Problem:
From a solid sphere of radius R, a quarter of the sphere is cut out as shown. The center of mass of the remaining part is at a distance of nR from the center of the sphere. Find the value of n.

Solution:
To find the value of n, we need to determine the location of the center of mass of the remaining part of the sphere. We can do this by considering the symmetry of the problem and using the concept of center of mass.

Step 1: Identifying the Symmetry
- The solid sphere is symmetric about its center, which means the center of mass of the sphere lies at the center.
- The quarter sphere that is cut out also has symmetry. The center of mass of the cut-out part lies on the line connecting the center of the sphere and the center of the cut-out part.

Step 2: Determining the Center of Mass of the Cut-Out Part
- The cut-out part is a quarter of a sphere, which has a well-known center of mass location.
- The center of mass of a quarter sphere is located at a distance of 3R/8 from the center of the sphere.
- Therefore, the center of mass of the cut-out part is located at a distance of 3R/8 from the center of the sphere.

Step 3: Determining the Center of Mass of the Remaining Part
- Since the remaining part is obtained by removing the cut-out part, its center of mass can be found by subtracting the center of mass of the cut-out part from the center of the sphere.
- Subtracting 3R/8 from R gives us (8R - 3R)/8 = 5R/8.
- Therefore, the center of mass of the remaining part is located at a distance of 5R/8 from the center of the sphere.

Step 4: Finding the Value of n
- The question asks for the value of n, which represents the distance of the center of mass of the remaining part from the center of the sphere as a fraction of the sphere's radius R.
- Comparing the distance of the center of mass of the remaining part (5R/8) to the radius of the sphere (R), we can see that n = 5/8.

Answer:
The value of n is 5/8.