Fractions, Improper Fractions, Mixed Numbers

Date: 02/01/97 at 23:36:43
From: John
Subject: Fractions, Improper Fractions, Mixed Numbers

How do you turn an improper fraction into a mixed number?

John

Date: 02/02/97 at 15:21:15
From: Doctor Reno
Subject: Re: pre-algebra

Hi!

Thanks for asking us for help, John. Once you truly understand what 
fractions are, it will become very easy for you to change improper 
fractions into mixed numbers! Just remember that *all* fractions are 
handled in the same way, whether they are "improper" or not.

I know you probably would like me to just give you some "rules" for 
doing this, but I have found it difficult to memorize all the "rules" 
in math. It is much easier for me to think about what I know about 
numbers so that I don't have to memorize stuff. When you memorize, you 
can easily forget the rules. When you understand a topic in math, when 
you know what is really going on, you *can't* forget what to do! My 
answer will take a while, though, so sit back with some paper and a 
pencil, relax, and enjoy!

The word fraction comes from an old Latin word that means "broken". 
Fractions break 1 of something (like an apple, a cake) into pieces, 
any number of pieces we need to break it into. Regular numbers are 
"whole", they are unbroken. So we can think of fractions as parts of 
whole numbers, or parts of whole things.

A half is what we have when we *divide* a whole into two pieces. A 
half, then, is 1 divided by 2.  A third is 1 divided by 3. We don't 
even do the division, we just say the fraction.  Instead of saying and 
writing "1 divided by 3", we say one-third, and we write 1/3. We don't 
need to do the division, because fractions can be manipulated, or 
used, as if they are the answer! We can add 1/3 to other numbers, or 
multiply it, or subtract or divide it. 

We have also just learned in the paragraph above that the "/" symbol 
is a shorthand way of saying or writing "divide". This will help us a 
lot when we work with fractions.

We need to know how to change improper fractions to mixed numbers 
because it is a lot easier for people to understand a mixed number 
than an improper fraction. For example, isn't it easier for you to 
understand me when I tell you that I have 3 candy bars than if I tell  
you that I have 6/2 candy bars? 

Let's look at the improper fraction 22/3. This means that some objects 
have been divided into thirds, and that we have 22 of these pieces in 
front of us. But it's difficult for me to know *exactly* what 22/3 
means! Let's say we're talking about chocolate bars. It will be easier 
for me to know how much candy I have if 22/3 is changed to a mixed 
number. There are several ways to figure out what it means. Let's take 
a look at 3 ways.

One way is to remember that 3/3 makes one whole: if we have cut a 
candy bar into three pieces, those three pieces together are the whole 
candy bar. We can then subtract 3 pieces from the 22 pieces of our 
22/3 and we have 1 whole plus 19 pieces left over: 1 19/3. 

If we do this again, we find that we have 2 whole candy bars and 16 
pieces left over: 2 16/3. And if we do this *again*, we see that we 
have 3 whole candy bars and 13 pieces left: 3 13/3. We can do this and 
do this until we find that we have 7 whole candy bars with 1 piece 
left over.

But this takes a long time, doesn't it? It would be a big help if we 
could find a quicker way to do this! Well, we could use our knowledge 
of arithmetic again to help us out. 22/3 = 21/3 + 1/3. And we know 
that 21/3 = 7 (21 divided by 3 is 7). And we then find that 22/3 is 
7 whole candy bars with 1 piece (1/3) left over. But even *this* 
method can take a long time, can't it?

The easiest method of all is to divide 22 by 3. Remember, we can do 
that because the "/" means divide. We did it in the paragraph above. 
22 divided by 3 is 7 with 1 left over. The only thing we have to be 
careful about is to remember exactly what that left over 1 means. It 
means 1 piece of a candy bar that has been cut into 3 pieces. 
Therefore, the 1 actually means 1/3 of a candy bar. 22/3 = 7 1/3.

Now that we understand fractions better, we can change *any* improper 
fraction into a mixed number! All we have to do is divide (and 
remember what the remainder means). 

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