The question states that the ratio of the radius of the Moon to the radius of the Earth is 1/4 and the ratio of the acceleration due to gravity on the Moon to the Earth is 1/6.

To find the ratio of the mass of the Earth to the mass of the Moon, we can use the formula for acceleration due to gravity, which is given by the equation:

g = GM/r^2

where g is the acceleration due to gravity, G is the gravitational constant, M is the mass of the celestial body, and r is its radius.

Now, since the acceleration due to gravity on the Moon is 1/6 of that on Earth, we can say:

1/6 = (GM_moon)/(r_moon)^2

Similarly, since the radius of the Moon is 1/4 of that of the Earth, we can say:

(1/4)^2 = (r_moon)^2/(r_earth)^2 = (r_moon/r_earth)^2

Now, substituting this into the equation for gravity on the Moon, we get:

1/6 = (GM_moon)/(r_moon)^2 = (GM_earth)/(4r_earth