For circuit (a),

$i_R=\left(\frac{10}{{\displaystyle \underset{\mathrm{Current} \mathrm{through} \mathrm{cell}}{\underset{\uparrow }{\frac{300R}{300+R}}}}+300}\right)\times \frac{300}{300+R}$

[Note : $300 \mathrm{\Omega } \& R$ are in parallel which is in series with $100 \& 200 \mathrm{\Omega }$]

$\therefore V_{R_a}=\frac{10\times 300R}{300R+{300}^2+300R}$

[$V_{R_a}$ is potential difference across resistance $R$]

For circuit (b),

$i_R=\left(\frac{10}{{\displaystyle \underset{\mathrm{Current} \mathrm{through} \mathrm{cell}}{\underset{\uparrow }{\frac{(200+R)\left(300\right)}{200+R+300}}}+100}}\right)\times \frac{300}{300+200+R}$

[Note: $R \& 200 \mathrm{\Omega }$ are in series which is in parallel with $300 \mathrm{\Omega }$ & again the combination is in series with $100 \mathrm{\Omega }$]

$\therefore V_{R_b}=\frac{100\times 300R}{300\times 200+300R+100\times 500+100R}$

[$V_{R_b}$ is potential difference across resistance $R$]

According to given situation

$V_{R_a}=V_{R_b}$

$\therefore 300R+9\times {10}^4+300R=6\times {10}^4+400R+5\times {10}^4$

$\Rightarrow 200R=2\times {10}^4\Rightarrow R=100 \mathrm{\Omega }$