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 Subject: RE: Formulas Author: Mathman Date: Apr 6 2005
On Apr  6 2005, jazy wrote:
> David, I am one of those students who knows to invert and multiply
> when when working with a problem that deals with the division of a
> fraction.  Could you please explain the principle on which this is
> based on.

Ingrid, I am referring to equivalence of fractions, related also to equality of
ratio and all other such relationships.  That is:

ma/mb <==> a/b

I also refer to the equivalence x/x=1 and x/1 = x.

So, when you have as a specific example (2/3)/(5/7), and if applying this
principle, you would multiply numerator and denominator by the same quantity.
That can be any quantity, but we choose the inverse of the lower fraction,
(7/5).  This is done to force the denominator to be a 1.

The result would be [(2/3)(7/5)]/[(5/7)(7/5)].  The second equivalence makes
this result in [2/3)(7/5)]/[1], and again simply (2/3)(7/5).

We avoid all of this writing, although the process of "invert and multiply"  is
based upon it, and write simply the end result (2/3)(7/5) from the initial
question, and carry on from there.

This is my point with transforming formulas.  The fact is that we "do the same
to both sides", but having done so many times we see that we don't need to, and
can simply transpose.  I believe that many students get frustrated by having so
much baggage to carry around wit hall of that writing, and that it makes them
more prone to error and disappointment.  A few examples would do to set the
principles, then lots of graded [very easy to difficult] questions for practice
applying the learned principles.  That is just my opinion and experience, and I
do not challenge in any way that of others who prefer to keep on writing
additional terms on both sides at every step.  However, I would suggest that it
is not done in higher education where it would be far too much of a burden.  But
techniques learned with simpler problems would still be applicable.  To me, it
is in a sense learning to "read" mathematics.  So, after some preliminary work
establishing the rules, I would concentrate on learning to resolve formulas and
equations by transposing terms.  It is in fact what I did, and it worked well
over the years.

David.