To find the ratio of the final volume of the gas to its initial volume, we use the ideal gas law:

\(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\)

Rearranging gives:

\(\frac{V_2}{V_1} = \frac{P_1 T_2}{P_2 T_1}\)

Substituting the values:
- \( P_1 = 200 \, N/m^2 \)
- \( P_2 = 100 \, N/m^2 \)
- \( T_1 = 400.15 \, K \)
- \( T_2 = 200.15 \, K \)

We have:

\(\frac{V_2}{V_1} = \frac{200 \times 200.15}{100 \times 400.15}\)

Calculating the ratio:

\(\frac{V_2}{V_1} = \frac{40030}{40015} \approx 1.0:1\)

Thus, the ratio of the final volume of the gas to its initial volume is: 1.0:1