Percentage Increase vs. Percentage

Date: 05/13/2003 at 13:57:36
From: Diane Portantiere
Subject: Percentage increase

If there were 10,000 claims in 2001, and that is a 300 percent 
increase since 1999, how many claims were there in 1999? 

My colleagues are giving me all different answers. I think the answer 
is 2,500. My colleagues say 3,333.

Please help.

Date: 05/13/2003 at 14:38:27
From: Doctor Ian
Subject: Re: Percentage increases

Hi Diane, 

A 300% increase means that 

    (claims in 2001) - (claims in 1999)   300
    ----------------------------------- = ---
               (claims in 1999)           100

That is, we're saying that the _increase_ is 300% of the original 
value.  

Since the numbers are so nice and round, we can do this:

    (claims in 2001)    (claims in 1999)   300
    ----------------- - ---------------- = ---
    (claims in 1999)    (claims in 1999)   100

                   (claims in 2001)        
                   ----------------- - 1 = 3
                   (claims in 1999)        

                       (claims in 2001)        
                       ----------------- = 4
                       (claims in 1999)        

So the number of claims in 2001 is 4 times whatever it was in 1999,
which means there were 2,500 claims in 1999. 

Note that if we change the wording slightly, we can come up with the
other answer. That is, if we say that the number of claims in 2001 is
300% of the number of claims in 1999, then we're saying

   (claims in 2001)   300
   ---------------- = ---
   (claims in 1999)   100

which we can rearrange to get

                      100 * (claims in 2001)
   (claims in 1999) = ----------------------
                              300

                      100 * 10,000
                    = ------------
                          300

                      10,000
                    = ------
                         3

                    = 3333 1/3

Let's put the two cases together, so you can compare them more easily:

   1) Claims in 2001 are an increase of 300% over claims in 1999.
      (The increase is 300% of the old value.)

      (claims in 2001) - (claims in 1999)   300
      ----------------------------------- = ---
                 (claims in 1999)           100


   2) Claims in 2001 are 300% of the claims in 1999. 
      (The new value is 300% of the old value.) 

      (claims in 2001)   300
      ---------------- = ---
      (claims in 1999)   100

Does this make sense? 

I find that it's useful to keep a few simple examples in my head. One
of my favorites is this: If I start with $1, an increase of 100% is
an increase of $1, which gives me $2, but $2 is twice as much as $1,
which means it's 200% of $1.

I hope this helps.  Write back if you'd like to talk more about this,
 or anything else.   

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/