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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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