By Scott S. Mitchell
One of my daughters, who loves math and majored in statistics in college, recently challenged me to solve a math word problem to which she had already figured out the answer. Since I generally enjoy word problems, and believed careful thinking on my part would reveal the solution, I had no reason not to accept her challenge. I also assumed that the problem was tricky and the answer counterintuitive, because otherwise my daughter would have no reason to use this problem to test me. So, with what I thought was the appropriate amount of confidence and wariness, I said “Sure, lay it on me.”
“You have 100 potatoes weighing 100 pounds,” she said, “and 99% of the weight of the potatoes is from water, and the remaining 1% of the weight is from what’s left when you take out all the water. Using a machine that dehydrates potatoes, you are able to reduce the percentage of water weight from 99% to 98% of the potatoes’ total weight. After this reduction, what is the new total weight of the 100 potatoes?”
The problem seemed simple, but I assumed it was deceptively simple, not obviously simple, so I set about trying to figure out why the problem seemed obviously simple when there had to be more to it. I tried to picture in my mind a potato dehydrator reducing the water weight of the potatoes from 99% to 98%, and what I’d be left with afterwards. In my mind, each potato would look almost exactly the same as before the seeming tiny amount of water weight was removed. The shrinkage would be minimal. 98 pounds of water weight and one total pound of potato weight, for a total of 99 pounds as the new weight, right? But that was too simple an answer for what had to be a tricky question. But how was I not picturing this problem accurately? The math had to be the easy part once the right equation was derived, I figured, but for a half hour I struggled to understand why a 1% dehydration would produce a counterintuitive equation. Finally I allowed my daughter to reveal the answer, which she’d eagerly offered to do three times before I relented.
The answer, it turns out, is 50 pounds. (How’s that for counterintuitive?) As I suspected, the easiest part of the problem is the algebra that produces this result, but there are two difficulties, or there were for me, that hampered the process of deriving the right equation. Picturing the problem in my mind was almost the worst thing I could do, because what I pictured was not remotely close to the amount of dehydration the scenario actually entails. The most fundamental mistake I made was overlooking the enormously helpful piece of information the problem supplies–the non-water potato weight increases to 2% of the new total weight when the percentage of water weight is reduced from 99% to 98%, but it remains constant at one pound (because dehydration doesn’t make the potato weight heavier, it only makes it proportionally heavier). Knowing that one pound of non-water potato material is 2% of the new total weight generates this remarkably simple equation, where x = the new total weight we’re trying to calculate:
.02x = 1
Multiplying both sides of the equal sign by 100, we get 2x = 100, or x = 100 divided by 2, which equals 50. This problem actually has a name in the math world, and it’s called The Potato Paradox. The word “paradox” suggests the answer is anything but expected.
Why was this problem so, well, problematic, to those of us who took longer than 45 seconds to resolve it, or never resolved it? It’s because it’s so hard to picture the full effect of what’s happening when superficially small numeric changes in proportionality are made. Here’s another example of a proportionality (i.e., percentage change) problem producing a counterintuitive result. I thought this one up myself during the last basketball season. I will call it the Free Throw Paradox:
BYU’s player leaves the basketball court with his team comfortably ahead of Utah in the last game of the year. For the year, he has made 99 of 100 free throws for a spectacular 99%. Utah also has a great free throw shooter on its team, who remains in the game hoping to pad his statistics with three minutes left, even though his team has no chance of winning. On the year, he has made 98 of 100 free throws for an amazing 98% percent, and before the game ends he hopes to get fouled so he can catch, and hopefully surpass, the free throw percentage of his BYU rival. But being a Utah player instead of a BYU one, he doesn’t realize that in order to raise his percentage 1% to catch the intrepid Cougar, he will have to make 100 free throws in a row in the next three minutes, and to surpass him he’ll have to make 101. The difference between 98% and 99% is now vast indeed. The Ute also doesn’t realize that the Cougar intentionally missed his last free throw of the year, just to entice the Ute into thinking he could attain the highest free throw percentage by making a couple more free throws before the game ended. But if the Cougar had missed just one more free throw during the season, the Ute would only have to make one more free throw to win the free throw title (which might lessen the sting of losing the game to BYU by about 1%).
Why did I fail to see how easy the Potato Paradox problem was, even when I accurately expected it to be deceptively simple? There are multiple reasons, but they all revolve around the rest of my expectations, which were all wrong. Mathematically, as stated above, I failed to focus on the most important piece of information supplied by the wording of the question–the fact that the pure potato weight stayed at one pound but was now 2% of the overall weight of potatoes. Conceptually, I failed to remember that percentage problems belong to that weird family of proportion math problems where, as you approach 100%, superficially tiny differences between one high percentage point and another high percentage point are shockingly large if you try to close the difference between them. Experiencially, I had never thought about dehydration math at all, but had thought plenty about free throw percentages in basketball and batting averages in baseball. If the problem had concerned sports percentages, my experience with those kinds of problems would have aided greatly in producing a quick solution.
What do the above two examples tell us about the way we approach scriptural interpretation? In Part 2 of this essay, we’ll discuss what Nephi is telling us in his famous Book of Mormon psalm in 2 Nephi 4, and how objectively interpreting that chapter helps us understand other LDS scriptures or history without erroneous suppositions throwing us off. Meanwhile, ask yourself if you have ever in your life heard a member of the Church of Jesus Christ of Latter-day Saints publicly express an opinion as to which sins implied in 2 Nephi 4 are repeatedly tempting him and threatening his self-esteem? I’m 65, but have never once heard or read any LDS teacher or writer tackle this subject. If your experience is similar to mine, ask yourself, Why is that? Why have we shied away from this topic? What about it scares us? Is it solely the lack of textual clues, or is there something about the subject matter that makes us uncomfortable analyzing the textual clues that are there?
Similarly, if you have heard or read a discussion of this issue, did the opinion expressed assume that Nephi set supremely high standards for himself and was doubtlessly troubled by something trivial, like swearing when he hit his finger with a hammer, or smiling at an off-color joke? Did this proffered explanation sound reasonable?
And here are my favorite questions to ask about this subject: Did Nephi intend that we shy away from deciphering his psalm and examing his personal life? If so, why did he write so much about his sinfulness in a book he knew so many of us were going to be reading?