What happens to the force between two objects, if the distance between the object is doubled?

Let the initial conditions be,

Mass of the first object = M

Mass of the second object = m

Distance between two objects = d

Force between two objects = F

By newtons law of gravitation ,

\(F_i=\frac{GMm}{d^2}\) --------- (1)

Now according to final condition, distance has been doubled i.e., distance = 2d

So, \(F_f=\frac{GMm}{{(2d)}^2}\)

Or, \(F_f=\frac{GMm}{4d^2}\)

Or, \(F_f=\frac14\frac{GMm}{d^2}\)

∴ \(F_f=\frac14F_i\)   -------- (using equation 1)

Hence final force of attraction reduces by four times if the distance between them is increased by two times keeping the masses constant.