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The question was:
Which is bigger: 25/28 or 15/17? [1]
Here's my take: Write it as
a b
------- ¿>? ------- [2]
a + 3 b + 2
We know the value of a and b, so we don't need to solve
for them, but for now let's leave them as symbolic rather
than numeric.
Cross multiply. Throw away the "ab" term from both sides.
This leaves us with
2a ¿>? 3b
Now plug in the numeric values and do the multiplication.
I can do all of the above in my head, in less time than it
takes to find a pencil and paper.
=================
The larger point here is that sometimes it is /easier/ to do
the general case rather than the specific case. It's just
plain easier, even if only one specific case is of interest.
-- The advantage is even greater if there are multiple specific
cases on the agenda.
-- The advantage is even greater if the generalization provides
some insight into the structure of the problem, into the
nature of the problem.
There is artistry involved in finding a "good" generalization.
Equation [2] is not the only possible generalization of
equation [1].
The artistry is not however a shot in the dark. Experience
suggests patterns that are worth looking at. In this case
there is an analogy to differential-mode signaling. On the
LHS "a" is the common-mode signal, common to both numerator
and denominator, while "3" is the differential-mode signal.
Rewriting it so as to focus attention on what's common and
what's different is a technique that you can use in lots of
situations. There is no chance that HS students will have
the experience and expertise to do something like this, which
is why this is not a placement-test question but rather a
just-for-fun question.
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