Percentage Problems

Date: 01/20/98 at 14:11:37
From: Tyrone J Billups
Subject: Percentage Proportion

My son came to me a with a math problem that is a little hard for me 
to understand. I am a formula man in that if I understand the formula, 
I can figure out how it works. Anyway, here is the formula:

   P/B = r/100 

where P = Percentage, B = Base and r/100 = rate. 

Now with this formula I need to figure out problems like this:

  a) 24 is 25% of what number?
  b) 24 is what percent of 25?
  c) What number is 25% of 24?

These are 3 different problems that have to be plugged into that 
formula to make it work. My problem is what numbers do I use for 
P, B, and r? Can you help me so that I can explain it to my son so 
that he can understand it?

P.S. Here is the answer that the book gave:

   a = 24/B = 25/100
   b = 24/25 = r/100
   c = P/24 = 25/100

I don't know where to put the numbers all the time.

Date: 01/22/98 at 12:34:10
From: Doctor Ezick
Subject: Re: Percentage Proportion

Good Question!

The way in which this particular formula works is as follows.

P/B = r/100  

  1. r = the percentage (i.e., the number with the percent sign.)

  2. B = base, the number which follows the word "of" - B is the thing 
         you are taking a percentage of.

  3. P = The value that proceeds the "is" - this is the value that is 
         the actual ammount you have.

An example:  10 is 20% of 50.

Then 10 = P
     50 = B
     20 = r

A great way to tackle these problems is to do the following:

  1. Treat the word "is" as an equal sign
  2. Treat the word "of" as a times sign
  3. Treat the percentage as a fraction r/100.

so, 10 is 20% of 50 becomes:

  10 = 20/100 x 50 

or, rewritten:  

  10/50 = 20/100

This is where the formula they gave comes from. Here I just divided 
both sides by 50.

P is r% of B becomes P/B = r/100, just as they have.

This parsing trick works wonders on these sorts of problems! Another 
addition to this trick is that the word "what" is usually the unknown.

To see how this works on an actual problem, consider "What number is 
25% of 24?" 

Broken down, this becomes:

What number is   25%   of  24
     P       =  25/100  x   24
     P / 24  = 25 /100
     P * 100 = 24 * 25   (cross multiply)
     P * 100 = 600
     P = 6               (divide both sides by 100)

That's all there is to it!

Hope this helps,

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