Obviously part of the problem is setting up the equation to start with (and understanding that).
Once you get to an equation like yours, I personally think it's easier to simplify it with the variable (and you might as well keep going and solve it). The effort of manipulating an expression with a variable is often barely different to manipulating it after substituting a particular number for the variable, so doing the latter 5 times is almost certainly a waste of time.
A slight variation of "multiply both sides by the product of (200 + w) and (200 - w)" would be to put the RHS over a common denominator. And I wouldn't call that "factoring".
As far as a general strategy for multi choice problems that amount to solving an equation: On the one hand there might be situations where it's better to actually solve it then select the correct answer, and on the other hand there might be situations where it's better to substitute in each of the given choices. I couldn't give any rule of thumb for this but would say that someone who could consistently choose the better strategy would be at a higher level than the level being tested.
Also don't just think about how to get through a test. You want your DS to be able to do the algebra (whether or not it is necessary for this particular question) and if he's not fluent in doing it mentally he should build up his fluency on paper (i.e. "walk before you run (or fly)"). You have to think what's better for long term learning.
