From the Comments: A Puzzle About Percentages

Dr Gibson hands out a tough quiz in the ‘comments’ thread of Jacques’s latest post on comparative advantage:

Quiz: last year I earned no money from writing. This year I expect to make $5,000. By what percentage will my writing income have risen?

Jacques is stumped. I am too, but I think I’ll take a stab at it anyway. The worst that can happen is that I’m wrong, right? Warren, by the way, has a PhD in engineering as well as an MA in economics, so math is his forte (he is also the math reader for Econ Journal Watch).

I speculate that the percentage of his writing income has risen by 100%. I don’t see how it could be anything else. If you start out at zero, then even if Warren only made $1 this year an increase from $0 to $1 would have to be 100%, right?

Am I right? I need help.

Digression: Jacques is right that the Romans got along fine without the zero, but that’s not saying much. Here is Tocqueville:

If the Romans had been better acquainted with the laws of hydraulics, they would not have constructed all the aqueducts which surround the ruins of their cities – they would have made a better use of their power and their wealth. If they had invented the steam-engine, perhaps they would not have extended to the extremities of their empire those long artificial roads which are called Roman roads. These things are at once the splendid memorials of their ignorance and of their greatness. A people which should leave no other vestige of its track than a few leaden pipes in the earth and a few iron rods upon its surface, might have been more the master of nature than the Romans.

4 thoughts on “From the Comments: A Puzzle About Percentages

  1. It is undefined guys. A percentage is essentially just a fancy way to calculate a division of something and you can’t divide by zero. Well you can but not for practical purposes.

  2. Answer: division by zero is undefined and therefore no percentage can be calculated. Also, one is not “close enough to zero” as Jacques says if, for example, you’re measuring the width of your pencil in meters.

    A related example from economics: suppose the price of a cup of coffee rises by $0.001 and you continue to purchase 20 cups per week despite this rise. Does this example violate the law of demand? No, but not because $0.001 is “close to zero.” It’s because $0.001 is small compared to the threshold of your subjective perception which might be $0.10.

  3. I am always eager to learn from my better. It the practical case that preoccupied me though the correct answer is not better than my own incorrect one.

    As to the Romans: with no zero, they managed to build bridges and aqueducts that did the job well and were pleasing to the eye. One beautiful example is still standing and it could be functional in southern France. (Le Pont du Gard.) However, if I had to vote, I would definitely vote in favor of the zero.

    Here is some useful French of breathtaking imagery: “Masturber les puces avec des gants de boxe.”

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