Calculate the force of gravity on a 1–kilogram box located at a point 1.3 × 107 meters from the center of earth if the force on a 2.5–kilogram box at the same point is 6.1 newtons.

Since you already gave us the weight of the 2.5-kg box,
we don't even need to know what the distance is, just
as long as it doesn't change.

Look at the formula for the gravitational force:

                           F = G  m₁ m₂ / R² .

If 'G', 'm₁' (mass of the Earth), and 'R' (distance from the Earth's center)
don't change, then the Force is proportional to  m₂ ... mass of the box,
and you can write a simple proportion:

                       (6.1 N) / (2.5 kg)  =  (F) / (1 kg)

Cross-multiply:  (6.1 N) (1 kg)  =  (F) (2.5 kg)

Divide each side by (2.5 kg):  F = (6.1N) x (1 kg) / (2.5 kg)  =  2.44 N .

You get 6 for mechanical