homework help; center of mass and planets?

Question

A planet with radius of 6.25 * 10^6 m has a mass of 5.95 * 10^24 kg, its moon has a mass of 7.43 * 10^22 kg, and their center-to-center distance is 3.75 * 108 m. (As the planet and moon orbit one another, their centers orbit about their common center of mass.)
How far from the center of the planet is the planet-moon center of mass?
Is the planet-moon center of mass above or below the surface of the planet?
By what distance?

Alright, my professor hasnt really gone over anything in class that has to do with any of this. Just thinking about it, it would seem that the center of the planet would be the center of mass, so the distance for the first question would be 0. For the second question, i know that it would be below the surface of the planet. Finally, for the 3rd question, if the center of mass was the center of the planet, wouldnt the distance just be the radius? I dont believe this is right though, seems too easy

Answer 1

The center of mass of the planet is at the center of the planet, and the center of mass of the moon is at the center of the moon, but the combined center of mass is somewhere in between. Think of it this way: if you put the planet on one end of a see-saw, and the moon on the other, where would you have to place the pivot so that the see-saw was level?

Only the masses and center-to-center distances matter to the first calculation:
The ratio of the masses is 5.95*10^24 divided by 7.43*10^22, or
595/7.43 - I'll let you do that part, but it's about 80.

So the "fulcrum" point needs to be 1/81 of the distance from the center of the planet to the center of the moon (or 80/81 of the distance from the center of the moon to the center of the planet).
1/81 times 3.75*10^8 (I'm assuming you left out the ^) is 4687500, or 4.6875*10^6

Since the radius is 6.25*10^6, this is outside the planet's surface, by (6.25-4.6875)*10^6 meters