Would you suspect the density of the mantle to be greater or less than the crust? why?

Yes I would suspect the density of the mantle to be greater than the density of the crust.

This is because the crust sits on top of the mantle. Its basically floating on the mantle. This would suggest that the density of the mantle is greater because it sinks below the crust. If a medium has a greater density than another medium, then that first medium will sink below the second because the particles are more tightly packed.

Hope this helps and have a nice day! :)