Making Fractions into Percentages

Date: 04/30/2003 at 22:19:52
From: Bob
Subject: Percentages

I am really confused about how to make fractions into percentages.  
For example, 3 over 5 is what percent? Can you show me a simple way?

Date: 05/01/2003 at 09:30:02
From: Doctor Ian
Subject: Re: Percentages

Hi Bob, 

There are three kinds of cases. 

(1) If the denominator of the fraction evenly divides 100, as in your 
    example of 3/5, then just convert to the equivalent fraction:

  3   20    60
  - * -- = --- = 60%
  5   20   100

Note that 60/100 _is_ 60%.  Those are just two different notations for
the same thing.

(2) If the denominator evenly divides some higher power of 10, start 
    by doing the same thing,

  3   125    375
  - * --- = ----
  8   125   1000

and then move the decimal point over in the numerator while lopping
zero's off the end of the denominator, until you get down to 100:

  3   125    375   37.5
  - * --- = ---- = ----- = 37.5%
  8   125   1000    100

But don't just learn this as a rule.  Make sure you understand _why_
this works.  

How do you _know_ when the denominator will evenly divide 100, or some
other power of 10?  The easiest way to figure this out is to break the 
denominator into prime factors.  Consider the following cases:

   Fraction   Denominator    Prime factors
   --------   -----------    -------------
     1/2           2         2
     1/5           5         5
     1/8           8         2*2*2
     1/20         20         2*2*5
     1/15         15         3*5

Now, for something to be a power of 10, it has to have _only_ pairs of
2's and 5's as prime factors:

     10 = 2*5

    100 = (2*5)*(2*5)

   1000 = (2*5)*(2*5)*(2*5)

So if you have a denominator whose prime factors are only 2's and 5's, 
you can make an equivalent fraction by supplying the 'missing' 2's and 
5's.

  1   1   2*5*5       50
  - = - * ----- = -----------
  2   2   2*5*5   (2*5)*(2*5)

  1   1   2*2*5       20
  - = - * ----- = -----------
  5   5   2*2*5   (2*5)*(2*5)

  1     1     5*5*5         125
  - = ----- * ----- = -----------------
  8   2*2*2   5*5*5   (2*5)*(2*5)*(2*5)

Does that make sense?  

(3) Now, what about a denominator like 15?  

It has a 3 as one of its prime factors, so there's _nothing_ you can 
multiply it by that will let you end up with _only_ 2's and 5's as 
prime factors.  In this case, you just have to go ahead and divide:
     
       0.0 6 6 ...
      __________
  15 ) 1.0 0 0 0       
         9 0
       -----
         1 0 0
           9 0
         -----
           1 0 0 

At some point, you have to truncate to get an approximation, e.g., 

  1/15 = 0.06667

Once you've done that, you're almost done, because a decimal is really 
just a way of writing a fraction with a power of 10.

    0.07 = 7/100

   0.067 = 67/1000
  
  0.0667 = 667/10000

and so on.  Let's say we settle on 667/10000.  Now we move the decimal
place and get rid of zeros, as before:

   667    66.7    6.67
  ----- = ---- = ------ = 6.67%
  10000   1000     100

This third case, where you just go ahead and divide, is the most 
general, and can _always_ be used, if you don't feel like taking the 
time to find the prime factors of your denominator.  

Does this all make sense?  Do you think you can work these kinds of
problems on your own now?  (A good way to find out whether you
understand something is to try to explain it to someone who doesn't
already know it.)  Write back if any of this wasn't clear, or if you
have other questions. 

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