Figuring Out Fractions

Date: 04/28/2004 at 12:43:57
From: Meghan
Subject: Fractions

Dear Dr. Math,

I need help with adding and subtracting fractions.

I just don't get it! It's so confusing.  I have to do page after page,
and it seems like doing one page takes forever. 

Please try to explain it so it's not difficult to understand.  Here is
an example:

   1     5
 1--- + --- = ?
   4     6

Date: 04/28/2004 at 13:37:46
From: Doctor Ian
Subject: Re: Fractions

Hi Meghan,

The place to start is by working with fractions that have the same
denominator.  Can you do this example?

  30   20
  -- + -- = ?
  24   24

Try that, and let me know what you come up with.  Then I'll show you
how to turn your problem into one that looks like this.

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

Date: 04/28/2004 at 16:19:17
From: Meghan
Subject: Re: Fractions

I get
 
   30     20     50       2       1
  ---- + ---- = ---- = 2---- = 2----
   24     24     24      24      12

But I still don't know how to do this starting with mixed numbers.

Date: 04/28/2004 at 21:44:51
From: Doctor Ian
Subject: Re: Fractions

Hi Meghan, 

Nicely done!

Note that mixed numbers and fractions are just two ways of looking
at the same thing.  They're interchangeable.  

So, here's why I picked this example.  Let's look at 1 1/4:

  1 1/4

As you seem to know already, you can convert this to an improper fraction:

  1 1/4 = 5/4

What you may not realize is that you can convert this to any number of
equivalent fractions, i.e., fractions that look different but have the
same value.  For example, suppose I multiply 5/4 by 1.  I end up with
the same thing, right?

  5       5
  - * 1 = -
  4       4

Okay, now note that any number (except zero) divided by itself is 
equal to 1:

  2
  - = 1
  2

  3
  - = 1
  3

and so on.  Does this make sense?  So, what happens if we do this?

  5   2
  - * - = ?
  4   2

Well, if we follow the rules of fraction multiplication, we get

  5   2   5 * 2   10
  - * - = ----- = --
  4   2   4 * 2    8

So, what does this tell us?  Since we multiplied by 1, we didn't 
change anything.  So 5/4 and 10/8 are equal.  That is, they look
different, but they have the same value; and anywhere we can use one,
we can use the other.  

This means that 

  5   5   10   5    
  - + - = -- + -
  4   6    8   6
  
Maybe we can't do either addition right now, but we know that the two
additions should give the same result.  

Okay, so here's the big trick:  We'd LIKE to convert the addition

  5   5
  - + - = ? 
  4   6

to something where both fractions have the same denominator.  For
example, suppose I multiply 5/4 by 6/6, and multiply 5/6 by 4/4?  Then
I get

  5*6   5*4   30   20
  --- + --- = -- + --
  4*6   6*4   24   24

and you already know how to do the addition on the right. We know 
that, because you already did it!

So the only piece you're missing is how to make the denominators the
same; and that's just a matter of multiplying each fraction by N/N,
where N is the denominator of the other fraction. 

Here's another example:

  1   2   1   7     2   3
  - + - = - * -  +  - * -
  3   7   3   7     7   3

          1*7   2*3
        = --- + ---
          3*7   7*3

           7    6
        = -- + --
          21   21

          7 + 6
        = -----
            21

          13
        = --
          21

Does this make sense?

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

Date: 04/29/2004 at 11:49:23
From: Meghan
Subject: Re: Fractions

I'm sorry, Dr. Math.  It doesn't make sense...

Date: 04/29/2004 at 12:47:26
From: Doctor Ian
Subject: Re: Fractions

Hi Meghan,

No need to be sorry.  But can you tell me _which_ part doesn't make
sense?  For example, do you understand how 

  2  4  6
  -, -, -, ...
  3  6  9

all have the same value, even though they look different? 

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

Date: 04/29/2004 at 16:45:20
From: Meghan
Subject: Re: Fractions

I sort of understand that part, but it's tricky for me.  I'm good at
math, but not fractions.

Date: 04/29/2004 at 19:16:49
From: Doctor Ian
Subject: Re: Fractions 

Hi Meghan,

There are a couple of ways to think about it.  One is this:  Suppose I
have an item, and cut it into three pieces:

   +---+---+---+
   |   |   |   |
   |   |   |   |
   |   |   |   |        3/3 of an item
   |   |   |   |
   |   |   |   |
   +---+---+---+

If I 'keep' 2 of these pieces, that represents 2/3 of the thing, right?  

   +---+---+---+
   |   |   |...|
   |   |   |...|
   |   |   |...|        2/3 of the item
   |   |   |...|
   |   |   |...|
   +---+---+---+

Now, what if I cut each of my pieces into two other pieces?

   +---+---+---+
   |   |   |...|
   |   |   |...|
   +---+---+---+        4/6 of the item
   |   |   |...|
   |   |   |...|
   +---+---+---+

Now I have 4 out of 6 pieces, instead of 2 out of 3.  But it's the
same part of the item!  I just cut it into more pieces, and kept more
of them.  But the _ratio_ of pieces kept to total pieces is the same. 

So 2/3 and 4/6 are two different fractions, that have the same value. 
Does that make sense?

Another way to think about it is to realize that when we write a
fraction like 3/5, what we really _mean_ by that is

  3
  - = whatever number we'd get by dividing 3 by 5
  5

That is, to find the value of 3/5, we do this:

      0.6
     ____
  5 ) 3.0               3/5 = 0.6
      3 0
      ---
        0

Now, suppose we divide 10 into 6:

       0.6
      ____
  10 ) 6.0              6/10 = 0.6
       6 0
       ---
         0

Or suppose we divide 15 into 9:

       0.6
      ____
  15 ) 9.0              9/15 = 0.6
       9 0
       ---
         0

In all three cases, we get the same value.  So we say that 

  3    6    9
  - = -- = --
  5   10   15

By the way, this is one way to check whether two fractions are equal.
Divide the numerators by the denominators, and see if you get the same
things.  So we might ask:  Is 40/75 the same as 3/5?  Let's find out:

        0.6
      _____
  75 ) 45.0
       45 0

It's the same!  What about 30/55?

        0.54
      _____
  55 ) 30.0
       27 5
       ----
        2 50
        2 20
        ----
          30

We don't even have to finish to see that they aren't the same. 

Let me know if this makes sense to you; or if it doesn't, which part
you find confusing.  

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

Date: 04/30/2004 at 16:44:59
From: Meghan
Subject: Thank you

Thank you, Dr. Math!

I finally get fractions! You helped me!

Yours truly,
Meghan