Introduction:
The moment of inertia of an object is a measure of its resistance to rotational motion. It depends on the mass distribution of the object and the axis of rotation. The moment of inertia of a disc is given by the formula:

I = (1/2) * m * r^2

Where:
I = Moment of inertia
m = Mass of the disc
r = Radius of the disc

In this problem, we are given two discs with different radii and we need to find the ratio of their moments of inertia about their axes.

Solution:

Step 1: Calculate the moments of inertia for each disc:

Let's denote the smaller disc with radius 0.2m as Disc A and the larger disc with radius 0.6m as Disc B.

The moment of inertia for Disc A is given by:
I_A = (1/2) * m_A * r_A^2

The moment of inertia for Disc B is given by:
I_B = (1/2) * m_B * r_B^2

Step 2: Determine the mass of each disc:

Since the discs are made of the same material and have the same thickness, we can assume that their densities are the same. Hence, the mass of each disc is directly proportional to its area.

The area of Disc A is given by:
Area_A = π * r_A^2

The area of Disc B is given by:
Area_B = π * r_B^2

Since the density is the same, the mass of each disc is proportional to its area. Therefore, we can write:
m_A/m_B = Area_A/Area_B

Step 3: Calculate the ratio of moments of inertia:

Substituting the values of mass from Step 2 into the formulas for moments of inertia, we get:
I_A/I_B = (1/2) * m_A * r_A^2 / (1/2) * m_B * r_B^2

Canceling out the common factors, we have:
I_A/I_B = (m_A/m_B) * (r_A^2/r_B^2)

From Step 2, we know that:
m_A/m_B = Area_A/Area_B

Substituting this into the previous equation, we get:
I_A/I_B = (Area_A/Area_B) * (r_A