Recall the formula for the period of a mass $m$ attached to a spring with spring constant $k$ whose other side is attached to a wall:

$\begin{align*}T = 2 \pi \sqrt{\frac{m}{k}}\end{align*}$
Also, recall that the effective spring constant of two springs in parallel is $k_{eff} = k_1 + k_2$ and the effective spring constant of two springs in series is $1/k_{eff} = 1/k_1 + 1/k_2$ . So, the effective spring constant for Figure $1$ is $2k$ and the effective spring constant for Figure $2$ is $k/2$

Therefore, the desired ratio is

$\begin{align*}\frac{\displaystyle 2 \pi \sqrt{\frac{M}{2 k}}}{\displaystyle 2 \pi \sqrt{\frac{M}{k/2}}} = \boxed{\frac{1}{2}}...$
Therefore, answer (A) is correct.

Solution to 2001 Problem 90