Multiplying and Dividing Fractions

Date: 01/23/2001 at 13:05:33
From: Adom and Jamie
Subject: Multiplying and dividing fractions

We were just trying to figure out how to understand division of 
fractions and multiplication of fractions. It is weird because when 
I divided 1/2 by 1/2 on the calculator, I got 1, but when I multiplied 
them, I got 1/4.... 

I want to be able to explain in a drawing. HELP!
Adom and Jamie

Date: 01/25/2001 at 21:12:05
From: Doctor Ian
Subject: Re: Multiplying and dividing fractions

Hi guys,

Let's start from the beginning.  When you multiply by a whole number, 
you replicate something some number of times:

    * * *  x  4  =  * * * 
                    * * * 
                    * * * 
                    * * *

And when you divide by a whole number, you cut something into some 
number of pieces, and throw away all but one of them:

    * * *  /  4  =  * * *
    * * *
    * * *
    * * *

When you multiply by a fraction, you do BOTH of these things.  For 
example, to multiply by 3/4, you divide by 4 and then multiply by 3:

  * * * * *  x (3/4)  =  * * * * *  x  3  =  * * * * *
  * * * * *                                  * * * * *
  * * * * *                                  * * * * *
  * * * * *

or you multiply by 3 and then divide by 4:                            

  * * * * *  x (3/4)  =  * * * * *  /  4  =  * * * * *
  * * * * *              * * * * *           * * * * *
  * * * * *              * * * * *           * * * * *
  * * * * *              * * * * *
                         * * * * *
                         * * * * *
                         * * * * *
                         * * * * *
                         * * * * *
                         * * * * *
                         * * * * *
                         * * * * *

Either way you end up with the same result.  

So there are no new ideas here, just a couple of old ideas bunched 
together. 

If the fraction is less than 1, you lose more in the division than you 
gain in the multiplication. (For example, when you multiply a dollar 
by 3/4, you break the dollar into 4 quarters, and triple one of them, 
leaving you with 3 quarters.)

If the fraction is greater than 1, you gain more in the multiplication 
than you lose in the division. (For example, when you multiply a 
dollar by 5/4, you break the dollar into 4 quarters, and quintuple one 
of them, leaving you with 5 quarters.)

If the fraction is equal to 1, the multiplication and the division 
cancel each other out, and you end up where you started. 

If it helps, you can think of 

  1/4 * 1/4

as being the same thing as

  1 * 1/4 * 1/4

So, you start with a whole something; cut it into four pieces, and 
keep one; then cut _that_ into four pieces, and keep one.  Think about 
doing that with a pizza. How many pieces of the final size would you 
need to make a whole pizza? You'd need 16 of them, right? 
Conveniently, 

  1/4 * 1/4 is 1/16. 

How about 2/3 * 3/2?  Again, you start with a whole something; cut it 
into three pieces, and double one of them. Then you take those two 
pieces, cut each one in half, and triple each one of them.  What do 
you end up with?  Six pieces, each of which is 1/6th of the original 
thing. That is, you end up where you started. 

You can actually do this with paper and scissors, and that's not a bad 
idea if it isn't yet clear how this works. If you don't absolutely 
understand what it means to multiply something by a fraction, you are 
going to have great difficulty in every math class that you take from 
now on. 

Okay, so what about dividing by a fraction? Well, there is no good way 
that I know of to illustrate that with the kinds of pictures that you 
can use for multiplication. But perhaps that's not such a big deal, 
because division is primarily just another way of looking at 
multiplication.  That is, once we know something like

  24 = 6 * 4

this is really exactly the same piece of information as

  24 / 6 = 4     and    24 / 4 = 6

isn't it?  In fact, we DEFINE division this way.  We say that

  a / b = c     WHENEVER   c * b = a

That's the definition of division. That's what division MEANS. So how 
does this apply to fractions? Well, now that you know how to multiply 
fractions, you understand why

  9 * 2/3 = 6

right?  We divide 9 by 3 to get 3, and multiply that by 2 to get 6; or 
we multiply 9 by 2 to get 18, and divide by 3 to get 6.  Either way, 
this isn't a surprising fact.  

Well, because of how we've DEFINED division, if we say

  9 * 2/3 = 6

this is really exactly the same piece of information as

   6 / 9 = 2/3       and      6 / (2/3) = 9

Make sure you understand why this is true.  If it helps, look at these 
patterns again, and match up the letters with the numbers:

  b * c = a      <------->     a / b = c, and a /c = b
            
So, what do we have to do to get 6 from 9?  Well, we could cut it in 
half, and then triple what we get; or we could triple it, and take 
half of what we get. In other words, we get from 6 to 9 in this way:

  6 * (3/2) = 9

But we also know that

  6 / (2/3) = 9

And when two things are equal to the same thing, they must be equal to 
each other, right?  That means

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

So when you want to divide by a fraction, you invert the fraction and 
multiply instead.  

This is really all that's going on.  Note that I've used some 
particular numbers: 6, 9, 2/3, etc. - but you can work it out using 
only letters, and you'll see that so long as we accept the DEFINITION 
of division, we have to invert and multiply in order to divide by a 
fraction. If we did anything else, we'd get crazy results.  

In a sense, this goofy rule for dividing by fractions is the price we 
pay to keep the rest of math running smoothly.  

I hope this helps.  Write back if there is any part of this that 
wasn't crystal clear, or if you have any other questions. 

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