A rope ladder of length L is attached to a balloon of mass M. As the man of mass m climbs the latter into the balloon basket, the balloon comes down by a vertical distance s. Then the increase in potentail energy of man divided by the increase in potential energy of balloon is : 

\(\frac{L-s}{s}\)

Work done by man : m g L = m g ( L − s ) + m g s

where m g ( L − s ) is the increase in potential energy of the man and mgs is the increase in potential energy of the balloon because the balloon would have been lifted up but for the climbing of the man. So

\(\frac{\text{The increase in PE of man}}{\text{Increase in PE of balloon}} = \frac{mg(L-s)}{mgs}\)

   =\(\frac{L-s}{s}\)\)