A body weighs 250 N on the surface of the earth. How much will it weigh half way down to the centre of the earth?

Let m be mass of a body.

   Weight of the body on the surface of the earth is W = mg = 250 N

Acceleration due to gravity at a depth d below the surface of the earth is

${\mathrm{g}}^{\text{'}} = \mathrm{g}(1-\frac{\mathrm{d}}{{\mathrm{R}}_{\mathrm{E}}})$

Weight of the body at depth d is

$\mathrm{W}\text{'} = {\mathrm{mg}}^{\text{'}} = \mathrm{mg}(1-\frac{\mathrm{d}}{{\mathrm{R}}_{\mathrm{E}}})$

Here, d = $\frac{{\mathrm{R}}_{\mathrm{E}}}{2}$

${\mathrm{W}}^{\text{'}} = \mathrm{mg}(1-\frac{{\displaystyle \frac{{\mathrm{R}}_{\mathrm{E}}}{2}}}{{\mathrm{R}}_{\mathrm{E}}}) = \frac{\mathrm{mg}}{2} = \frac{\mathrm{W}}{2} = \frac{250 \mathrm{N}}{ 2} = 125 \mathrm{N}$