In figure (A), mass 2m is fixed on mass m which is attached to two...

The time period of spring block system is given by $T = 2\pi\sqrt{\dfrac{m}{k}}$.

In both the cases springs are connected in parallel,

In case $1$: ${k}_{eq} = 2k$ and total mass is $3m$. Therefore,

${T}_{1} = 2\pi\sqrt{\dfrac{\left(3m\right)}{{k}_{eq}}} = 2\pi\sqrt{\dfrac{\left(3m\right)}{2k}}$

In case $2:$${k}_{eq} = 3k$ and total mass is $m$. Therefore,

${T}_{2} = 2\pi\sqrt{\dfrac{\left(m\right)}{3k}}$

$⇒ \dfrac{{T}_{1}}{{T}_{2}} = \sqrt{\dfrac{9}{2}} = \dfrac{3}{\sqrt{2}}$