The area of each of the four congruent L-shaped regions of this 100-inch by 100-inch square is 3/16 of the total area.  How many inches long is the side of the center square?
[asy] draw((2,2)--(2,-2)--(-2,-2)--(-2,2)--cycle); draw((1,1)--(1,-1)--(-1,-1)--(-1,1)--cycle); draw((0,1)--(0,2)); draw((1,0)--(2,0)); draw((0,-1)--(0,-2)); draw((-1,0)--(-2,0)); [/asy]
$\text{(A)}\ 25 \qquad \text{(B)}\ 44 \qquad \text{(C)}\ 50 \qquad \text{(D)}\ 62 \qquad \text{(E)}\ 75$
The area taken up by the L's is $4*\frac{3}{16}=\frac{3}{4}$ of the area of the whole square. What remains has $\frac{1}{4}$ of the area of the larger square. $\frac{100*100}{4}=\frac{100}{2}*\frac{100}{2}=50*50$ is the area of the smaller square, so its side length is 50. $\text{(C)}$