Dividing by a Fraction

Date: 7 Jun 1995 13:15:08 -0400
From: Cad09
Subject: Dividing by a Fraction

Hello, my name is Janet.  I am a sixth-grade student at Cadwallader
Elementary school in San Jose, California.  

I have a question: Why is it when dividing fractions, you have 
to multiply by the reciprocal? 

Thank you.

Date: 7 Jun 1995 13:54:24 -0400
From: Dr. Ken
Subject: Re: Dividing by a Fraction 

Hello Janet.

Well, the answer has to do with what division IS.  I'll bet that before you
learned division, you learned multiplication.  You learned that 6 x 7 = 42,
and then a while later you learned that 42 % 6 = 7.  In this sense,
multiplication and division do opposite jobs; the technical term for this is
that multiplication and division are "inverse" operations.  (This computer
doesn't have the normal division symbol that you may be used to, so I'm
going to use the % symbol when I mean divide)

You may also have learned that whenever you divide by a number, that's
really the same thing as multiplying by the "inverse" of that number.  When
I say inverse here, I mean "one over that number."  Like 42 % 6 is the same
as 42 x 1/6.  Technically, what the inverse of a number means is that it's a
number you can multiply your first number by to get 1, for instance the
inverse of 6 is 1/6 since 6 x 1/6 = 1, and the inverse of -3/4 is -4/3,
since -3/4 x -4/3 = 1.

Now look at this.  One way we can write division is to write it as a
fraction: the number 42 % 6 is the same as the number 

42
---
 6  .


So let's say we have the division problem 


42       7
---  %  ---
 6       3  .



Instead of writing that as a division problem, we can write it as a
multiplication problem: dividing by a number is the same as multiplying by
its inverse.  So what's the inverse of 7/3?  It's 3/7, since 7/3 x 3/7 = 1.
Now we can rewrite the problem as

42       3
---  x  ---
 6       7  ,



and then you probably know how to do it from there:


42       3       42 x 3        6 x 1       6
---  x  ---  =   -------   =  -------  =  ---   =  3.
 6       7        6 x 7        2 x 1       2
                           ^
                           |
                           |
                           |In this step, I cancelled the 42 with the 7, 
                            and the 3 with the 6.

Thanks for the question!

-Ken "Dr." Math