Changing Decimals to Fractions and Reducing

Date: 12/12/2002 at 20:42:29
From: Bihanca 
Subject: Changing decimals to fractions

Hi, 
     
Could you please help me to understand, in the simplest way, how to 
write a decimal as a lowest-term fraction or mixed number?

Example: 4.35

I'm stuck after this. I just don't get it.
Thank you!

Date: 12/12/2002 at 22:13:38
From: Doctor Ian
Subject: Re: Changing decimals to fractions

Hi Bihanca,

The absolute simplest way is this.  First, separate the integer and
decimal parts, like so:

                    375
  4.375   --->    4 ---

Now, for each digit in the decimal part, put a (2*5) in the 
denominator: 
                         375
  4.35   --->    4 -----------------
                    (2*5)(2*5)(2*5)

Now start trying to divide those 2's and 5's into the numerator, to
cancel them out. For example, 375 is divisible by 5, so go ahead and
do the division:

         75
  4 --------------
     (2*5)(2*5)(2)


75 is also divisible by 5:

          15
  4 --------------
     (2*5)(2)(2)


And 15 is divisible by 5:

         3
  4 -----------
     (2)(2)(2)


There are no more divisions to do, so you're done:

    3
  4 - 
    8


Does it look kind of like a magic trick? If so, think about what we
did with the fractional part. When we write something like 

  0.375

this is really just a very compact way of writing something much 
bigger:

                            3    7      5
  0.375   is the same as   -- + --- + ----
                           10   100   1000

(If you didn't already know this, make sure you understand it now. And 
if you're having trouble making sense of it, please let me know, 
because it's a really crucial thing to understand.) 

Now, note that 


   3    7      5      300    70      5
  -- + --- + ---- =  ---- + ---- + ----
  10   100   1000    1000   1000   1000


                      300 + 70 + 5 
                  =  --------------
                         1000

                       375 
                  =  ------
                      1000


                         375 
                  =  ----------
                      10*10*10


                            375 
                  =  ----------------
                      (2*5)(2*5)(2*5)

So all I did was skip a few steps and go right to the prime
factorization of the denominator, which makes it easier to do the
reduction.  

If you don't have to worry about reducing the fraction to lowest
terms, it's even easier. Just separate the integer and decimal parts, 

     375
  4 -----

and write a zero for each digit in the denominator:

     375
  4 -----
     000

Then put a '1' in front:

     375
  4 ----
    1000

There is one final thing to watch out for. If you have any leading
zeros, you have to include them as digits during the first step:

                         035
  4.035   --->   4 ---------------
                   (2*5)(2*5)(2*5)


Otherwise you'd end up treating 4.35 and 4.035 as if they were the
same number, which they're not.

Does this make sense? 

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