Comments on Dividing Fractions

Date: 03/15/2007 at 18:55:16
From: Tiffany
Subject: Is there a much simpler term in dividing fractions?

I was reading your posts on how to divide fractions and it took me a 
while to understand.  It's basically multiplying and find the 
reciprocal right?  My teacher told me you can use the KCF to help. 
KCF stands for "Keep it. Change it. Flip it."

ex. 4/5 divided by 1/5

 K   C    F
4/5  /   1/5

4/5  x   5/1  =  4

We can still use that too, right?

Date: 03/15/2007 at 20:09:01
From: Doctor Peterson
Subject: Re: Is there a much simpler term in dividing fractions?

Hi, Tiffany.

You didn't say which page(s) you read; sometimes we explain things the
long way in order to talk about why we do something, rather than just
tell you quickly what to do.  We usually avoid just giving mnemonics
like your KCF, but there's nothing wrong with it.  It's merely a very
short way to say the same thing we say.

My own "quick version", which is more in line with the way 
mathematicians like to think, is this: "division is defined as
multiplication by the reciprocal".  That is,

  a / b = a * 1/b

So, to divide a by b, we multiply a by the reciprocal of b, which
means exactly what your teacher says: we "change" the operation from
multiplication to division, and we "flip" the divisor to use its
reciprocal.

Similarly, I've heard subtraction explained as KCC, or "Keep Change
Change", meaning that you change the subtraction to an addition, and
change the sign of the second number (the subtrahend).  The
mathematician's version of that one is that subtraction is defined as
addition of the negative (additive inverse).  That is,

  a - b = a + -b

So really, subtraction and division are just addition and 
multiplication with the second operand "inverted" in an appropriate
way.  I rarely hear anyone point out the similarity of the two rules
(or definitions), but I think it's very useful to see it.

If you have any further questions, feel free to write back.


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