At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for $1000$ of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these $1000$ babies were in sets of quadruplets?
$\textbf{(A)}\ 25\qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 100\qquad\textbf{(E)}\ 160$
We can set up a system of equations where $a$ is the sets of twins, $b$ is the sets of triplets, and $c$ is the sets of quadruplets.
\[\begin{split} 2a + 3b + 4c & = 1000 \\ b & = 4c \\ a & = 3b \end{split}\]
Solving for $c$ and $a$ in the second and third equations and substituting into the first equation yields
\[\begin{split} 2 (3b) + 3b + 4 (0.25b) & = 1000 \\ 6b + 3b + b & = 1000 \\ b & = 100 \end{split}\]
Since we are trying to find the number of babies and NOT the number of sets of quadruplets, the solution is not $c$, but rather $4c$. Therefore, we strategically use the second initial equation to realize that $b$ $=$ $4c$, leaving us with the number of babies born as quadruplets equal to $\boxed{\textbf{(D)}\ 100}$.