According to the question, Area of both bodies $A$ and $B=A$
Temperature of body $A=27^{\circ} \mathrm{C}=27+273 \mathrm{K}$
Temperature of body $B=177^{\circ} \mathrm{C}=177+273 \mathrm{K}$
Now, by Stefan-Boltzmann law, thermal energy radiated per second by a body
$$
Q=\sigma A T^{4}
$$
where, $A=$ Area
$T=$ temperature and $\quad \sigma=$ Stefan-Boltzmam's constant
So, the ratio of thermal energy radiated per second by $A$ to that by $B$ is
$$
\frac{Q_{1}}{Q_{2}}=\frac{\sigma A}{\sigma A}\left(\frac{T_{1}}{T_{2}}\right)^{4}
$$
Now, $\frac{Q_{1}}{Q_{2}}=\left(\frac{T_{1}}{T_{2}}\right)^{4}$
$$
\begin{array}{l}
\frac{Q_{1}}{Q_{2}}=\left(\frac{273+27}{273+177}\right)^{4} \\
\frac{Q_{1}}{Q_{2}}=\left(\frac{300}{450}\right)^{4}
\end{array}
$$
$$
\frac{Q_{1}}{Q_{2}}=\left(\frac{2}{3}\right)^{4}=\left(\frac{16}{81}\right)
$$
Ratio of thermal energy radiated per second
$$
Q_{1}: Q_{2}=16: 81
$$