1. Write the required formulas

For a gas:

• Root mean square speed: $u_{\text{rms}} = \sqrt{\frac{3RT}{M}}$
• Most probable speed: $u_{\text{mp}} = \sqrt{\frac{2RT}{M}}$

Here, $M$ is molar mass.

2. Use the given condition

Given:

• Gas $X$: $M_X = 40$, $T_X = 400\,\text{K}$
• Gas $Y$: $T_Y = 60\,\text{K}$, $M_Y = ?$

According to the question, $u_{\text{rms}}(X) = u_{\text{mp}}(Y)$

So, $\sqrt{\frac{3R(400)}{40}} = \sqrt{\frac{2R(60)}{M_Y}}$

3. Square both sides

$\frac{3R(400)}{40} = \frac{2R(60)}{M_Y}$

Cancel $R$: $\frac{3\cdot 400}{40} = \frac{2\cdot 60}{M_Y}$

$\frac{1200}{40} = \frac{120}{M_Y}$

$30 = \frac{120}{M_Y}$

4. Solve for $M_Y$

$M_Y = \frac{120}{30} = 4$

5. Final answer

The molecular weight of gas $Y$ is: $\boxed{4}$