The ratio of the diameters of two metallic rods of the same material is `2 : 1` and their lengths are in the ratio `1 : 4`. If the temperature difference between their ends are equal, the rate of flow of heat in them will be in the ratio

Answer

To solve the problem, we need to determine the ratio of the rate of heat flow (thermal current) through two metallic rods of the same material, given their diameters and lengths.

### Step-by-step Solution:

1. **Identify Given Ratios:**
- The ratio of diameters of the two rods is \( D_1 : D_2 = 2 : 1 \).
- The ratio of lengths of the two rods is \( L_1 : L_2 = 1 : 4 \).

2. **Relate Diameter to Radius:**
- Since the radius \( R \) is half of the diameter \( D \), the ratio of the radii will be the same as the ratio of the diameters:
\[ R_1 : R_2 = D_1 : D_2 = 2 : 1 \]

3. **Calculate the Cross-Sectional Area:**
- The cross-sectional area \( A \) of a rod (cylinder) is given by:
\[ A = \pi R^2 \]
- Therefore, the areas for the two rods are:
\[ A_1 = \pi R_1^2 \quad \text{and} \quad A_2 = \pi R_2^2 \]
- Using the ratio of the radii:
\[ A_1 : A_2 = R_1^2 : R_2^2 = (2)^2 : (1)^2 = 4 : 1 \]

4. **Use the Formula for Rate of Heat Flow:**
- The rate of heat flow \( I \) through a rod is given by:
\[ I = \frac{K \cdot A \cdot (T_H - T_C)}{L} \]
- Since both rods are made of the same material, the thermal conductivity \( K \) and the temperature difference \( (T_H - T_C) \) are the same for both rods.

5. **Set Up the Ratio of Heat Flow:**
- The ratio of the rate of heat flow for the two rods can be expressed as:
\[ \frac{I_1}{I_2} = \frac{A_1}{A_2} \cdot \frac{L_2}{L_1} \]
- Substituting the values we found:
\[ \frac{I_1}{I_2} = \frac{4}{1} \cdot \frac{4}{1} = 4 \cdot 4 = 16 \]

6. **Final Result:**
- The ratio of the rate of flow of heat in the two rods is:
\[ I_1 : I_2 = 16 : 1 \]

### Conclusion:
The ratio of the rate of flow of heat in the two metallic rods is \( 16 : 1 \).