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Fraction Operations Explained

Date: 11/23/2003 at 15:42:45
From: Akiko
Subject: Math notes

Can you help me understand adding, subtracting, multiplying and
dividing fractions?

Would it be a good idea to make a note page where I can copy it down 
and use it to study?

Date: 11/23/2003 at 23:09:27
From: Doctor Ian
Subject: Re: Math notes

Hi Akiko,

I'd say it's probably a bad idea to make up a page of notes for this,
because if you need the notes, it means you don't really understand
what happens when we add and subtract fractions, or when we multiply
and divide them.

If you _understand_ what's going on with those operations, you can
make up your own notes on the fly.  But of course, you won't need to. 

The thing to understand about adding and subtracting fractions is that
we really just want to use the distributive property.  That is, if we
have something like 

  2/7 + 3/7

we can use the distributive property to do just one division:

  (2 + 3) / 7

Now, note that I haven't mentioned fractions at all here.  We just had
two divisions, and an addition.  But here's the thing to understand--
all a fraction IS, is a division that we haven't done yet.  There's 
nothing more to it, really.  The fraction 3/5 just means 'whatever
we'd get if we divide 3 by 5'.  Let's try it:

     ____       3/5 = 0.60
  5 ) 3.0
      3 0

Now, what if we're trying to divide by different numbers?  For
example, suppose we have

  2/5 + 3/4

Now we can't use the distributive property...unless we arrange to be
dividing by the same thing in both places.  Note that in any division,
I can multiply both numbers by the same thing, and I end up with the
same quotient:
       0.75                0.75
      _____              ______
   4 ) 3.00       (5*4) ) (5*3)   

Do you see why this is true? 

Okay, so I can make this change:

    (2*4) / (5*4) + (3*5) / (4*5)
    \___________/   \___________/
     same as 2/5     same as 3/4

  =   8/20 + 15/20

and now I can use the distributive property again:

  =   8/20 + 15/20

  = (8 + 15)/20

  = 23/20

This is what we're doing when we find 'common denominators'.  

Let me know if this makes sense, and if it does, we can talk about
what's going on with multiplication and division of fractions, okay? 
- Doctor Ian, The Math Forum 

Date: 11/24/2003 at 00:17:23
From: Akiko
Subject: Math notes

Thanks.  Can you tell me what I need to know for multiplying and 
dividing fractions?

Date: 11/24/2003 at 09:53:05
From: Doctor Ian
Subject: Re: Math notes

Hi Akiko,

Let's start by thinking the following situation:

    6 * 8 / 2  = ?

Does it matter whether we do the multiplication first, and then the

    (6 * 8) / 2

  = 48 / 2

  = 24

or do the division first, and then the multiplication?

    6 * (8 / 2)

  = 6 * 4

  = 24

No, it doesn't.  So if we want to multiply an integer by a fraction,
we can go ahead and multiply the integer by the numerator, and keep
the same denominator:

  6 * 8/2 = (6*8)/2

          = 48/2

          = 24

We can do this even if the division wouldn't work out 'nicely':

  6 * 4/5 = (6*4)/5

          = 24/5

Now, what if we have a couple of divisions?

  6/3 * 8/2

Again, the order doesn't matter.  In fact, we can do the
multiplications first, and then the divisions:

    6/3 * 8/2

  = ((6 * 8) / 3) / 2

  = (48 / 3) / 2

  = 16 / 2

  = 8

It doesn't matter which division we do first:

    6/3 * 8/2

  = ((6 * 8) / 2) / 3

  = (48 / 2) / 3

  = 24 / 3

  = 8

And in fact, if you think about it, if I divide something by 2, and
then divide it again by 3, I'm really dividing it by (2*3), or 6.  
Does that make sense?

So what I'm really doing is this:

    6/3 * 8/2

  = ((6 * 8) / 2) / 3

  = (6 * 8) / (2 * 3)

  = 48 / 6

  = 8

In other words, I'm multiplying the numerators to get the new
numerator and multiplying the denominators to get the new denominator. 

What about division?  To understand that, it's easiest to think about
how we _define_ division.  Recall that if

  3 * 4 = 12


   3 = 12 / 4    and     4 = 12 / 3

These are all just different ways of saying the same thing.  

So, let's say we're dividing one fraction by another:

  (3/5) / (4/7) = ?

From the definition of division, we know that this has to be the same 

  (3/5) = ? * (4/7)

Okay, so now what?  Well, if we multiply two things that are equal by
the same thing, they stay equal, right?  So let's multiply both sides
of the equation by the reciprocal of (4/7):

  (3/5) * (7/4) = ? * (4/7) * (7/4)

Now, what happens on the right?  We can use the rule for multiplying

  (3/5) * (7/4) = ? * (4/7) * (7/4)

                = ? * (4*7)/(7*4)

                = ? * 28/28

                = ? * 1

                = ?

So this is what we were looking for originally.  Which means that the
way we divide by a fraction is to multiply by its reciprocal instead!

It seems a little funny at first, but once you understand how to get
there, you can come up with the rule on your own, whenever you need it.  

So, the rules you need to know are

  1) To add or subtract fractions with the same denominator,
     use the distributive property:

        a/b + c/b = (a + c)/b

  2) To multiply fractions, just change the order of the 
     operations, and remember that dividing by a and then 
     by b is the same as dividing by (a*b):

        a/b * c/d = a * c / b / d

                  = (a*c)/(b*d)

  3) To add or subtract fractions with different denominators,
     find equivalent fractions that have the same denominator,
     so you can use the distributive property:

        a/b + c/d = (a/b)*(d/d) + (c/d)(b/b)

                  = (a*d)/(b*d) + (b*c)/(b*d)

                  = [(a*d) + (b*c)]/(b*d)

  4) To divide by a fraction, multiply by its reciprocal:

        (a/b) / (c/d) = (a/b) * (d/c)

Now, if you just try to write down and _memorize_ these rules, you're 
going to forget them under stress (e.g., when you're taking a test).  
That happens to everyone.  

So my advice to you would be this:  Try starting with a blank piece of
paper, and use examples the way I have to make up the rules from
scratch.  Then check to make sure that you've done it correctly... and
throw away the paper.  Then do it again the next day. 

After a few times, you'll find that your brain is skipping ahead to
the final rules, without your having to 'remember' them.  

And if you find that when you try this, you're coming up with the
wrong rules, be sure to write back to me, and show me your steps, so I
can help you figure out what you're doing wrong.  

Does this help? 

- Doctor Ian, The Math Forum 

Date: 11/25/2003 at 00:30:01
From: Akiko
Subject: Math notes

Thanks a lot! We had a test today and I didn't have too much trouble 
figuring the questions out.  The only thing I got stuck on was mixed 

Date: 11/26/2003 at 08:16:32
From: Doctor Ian
Subject: Re: Math notes

Hi Akiko,

One straightforward way to deal with mixed fractions is to convert
them to improper fractions, do all your work using the rules we just
discussed, and then convert them back to improper fractions (although
the last step is optional).  

To see how to do the conversions, look at

  Converting Mixed Numbers to/from Improper Fractions 

Does this help?

- Doctor Ian, The Math Forum 

Date: 11/28/2003 at 20:40:48
From: Akiko
Subject: Thank you (Math notes)

Thanks! This helped a lot.  Now we're learning about percents, and the
last bit of fractions we did were really easy.
Associated Topics:
Elementary Fractions
Middle School Fractions

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