OA Explanation:

Let us assume the slowest one amongst three is X, then Y and Z be the quickest one.
Also, let time taken to finish the jon by all three (working simultaneously) be \(x\) days
i.e. \(\frac{1}{X} + \frac{1}{Y} + \frac{1}{Z}= \frac{1}{x}\)

When the slowest two work together they take n days to finish a job
\(\frac{1}{X} + \frac{1}{Y} = \frac{1}{n}\)

When the quickest two work together they take m days to finish the same job
\(\frac{1}{Y} + \frac{1}{Z} = \frac{1}{m}\)

One of them if works alone, would take thrice as much time as it would take when all three of them work together

Here, the important thing to notice is - For three people who are all equally efficient, it would take thrice as much time as for all three together.
So, the person who takes thrice as much time cannot be the quickest one. By the same logic, the slowest one cannot be the person who is one-third as efficient as entire team.

i.e Y would take thrice as much time as would take when all three work together

\(\frac{1}{x} = \frac{1}{m} + \frac{1}{n} - \frac{1}{3x}\)
\(\frac{1}{x} + \frac{1}{3x} = \frac{(m+n)}{mn}\)
\(\frac{1}{x}(1 + \frac{1}{3}) = \frac{(m+n)}{mn}\)
\(x = \frac{4mn}{3(m+n)}\)

Hence, option C