Initial momentum of man and balloon is zero, therefore on climbing upwards by man, the balloon will move downwards with velocity u such that the combined momentum of balloon and man may remains zero. The velocity of man with respect to Earth will be (v - u), therefore
m(v - u) - Mu = 0 or mv - mu = Mu
or mv = (Mu + mu)
∴ mv = u(M + m)
\(\therefore\ \mu = \frac{mv}{M + m}\)