A, B and C are a disc, a solid sphere and a spherical shell respectively, with the same radii and the same masses. These are placed as shown in the figure. The moment of inertia of the given system about the PQ axis is x/(15)I, where I is the moment of inertia of the disc about its diameter. The value of x is ______.

Let each body have mass M and radius R. Read the three distances off the figure first — that is the whole question.

PQ lies along a diameter of the disc A, so for the disc the axis passes through its centre and lies in its plane.

PQ is tangent to the sphere B and tangent to the shell C, so the centre of each of those two is a perpendicular distance R from the axis.

Disc A, about a diameter: I_A=1/4MR². This is the reference I the question names.

Solid sphere B, by the parallel-axis theorem: I_B=2/5MR²+MR²=7/5MR².

Spherical shell C, likewise: I_C=2/3MR²+MR²=5/3MR².

Adding, with a common denominator of 60: MR²((15+84+100)/(60))=(199)/(60)MR².

The reference is I=1/4MR², so MR²=4I.

(199)/(60)(4I)=(199)/(15)I.

x=199.

Every one of the three terms is a different standard result, which is why this question is really three questions in one.