The Worthless Increase

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!

Telling a novice this little story is quite disconcerting, since they would expect that with the same percentage increase and decrease you should be back to where you started. This is intuitive thinking, but wrong. Let them convince themselves 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 than the beginning amount.

Some 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 apparently makes no difference.

A similar situation, one that is deceptively misleading, can be faced by a gambler. Consider the following situation.

You are offered a chance to play a game. The rules are simple.

There are 100 cards, face down. Fifty-five 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 10 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 10,000 x (¾)⁴⁵ x (³/₂)¹⁰

This is $1.38 when rounded off. Surprised?