$\mu_{\mathrm{A}}=0.2, \mu_{\mathrm{B}}=0.1$

Let both the blocks are moving with common

acceleration a. Then,

$a=\frac{\mu_{A} m_{A} g}{m_{A}}=\mu_{A} g=0.2 \mathrm{g}$

and $F-\mu_{\mathrm{B}}\left(\mathrm{m}_{\mathrm{B}}+\mathrm{m}_{\mathrm{A}}\right) \mathrm{g}=\left(\mathrm{m}_{\mathrm{B}}+\mathrm{m}_{\mathrm{A}}\right) \mathrm{a}$

$F=\left(\mathrm{m}_{\mathrm{B}}+\mathrm{m}_{\mathrm{A}}\right) \mathrm{a}+\mu_{\mathrm{B}}\left(\mathrm{m}_{\mathrm{B}}+\mathrm{m}_{\mathrm{A}}\right) \mathrm{g}$

$=\left(\mathrm{m}+\frac{\mathrm{m}}{2}\right)(0.2 \mathrm{g})+(0.1)\left(\mathrm{m}+\frac{\mathrm{m}}{2}\right) \mathrm{g}$

$=\left(\frac{3}{2} \mathrm{m}\right)(0.2 \mathrm{g})+\left(\frac{3}{2} \mathrm{m}\right)(0.1 \mathrm{g})=\frac{0.9}{2} \mathrm{mg}$

$=0.45 \mathrm{mg}$