The balloon, the light rope, and the monkey shown in the figure are...

Since the system (balloon and monkey) is initially at rest in the air, the net external force acting on it is zero. The buoyant force balances the total weight of the system.

Because the net external force is zero, the center of mass of the system remains stationary during the motion of the monkey.

Let the balloon descend by a distance $x$ relative to the ground. Since the monkey climbs up the rope by a distance $L$ relative to the balloon, the upward displacement of the monkey relative to the ground is $L - x$.

Using the principle of conservation of the center of mass:

$M(-x) + m(L - x) = 0$

$-Mx + mL - mx = 0$

$x(M + m) = mL$

$x = \dfrac{mL}{M + m}$

Therefore, the balloon descends by a distance $\dfrac{mL}{M + m}$.