Converting Mixed Numbers to/from Improper Fractions

Date: 10/28/2001 at 15:37:01
From: Jo
Subject: Improper fractions

I have to change mixed numbers like 1 3/4 into improper fractions, and 
write fractions like 7/5 in simplest form by changing them to a mixed 
numbers. I don't know how to do this.

Date: 10/29/2001 at 11:56:41
From: Doctor Ian
Subject: Re: Improper fractions


Hi Jo,

You can represent a proper fraction with a picture like this, 

   +---+---+
   |###|   |
   +---+---+     3/4
   |###|###|
   +---+---+

right?  Well, you can represent a unitary fraction with a similar 
picture:

   +---+---+
   |###|###|
   +---+---+     4/4 = 1
   |###|###|
   +---+---+

So if you have something like 2 3/4, you can represent it this way:

   +---+---+     +---+---+     +---+---+
   |###|###|     |###|###|     |###|   |
   +---+---+     +---+---+     +---+---+
   |###|###|     |###|###|     |###|###|
   +---+---+     +---+---+     +---+---+
  
and now you can just count the #'s:

   +---+---+     +---+---+     +---+---+
   |###|###|     |###|###|     |###|   |
   +---+---+     +---+---+     +---+---+  = 11/4
   |###|###|     |###|###|     |###|###|
   +---+---+     +---+---+     +---+---+


Of course, it's a hassle to draw pictures, so you can do the same 
thing with numbers:

  4 3/5 = 4 + 3/5 

        = 1 + 1 + 1 + 1 + 3/5

        = 5/5 + 5/5 + 5/5 + 5/5 + 3/5

        = (5 + 5 + 5 + 5 + 3)/5

        = (4*5 + 3)/5

This last line looks almost like a formula, doesn't it?  It's just the 
same numbers arranged in a different way. I'll bet that if you were to 
work enough examples like this, you could probably convince yourself 
that you can convert any mixed fraction into an improper fraction this 
way:

  A + B/C = (A*C + B)/C

Of course, if you just try to remember the formula without 
understanding (from practice) why it works, you might get it mixed up.  
So you want to be careful not to try to 'memorize' the formula.  

Now, what about going the other way? Well, again using pictures, 
suppose we have a fraction like 25/6. We can start breaking 25 items 
into groups of 6:

  #####    #####          #####
  #####    #####   ###    #####   ###   ###
  #####  = ##### + ### =  ###   + ### + ### = ...
  #####    ####
  #####
           \__________/   \________________/
    25         19 + 6           13 + 2(6)

Eventually, you end up with 

       ###   ###   ###   ###
   # + ### + ### + ### + ###
  \__________________________/
         1 + 4(6)

Do you see how this corresponds to the picture that you would draw for 
the fraction 4 1/6?  

Again, you can do this with numbers instead of pictures, which speeds 
things up:

  25/6 = 19/6 + 1

       = 13/6 + 2

       =  7/6 + 3

       =  1/6 + 4

And eventually it would occur to you to notice that if you divide 25 
by 6, you get 4 remainder 1, which is to say, 

     ___
  6 ) 25  = 4 remainder 1  =  4 + 1/6

But again, if you just try to remember this 'method', you're likely to 
get mixed up under pressure (for example, when you're taking a test). 
The safest thing to do is to work it out the long way until you find 
yourself jumping ahead to the answer because you know what it's going 
to turn out to be.  

Does this help?  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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