Pertaining to two planets, the ratio of escape velocities from respective surfaces is `1:2` , the ratio of the time period of the same simple pendulum at their respective surfaces is `2:1` (in same order). Then the ratio of their average densities is

Answer

To solve the problem, we need to find the ratio of the average densities of two planets based on the given ratios of escape velocities and the time periods of a simple pendulum.

### Step-by-Step Solution:

1. **Understanding Escape Velocity**:
The escape velocity \( v \) from the surface of a planet is given by the formula:
\[ v = \sqrt{2gR} \]
where \( g \) is the acceleration due to gravity at the surface of the planet, and \( R \) is the radius of the planet.

2. **Given Ratios**:
We are given that the ratio of escape velocities \( \frac{v_1}{v_2} = \frac{1}{2} \).
Therefore, we can write:
\[ \frac{\sqrt{2g_1R_1}}{\sqrt{2g_2R_2}} = \frac{1}{2} \]
This simplifies to:
\[ \frac{\sqrt{g_1R_1}}{\sqrt{g_2R_2}} = \frac{1}{2} \]
Squaring both sides gives:
\[ \frac{g_1R_1}{g_2R_2} = \frac{1}{4} \quad \text{(Equation 1)} \]

3. **Understanding Time Period of a Pendulum**:
The time period \( T \) of a simple pendulum is given by:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where \( L \) is the length of the pendulum. Since \( L \) is the same for both planets, we have:
\[ \frac{T_1}{T_2} = \sqrt{\frac{g_2}{g_1}} \]
We are given that \( \frac{T_1}{T_2} = \frac{2}{1} \). Thus:
\[ \frac{T_1}{T_2} = 2 \implies \sqrt{\frac{g_2}{g_1}} = 2 \]
Squaring both sides gives:
\[ \frac{g_2}{g_1} = 4 \quad \text{(Equation 2)} \]

4. **Substituting Equation 2 into Equation 1**:
From Equation 2, we can express \( g_1 \) in terms of \( g_2 \):
\[ g_1 = \frac{g_2}{4} \]
Substituting this into Equation 1:
\[ \frac{\frac{g_2}{4}R_1}{g_2R_2} = \frac{1}{4} \]
Simplifying gives:
\[ \frac{R_1}{4R_2} = \frac{1}{4} \implies R_1 = R_2 \]

5. **Finding the Ratio of Densities**:
The acceleration due to gravity \( g \) can also be expressed in terms of density \( \rho \) as follows:
\[ g = \frac{GM}{R^2} = \frac{G(\rho \cdot V)}{R^2} = \frac{G(\rho \cdot \frac{4}{3}\pi R^3)}{R^2} = \frac{4}{3}\pi G \rho R \]
Thus, we find that:
\[ g \propto \rho R \]
Therefore, we can write:
\[ \frac{g_1}{g_2} = \frac{\rho_1 R_1}{\rho_2 R_2} \]
From Equation 2, we know \( \frac{g_1}{g_2} = \frac{1}{4} \) and since \( R_1 = R_2 \), we have:
\[ \frac{1}{4} = \frac{\rho_1}{\rho_2} \]

6. **Final Result**:
Thus, the ratio of the average densities of the two planets is:
\[ \frac{\rho_1}{\rho_2} = \frac{1}{4} \]

### Conclusion:
The ratio of the average densities of the two planets is \( \frac{1}{4} \).