1. Use the SHM time period formula

For a mass-spring system, $T=2\pi\sqrt{\frac{M}{k_{\text{eff}}}}$ where $M$ is the oscillating mass and $k_{\text{eff}}$ is the effective spring constant.

2. Case (A)

The block of mass $2m$ is fixed on the block of mass $m$, so both move together as a single body.

Hence total oscillating mass is $M_1 = m+2m=3m.$

The mass is attached between two springs, each of spring constant $k$. When displaced horizontally by $x$, one spring is compressed by $x$ and the other is stretched by $x$, so restoring forces add.

Thus, $k_{\text{eff,1}}=k+k=2k.$

Therefore, $T_1=2\pi\sqrt{\frac{3m}{2k}}.$

3. Case (B)

Here the oscillating mass is only $m$. So, $M_2=m.$

It is attached between two springs of constants $k$ and $2k$. Again, on displacement by $x$, restoring forces add.

Thus, $k_{\text{eff,2}}=k+2k=3k.$

Therefore, $T_2=2\pi\sqrt{\frac{m}{3k}}.$

4. Find the ratio

$\frac{T_1}{T_2} = \frac{2\pi\sqrt{\frac{3m}{2k}}}{2\pi\sqrt{\frac{m}{3k}}}$

$=\sqrt{\frac{3m}{2k}\cdot\frac{3k}{m}}$

$=\sqrt{\frac{9}{2}}$

$=\frac{3}{\sqrt{2}}.$

5. Match with options

Thus, $\frac{T_1}{T_2}=\frac{3}{\sqrt{2}}$ which corresponds to Option A.

6. Comparison with stored answer

Stored correct answer: A

Our derived answer: A

So the answer agrees with the stored correct answer.