The height at which the gravitational field of the earth becomes one-fourth the field at the surface is (where,  $R=$ radius of the earth)

The acceleration due to gravity is the measure of the gravitational field.  Let the gravitational field is $\frac{1}{4}$ th of the field at the surface at a height $h$ above the surface of the earth.

$g_h=\frac{g}{4}=\frac{g{R}^2}{{(R+h)}^2}\Rightarrow \frac{1}{4}={\left(\frac{R}{R+h}\right)}^2$

$\frac{R}{R+h}=\frac{1}{2}\Rightarrow 2R=R+h$

$h=R$