I have a question regarding gravitational force between mmon and earth.

If the moon losses its mass at a constant rate and the earth gains mass at this constant rate (so the sum of there masses is equal), then what happened with the force between both of them, would it stay same or not? if not then would it decrease or increase? if force is as follows:

F = GMm/d²

And also What happens to the moon?

Does the moon move away, stay the same or come closer to the earth.

This is a very good question, totaly based on the mathematical concepts, If you are good in calculations then think how:

The force does decrease,

as the product of the masses decreases  (how : as mass is moved from the smaller mass to

the larger mass).

The distance also decreases, as the speed needed fro orbit a a given distance (taking the

mass of both bodies into account is found by:

where M would be the mass of the Earth while m is the mass of the Moon. Note that while M+m remains constant, M increases as mass moves from m to M. Since we have to assume that the speed of m does not change, it would be less than the new speed needed to maintain that orbit and it will fall into a lower one.