Fraction Multiplication and Division

Date: 02/10/2001 at 22:21:50
From: (anonymous)
Subject: Fractions are killing me

I don't understand problems like 2 1/10 * 7 5/8 or 27/30 divided by 
75/100. I try to work the problem by dividing the bottom by the LCD 
and just multiplying or dividing the top.

Date: 02/12/2001 at 12:28:48
From: Doctor Ian
Subject: Re: Fractions are killing me

Hi Tiffany,

Maybe we should start at the beginning. When we have a fraction like 
3/5, that is really shorthand for a pair of operations - 'cut 
something into 5 pieces, and keep 3 of them.'

So, let's say that we want to take 3/5 of six candy bars. We cut each 
of the bars into five pieces:

     +-+-+-+-+-+    +-+-+-+-+-+    +-+-+-+-+-+  
     | | | | | |    | | | | | |    | | | | | |
     +-+-+-+-+-+    +-+-+-+-+-+    +-+-+-+-+-+  

     +-+-+-+-+-+    +-+-+-+-+-+    +-+-+-+-+-+  
     | | | | | |    | | | | | |    | | | | | |
     +-+-+-+-+-+    +-+-+-+-+-+    +-+-+-+-+-+  

Then we keep three from every bar:

     +-+-+-+    +-+-+-+    +-+-+-+  
     | | | |    | | | |    | | | |
     +-+-+-+    +-+-+-+    +-+-+-+  

     +-+-+-+    +-+-+-+    +-+-+-+  
     | | | |    | | | |    | | | |
     +-+-+-+    +-+-+-+    +-+-+-+  

So we have 18 pieces, any five of which could be put together to make 
up a whole candy bar. In other words, we have 18 'fifths' of a candy 
bar...or 18/5 of a candy bar. If we try to combine the pieces into 
whole bars, 

     +-+-+-+    +-+-+-+    +-+-+-+  
     |1|1|1|    |1|1|2|    |2|2|2|
     +-+-+-+    +-+-+-+    +-+-+-+  

     +-+-+-+    +-+-+-+    +-+-+-+  
     |2|3|3|    |3|3|3|    | | | |
     +-+-+-+    +-+-+-+    +-+-+-+  

we get three whole bars, plus three extra pieces.  In other words, 

     18/5 = 3 3/5

This is what we're doing when we say that 

         3     18       3
     6 * -  =  --  =  3 -
         5      5       5

Okay, so what if we have something like 2/3 * 3/4?

Well, imagine that we have something like a cake. Can we take 2/3 of 
the cake?  Sure - cut it into three pieces, and keep two of them:

     +----------------+        +----------------+
     |                |        |                |
     +----------------+        +----------------+
     |                |  =>    |                |
     +----------------+        +----------------+
     |                |
     +----------------+

So, can we take 3/4 of the remaining part? Sure - cut each of those 
pieces into four smaller pieces, and keep three out of each four:


     +----------------+        +---+---+---+---+      +---+---+---+
     |                |        |   |   |   |   |      |   |   |   |
     +----------------+  =>    +---+---+---+---+  =>  +---+---+---+
     |                |        |   |   |   |   |      |   |   |   |
     +----------------+        +---+---+---+---+      +---+---+---+

So we end up with six small pieces. But how big are these pieces? In 
fact, 12 of these pieces would make up one whole cake. So each piece 
is 1/12 of a cake. And we have six of them, so we have 6/12 of a cake. 

Note that this happens to be the same thing as having 1/2 of a cake.  

Now, can we get this same answer without drawing pictures? In fact, 
we can, by multiplying the numerators and multiplying the 
denominators to get a new fraction:

     2   3     2 * 3      6     
     - * -  =  -----  =  --          
     3   4     3 * 4     12

We don't really have to worry about things like LCDs when we're 
multiplying or dividing by fractions. We just multiply the numerators 
and denominators.

So what about something like 2 1/10 * 7 5/8?  Well, there are a 
couple of things that we need to keep in mind.  The first is that 
2 1/10 is the same thing as 2 + 1/10... the '+' is implied.  

(Mathematicians are always making up notations that let them cut 
down on the amount of writing they have to do. So they write '4*5' 
instead of '5+5+5+5' or '4+4+4+4+4,' and they write '2^6' instead of 
'2*2*2*2*2*2,' and so on.)

The second thing we need to keep in mind is that 2 is the same as 
20/10 because 20 divided by 10 is 2. And 7 is the same as 56/8 because 
56 divided by 8 is 7.  

So we can rewrite the problem this way:

     2 1/10 * 7 5/8 = (2 + 1/10) * (7 + 5/8)

                    = (20/10 + 1/10) * (56/8 + 5/8)

                    = 21/10 * 61/8

And now we just have a regular fraction multiplication problem, which 
works the same way as 2/3 * 3/4. Multiply the numerators, and 
multiply the denominators:

                      21 * 61
                    = -------
                      10 *  8


So, what about dividing by fractions? 

There is a trick to dividing fractions, which is this: To divide by a
fraction a/b, you multiply by the fraction b/a. So, for example, 

     2/3
     --- = (2/3) * (4/3) = (2*4)/(3*3) = 8/9
     3/4 

Why does this work? It's not easy to come up with a picture to explain
this, but the reason it works is that you can always multiply 
something by 1 without changing it, and '1' comes in many disguises!

     2/3         2/3   4/3
     --- * 1  =  --- * ---
     3/4         3/4   4/3


                 2/3 * 4/3
              =  ---------
                 3/4 * 4/3


                   2/3 * 4/3
              =  -------------
                 (3*4) / (4*3)                 


                 2/3 * 4/3
              =  ---------
                  12 / 12 


                 2/3 * 4/3
              =  ---------
                     1 


              =  2/3 * 4/3


You could go through these steps every time you wanted to divide by a
fraction, e.g., 

     5/6    5/6  * 17/9
     ---- = ----------- = 5/6 * 17/9
     9/17   9/17 * 17/9

but in the end, it's easier just to remember the trick, and go right 
to the final step.  

If you understand _why_ the trick works, then you can always figure it 
out again if you forget. This is one reason that you should never be 
content just to memorize something without understanding it. If you 
don't use the things you've memorized, they fade in your memory, and 
then when you can't remember them, you're just stuck.  

I hope this has been helpful. Write back if any of this wasn't clear, 
or if you're still stuck.  

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