Answer
The ratio of the sun's gravitational force on the moon and the earth's gravitational force on the moon is 2.18
Work Step by Step
Let $M_s$ be mass of the sun.
Let $R_s$ be the distance from the moon to the sun.
Let $M_m$ be the mass of the moon.
We can write an expression for the force of gravity $F_s$ that the sun exerts on the moon.
$F_s = \frac{G~M_s~M_m}{R_s^2}$
Let $M_e$ be the mass of the earth. Let $R_e$ be the distance from the earth to the moon. We can write an expression for the force of gravity $F_e$ that the earth exerts on the moon.
$F_e = \frac{G~M_e~M_m}{R_e^2}$
We can find the ratio of $\frac{F_s}{F_e}$.
$\frac{F_s}{F_e} = \frac{(\frac{G~M_s~M_m}{R_s^2})}{(\frac{G~M_e~M_m}{R_e^2})}$
$\frac{F_s}{F_e} = \frac{M_s~R_e^2}{M_e~R_s^2}$
$\frac{F_s}{F_e} = \frac{(1.989\times 10^{30}~kg)(3.84\times 10^8~m)^2}{(5.98\times 10^{24}~kg)(1.50\times 10^{11}~m)^2}$
$\frac{F_s}{F_e} = 2.18$
The ratio of the sun's gravitational force on the moon and the earth's gravitational force on the moon is 2.18
