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MatheMUSEments
Upgrading
By Ivars Peterson
Muse, October 2006, p. 33.
So it's already been a pretty tough school year? Some
of your grades are OK, but others are quite dismal (especially that
test you took the day after staying up way too late playing Guild
Wars online)? Luckily, your teacher says you can drop a single
test. Alright! But wait, which one?
The answer is easy if all of the tests are worth the
same number of points. You simply drop the lowest score. But, if the
tests are worth different numbers of points, dropping the lowest score
isn't always the best strategy. Consider the following example.
On the first quiz, you score 80 out of 100 points,
and get an 80 percent. On the second quiz, you score 20 out of 100:
20 percent. On the third quiz, you score 1 point out of 20: a miserable
5 percent. Without the option to drop one quiz, your final grade would
be 101/220, or 46 percent. Ouch.
If you drop your lowest score (1 point out of 20),
your final grade would be 100/200, or 50 percent. Ouch again. But,
if you drop the score on the second quiz instead, your final grade
would be 81/120, or 67.5 percent. Still not rocketing your way to
the academic honors list, but, OK, you're passing.
In this case, it pays to work out all possibilities
before deciding which grade to drop. Dropping your lowest score (or
your lowest percentage) wouldn't guarantee that you'll end up with
the best possible result.
It gets even trickier if you're allowed to drop two
or more scores from your total. And if you're dealing with a lot of
scores, it could take you a long time to calculate all the possibilities
to find the result that helps you the most, and who wants that?
Math professor Jonathan Kane of the University of
Wisconsin at Whitewater and his son Daniel, a student at the Massachusetts
Institute of Technology and twice a member of the U.S. Mathematical
Olympiad team, to the rescue. After the problem came up in conversation
during a 10K run, they worked out a fancy formula that pinpoints the
scores you should drop to get the best possible result, and created
a computer program that allows your teacher to do these calculations
quickly and easily.
AND, if you just mention casually that you're aware
of their work, that will probably get you five bonus points.
Read about the fancy formula here: web.mit.edu/dankane/www/droplowest.pdf.
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