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 MARCH 2006

The Dean's Column:
The Worthless Increase

By Alfred Posamentier, Ph.D.

Suppose you had a job where you received a 10% raise. Because business was falling off, the boss was soon forced to give you a 10% cut in salary.  Will you be back to your starting salary? The answer is a resounding (and very surprising) NO!

This little story is quite disconcerting, since one would expect that with the same percent increase and decrease your should be back to where you started.  This is intuitive thinking, but wrong.  Convince yourself of this by choosing a specific amount of money and trying to follow the instructions.

Begin with \$100.  Calculate a 10% increase on the \$100 to get \$110. Now take a 10% decrease of this \$110 to get \$99—\$1 less that the beginning amount.
You may wonder whether the result would have been different if we had first calculated the 10% decrease and then the 10% increase. Using the same \$100 basis, we first calculate a 10% decrease to get \$90. Then the 10% increase yields \$99, the same as before. So order makes no difference.

A similar situation, one that is deceptively misleading, can be faced by a gambler. Consider the following situation.  You may want to even simulate it with a friend to see if your intuition bears out.

You are offered a chance to play a game.  The rules are simple.  There are 100 cards, face down. 55 of the cards say “win” and 45 of the cards say, “lose.”  You begin with a bankroll of \$10,000. You must bet one half of your money on each card turned over, and you either win or lose that amount based on what the card says. At the end of the game, all cards have been turned over.  How much money do you have at the end of the game?
The same principle as above applies here.  It is obvious that you will win ten times more than you will lose, so it appears that you will end with more than \$10,000.  What is obvious is often wrong, and this is a good example. Let’s say that you win on the first card; you now have \$15,000.  Now you lose on the second card; you now have \$7,500. If you had first lost and then won, you would still have \$7,500.  So every time you win one and lose one, you lose one-fourth of your money. So you end up with….
This is \$1.38 when rounded off.  Surprised?#

You may find other such examples in Math Wonders: To Inspire Teachers and Students, by Alfred S. Posamentier (ASCD, 2003) see: www.ascd.org. or Math Charmers: Tantalizing Tidbits for the Mind, by Alfred S. Posamentier (Prometheus Books, 2003) see: www.prometheusbooks.com. If you wish to learn more about Pi, see: Pi : A Biography of the World’s Most Mysterious Number, by Alfred S. Posamentier (Prometheus Books, 2004) see: www.prometheusbooks.com.

Dr. Alfred S. Posamentier is Dean of the School of Education at City College of NY, author of over 35 books on math, and member of the NYS Standards Committee on Math.