Multiple Percentages

Date: 1/26/96 at 23:10:18
From: Anonymous
Subject: Algebra (Percentages)

You have a pair of jeans.  Every time they are washed, they lose 
5 percent of their color.  After 12 washings, what is the percent of
their original color?  

The technique I used was so simple that I don't think it was right.  
I used 100 as the "color" of the jeans and multiplied it by .95, 
which is 95 (duh!).  Then, multiplying 95 by .95, the answer is 90.25. 

Going through this procedure 12 times, the final number is about 54, 
which I converted into 54%. Is this correct? 

By the way, can I set this problem up with the explicit formula?

Date: 5/30/96 at 14:51:25
From: Doctor Gary
Subject: Re: Algebra (Percentages)

Great work (although I think you meant to say that your final product 
was about .54, rather than 54).  I was especially impressed by your 
ability to "translate" the 5% color loss into 95% color retention, and 
your re-expression of a percentage into decimal notation.   

Using common sense, which is the most important tool in all of math, 
you have intuited a formula which is very important in fields as diverse 
as probability and finance. Whenever idependent events have the 
identical "geometric effect" (a fancy way of saying that you are 
continually mutiplying by the same number), a solution can be 
expressed exponentially.

Here, the formula would have been .95 (as you know, "per cent" is 
just a fancy way of saying "divided by 100") raised to the power of 12.

The only shortcut I might have used would have been based on the rule 
of exponents that:

   (b^m)^n  =  b^(mn)

I would have done a bit less multiplying and squared the result of 
3 "washings" (i.e. .95^3), to get the result of 6 "washings"(i.e.  
.95^6).  I would then have squared THAT number to get the result of 
12 "washings" (i.e. .95^12).

 Keep up the great work.

-Doctor Gary,  The Math Forum