Unfortunately, I did this problem by picking numbers, which took me much longer.
Given the problem and statements are asking about the combined work for both Mary and Nadir, I decided to use the Combined Work Formula, which is \(\frac{mn}{(m+n)}\)
S1) I decided to pick numbers originally I chose m=3 and n=5, but then I realized in order for statement 1 to be true I needed m=4 and n=5. This was to test for m<n so an always or sometimes Yes.
I then flipped those numbers to m=5 and n=4, to test for an always or sometimes No. However, when I tried to test this the numbers didn't comply with statement 1 and so the answer became an Always Yes, which is Sufficient.
\(\frac{mn}{(m+n)}>\frac{m}{2}\)
\(\frac{mn}{(m+n)}=\frac{4*5}{(4+5)}=\frac{20}{9}=2\frac{2}{9}>\frac{4}{2}\)
Yes
\(\frac{mn}{(m+n)}=\frac{5*4}{(5+4)}=\frac{20}{9}=2\frac{2}{9}<\frac{5}{2}\)
Since \(2\frac{2}{9}<2\frac{1}{2}\), the statement is not True and these numbers can't be chosen
S2) I used the same numbers m=4 and n=5 to test for an always/sometimes Yes and I used m=5 and n=4 for an always/sometimes No. Once I put the numbers into the combined work formula only m=4 and n=5 complied with Statment 2. I also tried m=5 and n=6, which also turned out to work with Statement 2 and gave me an Always Yes, which makes the statement Sufficient
\(\frac{mn}{(m+n)}<\frac{n}{2}\)
\(\frac{mn}{(m+n)}=\frac{4*5}{(4+5)}=\frac{20}{9}=2\frac{2}{9}<\frac{5}{2}\)
Yes
\(\frac{mn}{(m+n)}=\frac{4*5}{(4+5)}=\frac{20}{9}=2\frac{2}{9}>\frac{4}{2}\)
Since \(2\frac{2}{9}>2\), the statement is not True and these numbers can't be chosen
Both statements also work when m=5 and n=6, but doesn't work when m=6 and n=5
The answer IMO is Either or D