An Idiot’s Guide to Mental Math

Dear Hope Nation,

I try to write for the next-to-lowest common denominator. That is, I usually write these letters with the entire readership in mind. Today, though, that herd will be culled. Instead, I’m writing for one person in particular, and I don’t know who it is. I met a higher-level math student at Hope within the past year, and I think he may like this, even though it has nothing to do with any arithmetic higher than long division, usually mastered by fourth grade.

By the third or fourth paragraph, you’ll know if this is for you, likely because you’ll be saying, “What the hell is all this about? I don’t get it.” In the dueling words of the Grateful Dead and Frank Zappa, at this point you should “hang it up and see what tomorrow brings,” because “someone out there knows what we’re doing and is getting off on it beyond his wildest comprehension.” 

Enjoy, if you can. If you can’t, it’s me, not you, and tomorrow will bring a new letter.

You matter. I matter. We matter.


Last night, I had dinner with my friends Matt and Rob, both of whom, like me, havare in recovery.  Matt is currently reconstructing his cottage, and acting as his own contractor.  While most of the conversation was scintillating, there was a point where Matt wanted Rob, a former math teacher, to calculate Matt’s discount on a recent purchase of lumber. 

While I don’t remember the exact numbers, Matt’s question went something like:

“If my original bill was $23.50 and my final bill was $21.62, what was my discount?”

Rob treated this question like a math problem.  He determined that the actual discount was $1.88 and then prepared to do the long division, dividing $1.88 by $23.50.  Even a math teacher can’t do that kind of division in his head, so Matt got out his telephone so he could use the calculator. 

Rob started reading off the numbers.  While Matt was still entering the first number, I quietly said, “Eight.”

Rob, who is a very kind man, said, “It can’t be eight.  It’s got to have a decimal.”

As Rob finished giving Matt the numbers, I again quietly said, “Percent.”

As I said this, Matt held the telephone in front of my face.  On the screen was “0.08.”

“How did you do that?” demanded Rob.  “Are you some kind of savant?’

“I just knew,” I said gnomically and truthfully.

Today, Rob emailed me about something else, but that parlor trick of mine still had him bamboozled.

Today, while walking, I practiced a little introspective metacognition, which, of course, is a fancy way of saying I thought about the way I think.  In doing so, I recognized how I had managed to figure that problem out so quickly.  Sadly I had to conclude that I was not a genius, but that I had simply defined the problem differently from Rob.

Let me explain.  In Rob’s world, the world of math, this was simply a matter of computation, with an infinite number of possible answers.  Once the division was done, the answer would present itself at the end of the process.  At the beginning, though, the answer could be 7,930.45.  It could be 666.  It could be 2.  Until the computation is complete, the quotient of any two positive numbers can be any positive number.  Rob still needed to crunch the numbers to find that answer.

My system didn’t start with stating the problem.  Instead, I assumed that if I were going to calculate a discount rate in my head, I needed to first determine the limits of the universe of possible answers.  I assumed that the answer had to lie somewhere between five and fifteen percent.  Any less, and the discount would be almost an insult.  Any more, and the retailer would go out of business, unless his prices, like most jewelry stores, were so inflated that nobody paid full price. 

While I have not bought lumber more than three times in my life, I do go to hardware stores, which, in my brain, are similar.  No hardware stores use inflated prices, so I assumed neither did lumber yards. 

Because of my experience with retailers of all kind, I assumed that the discount rate would be either a whole number or a number ending in point five.  I mean, stores offer “Ten percent off!”, not “Eleven-point-three-six-seven-four percent off.”

Given the limits of my mental calculations, I dropped the notion of point five answers.  I just didn’t think I’d be able to do the math.

So, I mentioned the point where Rob needed to divide $1.88 by $23.50.  At that moment, Rob’s universe of possible answers was infinite, while mine consisted of eleven whole numbers, almost half of them single digits. 

Rob’s problem consisted of crunching a long division problem, while mine was simply a matter of looking for a relationship between the two numbers.  He had to input numbers and find an answer, while all I had to do was play around.

Knowing that dividing $1.88 by $23.50 is another way of saying, “What number times $23.50 gives you $1.88?”, which rules out all the numbers greater than nine, because ten percent of $23.50 is more than $1.88.  Starting at the bottom, I also ruled out five and six, because six percent of  $30 is $1.80, and the difference between $30 and $23.50 is too great,

So, while Rob was reading off his first number in a three-step process, I had reduced the universe of possible answers to two numbers, eight and nine.  Eyeballing things, I rejected nine because nine times two is eighteen, and as with six above, the difference is too great.  So, the answer, in my head was eight.  Checking the answer I multiplied eight time $23.50.  More accurately, I massaged the two numbers, and found that they fit together well.  Eight times zero.  Check.  Eight times five gives you 40 and another zero, always handy in percentage problem.  Eight times three is 24, plus the four from 40, adds up to an eight.  Good enough.  Call it a day.  Eight percent is the answer.

This system, of course, is not a way to build a bridge, engineer a space program, or even calculate gas mileage.  Still, it’s interesting to me to see how a combination of sociology and consumer science can beat a math guy at his own game.

Mainly, though, I’m glad to learn that, while I may be an idiot, I’m not an idiot savant.

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