If there was one thing I excelled at in math, it was clumsy errors. These are the type of bonehead mistakes caused by haste or stress or both. You know the correct way to solve a problem, but get the wrong answer due to simple inattention. Clumsy errors are the mathematics equivalent of putting the batteries in backwards.
Such was the case on the evening of Tuesday, March 3. 2020, during coverage of the Super Tuesday primary results. Mike Bloomberg didn’t do well at all, and a Ms. Mekita Rivas tweeted that he spent $500 million in ads. Ms. Rivas observed that there are 327 million Americans, and if he’d given each $1 million, he’d had money left over. MSNBC’s Brian Williams and New York Times editorial board member Mara Gay, who were covering the primary results, commented that this was an incredible way of putting it.
Times being what they are, the error, and Williams’ and Gay’s failure to see it, brought a swift response. Mr. Rivas later posted that she knew (of the error), that she was bad at math, and Williams made a similar statement. That didn’t stop criticism or political commentary.
As one plagued by such clumsy errors in school, I can’t be smug about it. It does perhaps give me a little insight into how this error likely came about. I doubt I was the only one who read about it and didn’t have “wait a minute” kick in until a second or two later. Looking at how this sort of error happens can me most instructive.
The clue is in how the problem is phrased: $500 million and 327 million. Technically, there’s nothing wrong with that. $500 million means $500,000,000 and 327 million means 327,000,000. All of us know that. But to understand how this can lead to a clumsy error, consider 500 miles divided by 327 miles. This gives you a tad under 1.53. The clumsy error in Ms. Rivas’ post, and missed by Mr. Williams and Ms. Gay, is mentally placing million, a quantity, in the same category as units. Even if you know the correct meaning of the word million, as no doubt Ms. Rivas, Mr. Williams, and Ms. Gay do, it’s all to easy to get in a hurry, maybe while thinking of something else, and make this sort of error.
Such is the basis for trick math questions usually told as jokes. They work by not being what our minds expect, such as this gem from Martin Gardner’s The Unexpected Hanging: “How many months have 30 days?” It’s easy to answer “Four,” but the correct answer is “Eleven.” It’s all too easy to interpret the question as “How many months only have 30 days?” but that’s not what is asked. Word problems do something similar by adding unnecessary information to teach focusing on what is important to solving the problem. If getting sidetracked on such things were uncommon, we’d never have such problems to begin with.
Here’s an oldie but goodie, the source of which I can’t recall:
A boat floating at a dock has a rope ladder handing down the side. The side of the boat is 12 feet above the water. The rope ladder is 10 feet, six inches. The tide is coming in at a rate of 1 foot, six inches per hour. How long will it take for the water to reach the bottom of the ladder?
The answer is pretty obvious if you take time to think about it first. That’s the thing. As simple as this problem is, it wouldn’t exist if it wasn’t easy to get lost in the details. Solving problems is as much knowing what is important and keeping things straight as known how to do the math.
This brings us back to the original problem. All it takes is a moment’s inattention to make a clumsy error, unconsciously regarding “millions” as a unit of measure instead of a quantity. I’m pretty certain that if any of the three had written out the numbers, they would have arrived at the correct answer. Or they could have solved it by ($500 x 1 million) / (327 x 1 million), which means the 1 millions cancel out, leaving the problem with $500 / 327, which works out to almost $1.53 dollars per person. Those that caught the error probably did this unconsciously by not confusing million as a unit.
Does this mean those of us subject to clumsy errors are bad at math? I can give a definitive maybe. The thing to keep in mind, especially if you, like me, are subject to clumsy errors, is that they can be prevented by simply being more attentive. That’s something I learned the hard way, and still have to keep in mind.
As to the boat problem, the answer is “Never.” A floating boat rises as the tide comes in.