Answer
$m_M=\frac{1}{96}m_E$
Work Step by Step
We know that
$\frac{m_M}{m_E}=\frac{F_Mr_M^2/Gm_1}{F_Er_E^2/Gm_1}$ Where $m_M$ and $m_E$ are mass of the moon and mass of the earth respectively.
$\implies \frac{m_M}{m_E}=\frac{F_Mr_M^2}{F_Er_E^2}=\frac{(\frac{1}{6}F_E)(\frac{1}{4}r_E)^2}{F_Er_E^2}=\frac{1}{96}$
$\implies m_M=\frac{1}{96}m_E$
