The [average](https://artofproblemsolving.com/wiki/index.php/Average) weight of $6$ boys is $150$ pounds and the average weight of $4$ girls is $120$ pounds.  The average weight of the $10$ children is 
$\text{(A)}\ 135\text{ pounds} \qquad \text{(B)}\ 137\text{ pounds} \qquad \text{(C)}\ 138\text{ pounds} \qquad \text{(D)}\ 140\text{ pounds} \qquad \text{(E)}\ 141\text{ pounds}$
Let the $6$ boys have total weight $S_B$ and let the $4$ girls have total weight $S_G$.  We are given 
\begin{align*} \frac{S_B}{6} &= 150 \\ \frac{S_G}{4} &= 120  \end{align*}
We want the average of the $10$ children, which is \[\frac{S_B+S_G}{10}\]  From the first two [equations](https://artofproblemsolving.com/wiki/index.php/Equation), we can determine that $S_B=900$ and $S_G=480$, so $S_B+S_G=1380$.  Therefore, the average we desire is \[\frac{1380}{10}=138 \rightarrow \boxed{\text{C}}\]