It's more just an estimation problem but I thought the densities could be related using some other phenomenon such as an orbital period or if I knew the distance between the moon and the Earth. I wasn't quite sure what I was allowed to assume and what would be taken as common knowledge.
I think this is the right reasoning based off of the answers so far. It came at the end of a University of Cambridge Natural Sciences Interview Preparation paper so I thought it would require a bit more than this as it's just one step (and what I assumed to begin with).
I did initially think that, but I know that the acceleration on the moon is less than 1/6 of Earth's. This assumption would give 1/4 as the answer which I'm not sure is accurate enough, I don't actually know whether it just wanted a very rough approximation.
Surface acceleration is proportional to density and radius of planet (as 2 powers of R cancel with the volume)
g(moon)/g(earth) = density(moon)*radius(moon)/density (earth)*radius(earth) = (1/4)*density(moon)/density(earth)