I think the idea of thinking about what problem-solving tactics you use and having the capacity to be systematic about it is fantastic. It's fine to apply the tactic that simply occurs to you and if that works, you're done. But after trying and failing with a couple, it is clearly advantages to be able to go systematically through your toolbox trying each tool one at a time. I think we all use the "solve the general problem first" tactic in complicated problems and sometimes find that the algebra yields a simple solution in which some unknowns have canceled out. But I must admit, I don't believe I tend to apply the "solve the general problem first" tactic to problems that I can do in my head.
With your problem, noting that both values are less than 1, I asked myself the question, "Which is closer to 1?" This is equivalant to the question, how does (3/28) / (2/17) compare with 1? Since 51/56 < 1 I decided that 25/28 is closer to 1 so it must be the bigger number. I like your tactic better because your comparison did not get flipped--one less thing to keep track of in your head.
Up until recently, I had two tactics I tended to use for attacking such comparison problems. Divide one by the other and see how the ratio compares with 1, and subtract one from the other and see how the difference compares with 0. I learned another tactic from a student from Viet Nam this semester. He came up with an answer to a question I posed to the class more rapidly than I expected and I asked him to show his work on the board. The question was, for sound produced by an oscillator vibrating at a fixed frequency in still air, which observed frequency is greater, that heard by the observer when the source is at rest with respect to the air and the observer is moving toward the source at speed u, or that heard by the observer when the observer is at rest with respect to the air and the source is moving toward the observer at the same speed u? The tactic was to multiply by 1 written in such a fashion as to make it obvious how the two expressions for the observed frequency compared.
Another Vietnamese student made an unknown volume go away in an Archimedes' principle problem by multiplying by 1 written as mass / ( density times volume). It shortens the solution by about one line.
It's a handy tactic that I have underutilized heretofore.
It doesn't appear particularly advantageous in the problem you posted but this discussion made me think of it so I thought I'd throw it out there..
________________________________________
From: Phys-l [phys-l-bounces@phys-l.org] on behalf of John Denker [jsd@av8n.com]
Sent: Thursday, December 19, 2013 9:04 AM
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] just for fun
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].