Doctorates

this problem is fundamentally about inequalities - judging whether a particular number falls within a particular range - so you should consider EXTREME VALUES when appropriate.

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statement (1)
doesn't mention little 'w'
insufficient

statement (2)
consider extreme values.
one extreme: W = 5, w = 0. this is a 100% decrease, which is, obviously, greater than 10%.
other extreme: W = 367, w = 362 (can't go higher than this for 'w'). this is a decrease of 5/367, which is between 1% and 2%. (no need to calculate: 10% of 367 is 36.7, so this figure is way below 10%)
insufficient

(1) and (2) together
consider extreme values.
the lowest and highest possible values of W are 78 and 85 (because 1/10 < W/774 < 1/9 means 77.4 < W < 86, and it's "<", not "<").
if W = 78 and w = 73, then the decrease is 5 out of 78, which is less than 10% (note that 10% would be 7.8)
if W = 85 and w = 80, then the decrease is 5 out of 85, which is less than 10% (note that 10% would be 8.5)
so, less than 10% no matter what
sufficient

ans = c

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incidentally, the wording of this question is poor, especially by the unusually rigorous standards of the gmat. in particular, "such doctorates" is ambiguous; i had to read this question 3-4 times to make sense of it.