Putting Bill Gates in Proportion

bill-gates-hombre-mas-rico

[Feature photo above by Baluart.net.]

A friend gave me permission to turn our email discussion into an article…

Can you help us figure out how to figure out this problem? I think we have all the information we need, but I’m not sure:

The average household income in the United States is $60,000/year. And a man’s annual income is $56 billion. Is there a way to figure out what this man’s value of $1mil is, compared to the person who earns $60,000/year? In other words, I would like to say — $1,000,000 to us is like 10 cents to Bill Gates.

Let the Reader Beware

When I looked up Bill Gates at Wikipedia, I found out that $56 billion is his net worth, not his income. His salary is $966,667. Even assuming he has significant investment income, as he surely does, that is still a difference of several orders of magnitude.

But I didn’t research the details before answering my email — and besides, it is a lot more fun to play with the really big numbers. Therefore, the following discussion will assume my friend’s data are accurate.

This Is a Ratio Problem

More specifically, to solve this problem we need to write a proportion, which means one ratio is equal to another.

Here are a few things you need to know about ratios:

(1) A ratio is simply a fraction: “Numerator is to denominator.” For instance, the ratio of the average income to the huge income would be \frac{60,000}{56,000,000,000} .

(2) A proportion sets one ratio equal to another ratio. In a proportion, you need to be sure you are comparing similar things in the same order.

For instance, $1,000,000 to us is a whole bunch more than one year’s income. The thing that is like that to Bill Gates would be something closer to a trillion dollars, wouldn’t it? I think what you mean is: “$1,000,000 is to Bill Gates like 10 cents is to us.” On the other hand, ten cents to Bill Gates would be much less than a penny on the parking lot is to us.

In other words:

\frac{Big \: Number}{Big \: Income}   =   \frac{Little \: Number}{Little \: Income}

Or:

\frac{Little \: Income}{Big \: Income}   =   \frac{Little \: Number}{Big \: Number}

You can arrange the proportion in different ways, some of which are easier to solve than others. It is always nice to try to arrange a proportion so that your unknown number is in the numerator of one of the fractions, if you can — and you always can with a simple relationship like this, if you’re careful. The main point is, both sides of the proportion have to be set up so that similar thing compares to similar thing in the same order.

(3) After you get the proportion set up the way you want it, you need to solve for your unknown number using cross-multiplication.

[Cross-multiplication is confusing to many students. If you can’t remember how to do it, I can send you a review.]

Now to Try Your Problem

Little income = $60,000
Big income = $56,000,000,000
Big number = $1,000,000
Little number = x

I will use the proportion, “Big number compares to big income as little number compares to little income.”

\frac{1,000,000}{56,000,000,000}   =   \frac{x}{60,000}

Multiply both sides by 60,000:

\frac{1,000,000 \times 60,000}{56,000,000,000}   =   x

Simplify fraction by canceling zeros:

\frac{60}{56}   =   x

Put in lowest terms:

\frac{60}{56}   =   \frac{30}{28}   =   \frac{15}{14}   =   x  \approx 1.07

A million dollars is to Bill Gates about the same as a dollar and seven cents is to you and me.

Does That Make Sense?

Absolutely! This is for C’s speech. So, can he say that $1,000,000 to Bill Gates has the same value as $1 does to us? If he flips it, that $1 to Bill Gates is like $1,000,000 to us, I think that would confuse the audience because it sounds as if we have more money than Bill.

Or am I too confused?

You are a little confused. The first thing you said is right, the “flipped” one is wrong. To Bill Gates, $1 million is pocket change. It is about the same as a buck is to us. And to Bill Gates, $1 would not even be worth noticing. It would be about like $ 0.000001 would be to us. That is about 1/10,000 of a cent.

I have heard speakers use an analogy like this: If Bill Gates were walking down the street and saw a $100 bill on the ground, it would not be worth his time to pick it up. Compared to his net worth, that would be like one of us bending down to pick up 1/100th of a penny.

Of course, even Bill Gates could probably think of some use for a spare $100, right? Leave it as a lunch tip, or something…

Thank you! I understand now.

The “Real World”

Okay, so what should the calculation be, really? Let’s guess that Bill Gates’ investment income is about triple his salary income. I have no idea if that is accurate, but I couldn’t find any data online, and at least the assumption gives us something to work with. Rounding everything off would give us a total annual income of $4 million.

In that case, $1 million would be about 1/4 of Gates’ annual income, and the equivalent for a worker earning $60,000 is:

\frac{1}{4}  \times  60,000 =   15,000

And a $100 bill would be like $1.50 to us. That is still a big difference in perspective, but it sure doesn’t make the same impression in a speech.

[Edited to add: I found some additional data. Check out the revised figures in Bill Gates Proportions II.]

Bill Gates's house photo by jeffwilcox (CC BY 2.0).
Bill Gates’s house photo by jeffwilcox (CC BY 2.0).

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5 thoughts on “Putting Bill Gates in Proportion

  1. Comparing Bill Gates, $56 Billion to one’s $60K?
    Gates annual compensation is less then a million. But his net worth is $56 billion.
    Question; what is the person’s net worth including investments who makes $60K. I suggest you go back to the drawing board on this one.
    Most likely chump change but you are comparing apples to oranges.

  2. Roger,
    As I said, this came from an email correspondence, and the data are not well researched. The point of this exercise was to learn how to build a proportion. If you need actual figures, the follow-up post, Proportions II, compares Bill’s net worth to the net worth of us normal folks.

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