Calculating Percentage Increase

Date: 07/09/2004 at 15:38:52
From: Sharon
Subject: (no subject)

If my salary increased from 45k to 65k over a period of 10 years, what 
percentage increase did I receive per year?

Date: 07/10/2004 at 12:24:57
From: Doctor Ian
Subject: Re: (no subject)

Hi Sharon,

It depends on how you're defining things.  To show you what I mean, 
suppose you're making $45K in 1990.  10% of that is $4.5K.  If you
receive an increase of $4.5K per year for 10 years, 

  1990   $45,000
  1991   $49,500         add 10% of $45,000
  1992   $54,000         add 10% of $45,000
  1993   $58,500         add 10% of $45,000
  1994   $63,000
   .
   .
  2000   $90,000

But suppose each year, you receive an increase amounting to 10% of
what you made that year.  Now each increase is a little bigger than
the last:

  1990   $45,000
  1991   $49,500         add 10% of $45,000
  1992   $54,450         add 10% of $49,500
  1993   $59,895         add 10% of $54,450

What's going on in this case is that each year, the previous salary is
multiplied by 1.10 (100% plus 10%), so after 10 years the salary would be

               10
  45,000 * 1.10   = 45,000 * 2.59

                  = 116,550

So to answer your question, if you're talking about the first case, 
you just take the total difference and divide by 10 to get the yearly
increase,

  65,000 - 45,000   20,000
  --------------- = ------ = 2,000
        10            10

and then you figure out what percentage that is of the original 
salary:

   2,000    ?
  ------ = ---
  45,000   100


   2    ?
  -- = ---
  45   100


  2*100 = 45*?

  200
  --- = ?
   45

which comes out to about 4.4%.  That is, if you increase your original
salary by 4.4% OF THE ORIGINAL SALARY for 10 years, you end up with $65K. 

In the second case, the situation is this:

            10
  45,000 * ?   = 65,000

We can start by dividing each side by 45,000:

            10
           ?   = 65,000 / 45,000
            10
           ?   = 1.44

Now what we want to know is:  What number, raised to the 10th power,
gives us 1.44?  It turns out that

       10
  1.037  = 1.438

so if you increase your original salary each year by 3.7% OF THE
PREVIOUS YEAR'S SALARY, after 10 years you end up with $65K.  

Which method you prefer might depend on whom you're trying to convince
of what.  :^D

Does this help? 

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