Accleration due to gravity at a height $h$ from the suface of earth is 

$g' = \frac{g}{{{{\left( {1 + \frac{h}{R}} \right)}^2}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,…\left( i \right)$

Where g is the acceleration due to gravity at the surface of earth and $R$ is the radius of earth.

Multiplying by $m\,\, (mass\,\, of\,\ the\,\, body)$ on both sides in $(i)$, we get

 $mg' = \frac{{mg}}{{{{\left( {1 + \frac{h}{R}} \right)}^2}}}$

$\therefore $ Weight of body at height $h,\, w'\, =\, mg'$

Weight of body at surface of earth, $W\,=\,mg$

According to question, $W' = \frac{1}{{16}}W$

$\therefore \frac{1}{{16}} = \frac{1}{{{{\left( {1 + \frac{h}{R}} \right)}^2}}}$

${\left( {1 + \frac{h}{R}} \right)^2} = 16\,\,or\,\,1 + \frac{h}{R} = 4$

$or\,\frac{h}{R} = 3\,\,or\,\,h = 3R$.