It seems all too often that we get tangled up in a myriad of potential subjects to study, and in the process, spend more time deciding than actually studying. At least, I know for myself, the concept of the opportunity cost scares me.

So tonight, Vicki was asking me what she should study. And though I was reluctant to answer at first, I eventually formulated a solution. A mathematical solution. (This following formula will be slightly different from the one I gave her, but the general idea will be the same.)

Let’s define a calmness C, inversely proportional to urgency U, which will determine what you should study and what you can hold off. In other words

Now what could calmness, expressed as inverse of the importance of studying for a certain class, be parameterized by? So let’s brainstorm:

-Amount of knowledge you have now. Let’s called this *K*.

-Amount of studying you have already done. Let’s call this *S*. This may not matter much if *K *is small, but you’ll feel more calm and that may make an impact on exam day.

-Time between a set time *t _{0} *and time of exam

*t*Let’s just allow

_{e}.*t*to be 0 (“now”), and

_{0}*t*to be

_{e}*t.*

*–*The probability that you will study a given subject when you say you will study it. Let’s call this *P(S). *Let’s face it. If you hate the subject you will spend half of your time “studying” on Facebook. For now, I will ignore this term because this has a sort of time dependence. We’ll assume that this factor is included in *dK*/*dS*, described below; in other words, studying will be done no matter what, but it will be done slower. (Actually, I’m ignoring it because I forgot to put this factor in my original equations.)

*–*Rate of knowledge acquired per unit time studying. In other words: *dK/dS*. Some stuff is just hard to learn. Other stuff is just boring to learn. If you know you won’t learn much in 4 hours of studying, you will likely not spend those 4 hours studying! So this factor is INVERSELY proportional to the “calmness”, or proportional to urgency.

-Grade increase per unit knowledge gained, *dG/dK*. In some classes, the final is so ridiculously easy that you can know almost nothing and still get a decent grade. In others, the final is so ridiculously hard that you can know almost everything and still bomb the test. So this factor is also INVERSELY proportional to the the “calmness”, or proportional to urgency.

So then, how can we express U? There are two ways. I will express both.

1) As the product of the parameters:

or, alternatively and more conveniently,

The advantage to this is you can see some interesting trends. For instance, the knowledge *now *matters, but possible knowledge to be acquired during the study process (i.e. *dK*) does not show up in the formula. Rather, the grade increment per unit studying takes precedence. Which begs the question… do students really care about what they learn, or do the ends of a good GPA justify the means?

2) As a weighted sum (linear combination) of the parameters:

(Note that to be inversely proportional, the last two weights will have to be negative. Alternatively, one can write the last two variables as (dK/dS)^-1, etc, but I’m too lazy to update equations. We’ll assume we can have negative numbers in this formula.)

No way to reduce this into an elegant, self-explanatory form, but the advantage here is any individual can choose to put different weights into the different parameters, weighing them more heavily relative to other parameters. This may be the more practical formula for the confused individual deciding what to studying. And, it’s linear, so that’s kinda nice.

So now, off to more finals studying!