeter of water is about a gram, so a million cubic centimeters is about a million grams, or about a thousand kilograms. That's another 3 zeroes, so the mass of the Earth must be about 1 with 21+3=24 zeroes.

Of course, that assumes that the Earth is basically water, but the Earth is actually heavier than water -- about 7 times heavier, which gives the total mass of the Earth as about 7 million million million million kilograms (that's a 7 with 24 zeroes).

My physics text gives 6 with 24 zeroes, so it's a pretty good approximation.

To review the approximations, we rounded 13 down to 10 before multiplying. Then we multiplied the diameter by itself three times to get the volume of the Earth. Fortunately, the first one caused our answer to be about half what it should be, and the second made our next answer double what it should be, so they worked out, but even if they hadn't we'd only be off by maybe a factor of 4, which still tells us the Earth has a lot of mass!

The mass of the sun is about a million times more. And now we can put these facts with the rest of our math facts.

Make A Million Dollars Fast

In the movie "Blank Check", Brian Bonsall asks his computer how long it will take $11 to become $1,000,000 at 3.45% interest. The computer works for a few seconds, and spins up the number 342,506 years.

Is it right? Nope. We know those movie people know nothing about computers -- see our movie mistakes page -- so they simply took a wild guess. But you and I can do better. Let's figure out the real answer.

There's a rule for calculating interest called the "Rule Of 72". It tells you that for any given interest rate, you can find out how many years it will take your money to double by dividing the interest rate into 72. If you can't remember the '72', don't worry. For quick check calculations, just use 100.

Okay, in this case, we would have to divide 3.45 into 72. But those extra digits make it harder. Let's just try it with 3 percent to get an approximate answer. That means it will take about 72/3, or 24, years (or 33 if you used 100) for the money to double. For simplicity, just use 25 or 30.

So, how many times does it need to double? Let's make it easier on ourselves by starting with $10. If it doubles once, we'll have $20. Again, and we'll have $40. Then $80, $160, $320, and now the numbers are getting large.

After doubling 4 times, we had $160. So if we round that off to $100 we can get a quick answer. Doubling 4 times gives us $100, so another 4 times will give us $1000, then another 4 gives $10,000, 4 more gives $100,000, and, finally 4 more gives $1,000,000. We're there.

Now we just need to see how long all that took. Okay, we had to double our money 20 times (4 times for each additional digit), and each doubling takes about 25 years (30 if you used 100), so it will take about 500 years (or 600 years) to get a million dollars. The actual answer is about 337 years, so our answers aren't vey close, but they're a lot closer than the 342,000 years calculated by the computer in the movie!

We got here by using these two rules:

Then just multiply them to get the answer, 500.

By the way, a more accurate version of the second rule says that you can add 3 digits whenever you double 10 times. That would have given us an answer of 18 doublings, or 450 (or 540) years, which is a bit closer.

If we'd done all the calculations accurately (still using the Rule Of 72), we would have found that it actually takes 20.87 years to double, and needs to double 16.5 times, giving 344 years, which is close to the correct answer of 337. Those movie people were off by more than three hundred thousand years! That's several ice ages!

There are three kinds of people: Those who check the numbers, and those who don't.

- Cool tricks with numbers.
- Some guidelines for 'quick check' calculations.
- Some useful number facts for quick checks.
- Take the probability quiz.
- Wordlore and wordplay.
- Return to the front gate of Esmerel.

Page maintained by the Mathemagician.