A balloon with mass $m$ is descending down with an acceleration$a (where a<g)$. How much mass should be removed from it so that it starts moving up with an acceleration $a$?

Let $F$ be the upthrust of the air. As the balloon is descending down with an acceleration $a$ 

$mg - F = ma ....\left(i\right)$

Let mass $m_0$be removed from the balloon so that it starts moving up with an acceleration $a$. Then, 

$$\begin{gathered} F -(m-m_0)g = (m-m_0)a \\ F - mg + m_{0}g = ma - m_{0}a ...\left(ii\right) \end{gathered}$$

Adding eqn (i) and (ii) 

$$\begin{gathered} m_{0}g = 2ma -m_{0}a ; m_{0}g +m_{0}a = 2 ma \\ {m}_0(g+a) = 2 ma \\ {m}_0 = \frac{2ma}{a+g} \end{gathered}$$