The sun pulls the moon harder than the earth does. So why does the moon stay with earth?

Because the Moon is much, much closer to the Earth. The Moon is within the Earth's Hill sphere, which is basically the area around each celestial body where that body is gravitationally dominant.

http://en.wikipedia.org/wiki/Hill_sphere

The correct answer is that it's motion and speed is perpendicular to the effects of the gravity of the sun. This is what keeps any object in orbit. True, the moon is closer to the earth than the sun. But this is not the reason why the moon does not fall into the sun. If the moon's orbit around the sun slowed down enough, then it would fall into the sun no matter what the distance was.

unmasked is right. The moon actually does orbit the sun. At the same time it also orbits the earth. As it circles the earth, both the moon and the earth make a much bigger circle around the sun. If you tracked the moon's motion for a year it would look something like this:

http://www.astronomyforbeginners.com/images/orbit-...

(The sun would be in the center of that picture.)

@RickB Heh, yeah, I know. The basic problem with the diagram is that it greatly exaggerates the size of the moon's orbit relative to the sun's (hence "something like this:"). This image shows what RickB is talking about: http://blogs.discovermagazine.com/badastronomy/fil...

But it gives an idea of how the moon can do both motions at once. Here's one I like better than my original, though the shape is still wrong (the last link I gave shows the correct shape). But it includes the moon, sun, and earth, in case anyone's confused about what the diagram is showing: http://www.answering-christianity.com/moon_s-shape...

Yes, the sun pulls on the moon about twice as hard as the earth does. However, the center of gravity of the earth mooon "system" is inside the earth.

But you know, if you look at the motion of the moon from a far enough vantage point, it's motion can be approximated as being in orbit around the sun, very similar to the earth's. It's motion about the earth is very small compared to its motion about the sun. So why would you expect the sun to rip the moon away?

This is simply because the moon is near the earth. And it shares to motion of the earth around the sun.

You might as well ask why does not the earth crash into the sun

I think you have it wrong...

The sun's gravity is very strong, but gravity fades with the square of the distance. As the sun is 93 million miles away, and the Earth is only 238,000 miles away, the Earth's pull on the moon is much stronger, comparatively, to the sun's.

Now, the moon *is* moving away from Earth, slowly, every day - so the Earth's influence on the moon is decreasing, while the sun's influence is staying more or less constant...

The Sun pulls equally on both the earth and moon. But since the moon is much (much) closer to the earth it is also captured by earth's gravitational field.

Suns pull on moon is never more than that of earth because Sun is very far away.

The other answers you've gotten so far are very good. Do the math for verification of what they're saying --

f = G * m1 * m2 ÷ r²

'G' gravitational constant

'm1' mass of object 1

'm2' mass of object 2

'r' distance between m1 and m2