On the line of centres between the Earth and the Moon, there is a point where the net gravitational force is zero. Given that the distance between the two is 385,000 km, and that the Earth has a mass 81x that of the Moon, how far is this point from Earth?

Here, we must consider Newton's Law of Universal Gravitation. This states that the gravitational force acting between two bodies is proportional to the masses of each body and inversely proportional to the square of the distance between them, F=Gm₁m₂/r²  As the Earth is 81 times the mass of the moon, the distance of this point from the moon must be 1/81^1/2 = 1/9 that of the distance to Earth. This means that we are dealing with a ratio of 1/9 of our distances. We therefore take 385000 * 9/(9+1) to find the distance, which is equal to 346,500 km, or 347,000 km to 3 significant figures.