The sum of two numbers is $S$. Suppose $3$ is added to each number and then
each of the resulting numbers is doubled. What is the sum of the final two
numbers?
$\textbf{(A)}\ 2S + 3\qquad \textbf{(B)}\ 3S + 2\qquad \textbf{(C)}\ 3S + 6 \qquad\textbf{(D)}\ 2S + 6 \qquad \textbf{(E)}\ 2S + 12$
Suppose the two numbers are $a$ and $b$, with $a+b=S$.
Then the desired sum is
$2(a+3)+2(b+3)=2(a+b)+12=2S +12$, which is answer $\boxed{\textbf{(E)}}$.