jmc

Let the $6$ boys have total weight $S_B$ and let the $4$ girls have total weight $S_G$. We are given $$\begin{array}{rl}\frac{S_B}{6}& =150\\ \frac{S_G}{4}& =120\end{array}$$ We want the average of the $10$ children, which is $$\frac{S_B+S_G}{10}$$ From the first two equations, we can determine that $S_B=900$ and $S_G=480$, so $S_B+S_G=1380$. Therefore, the average we desire is $$\frac{1380}{10}=138\to \boxed{138\text{ pounds}}$$