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### 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.

more, or if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Fractions

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