Restoring an Original Price

Date: 8/6/96 at 15:56:48
From: Anonymous
Subject: What Percent Markup do you Use?

I know there are formulas for restoring a price to its original value 
after taking a discount on it. How do you determine what the 
multiplier is for getting back to your original number?

Example: I start with a cost of $48,695. I discount it by 30 percent, 
bringing it to $34,087. What do I multiply $34,087 by to get back
to $48,695?

Your help in this matter would be very much appreciated. Please 
explain the method of getting the formula.

Thank you.

Date: 8/6/96 at 18:18:40
From: Doctor Mike
Subject: Re: What Percent Markup do you Use?

Hello, 
  
If you discount by a certain percentage, like 30 percent, the 
discounted price is 70/100 (or 70 percent) of the original.  
You then multiply the discounted price by 100/70 (about 1.4286) 
to get the original price back.  This multiplier is equivalent to 
142.86 percent of the discounted price. 
  
The 142.86 breaks down to 100 + 42.86, which means that to get back up
to the original price you have to take 100 percent of the discounted 
price and add to that 42.86 percent of the discounted price.  The 
42.86 percent part of it is the "percent of increase" over and above 
the discounted price.  This can be surprising at first; that you 
decrease by 30 percent but then must increase by about 43 percent to 
get back up to where you started. This is just another way of saying 
that "30 percent of the original price" is about the same as 43 
percent of the discounted price."
  
What we did for 30 percent we can do for any percentage "P".  The 
discounted price is (100-P)/100 of the original.  You multiply the 
discounted price by 100/(100-P) to get back up to the original price.  
Multiply this by 100 to get the equivalent percentage (like 142.86 
above), and then subtract 100 to get the percent of increase (like 
42.86 above).
  
        /   100         \            100*P
        | ------- * 100 | - 100  =  -------  = Grossing Percentage
        \  100-P        /            100-P 
  
Checking for P = 30% , (100*30)/(100-30) = 3000/70 = 42.85714.... 
  
If you want to work with multipliers rather than percentages of 
increase, just use "100/(100-P)" for the "grossing up" multiplier.
  
I hope this clears the process up for you.   

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/