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### Fractions, Improper Fractions, Mixed Numbers

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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
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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/
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Associated Topics:
Elementary Fractions
Middle School Fractions

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