A and B are two cylinders identical in size but made of different materials. A floats with half of itself immersed in a liquid. When B is placed on A, both the cylinders remain floating in the liquid, but completely immersed in it. Compare the density of the liquid and the densities of the materials of A and B.

Explanation

Let the volume of each of the cylinders A and B be V. Also, let the densities of the materials of A and B and that of the liquid be `d_(1),d_(2)andd_(3)` respectively.
When cylinder A alone floats on the liquid, then
volume of the cylinder `xx` density of its material = volume of displaced liquid `xx` density of the liquid
or, `Vxxd_(1)=V/2xxd_(3)or,d_(1)=d_(3)/2`
When cylinders A and B together float in the liquid, then volume of the cylinder A `xx` its density + volume of the cylinder B `xx` its density = volume of displaced liquid `xx` its density
or, `Vd_(1)+Vd_(2)=2Vxxd_(3)or,d_(1)+d_(2)=2d_(3)`
or, `d_(3)/2+d_(2)=2d_(3)or,d_(2)=2d_(3)-d_(3)/2=(3d_(3))/2`
`therefore" "d_(1):d_(2):d_(3)=d_(3)/2:(3d_(2))/2:d_(3)=1:3:2`.