A pyramid is constructed on the base of a cube such that its base coincides with the base of the cube and its vertex is at the centre of the cube. What fraction of the volume of the cube is occupied by this pyramid?

Introduction / Context:
This problem involves three dimensional geometry and compares the volume of a pyramid to the volume of a cube. The pyramid has the same square base as the cube, and its vertex is at the centre of the cube. This configuration appears frequently in solid geometry, and the question tests whether you can correctly use the formula for the volume of a pyramid and relate it to the volume of the cube.

Given Data / Assumptions:

- The solid is a cube with side length a units.
- A pyramid is built with base equal to the base face of the cube.
- The vertex of the pyramid is located at the centre of the cube.
- The height of the pyramid is the distance from the base of the cube to its centre, which is half the side length of the cube.
- We are required to find the fraction of the cube volume that the pyramid occupies.

Concept / Approach:
The volume of a cube with side a is V_cube = a^3. The volume of a pyramid is given by V_pyramid = (1 / 3) * base area * height. In this configuration, the base area of the pyramid is the area of the square base of the cube, and its height is half the side of the cube. By writing both volumes in terms of a and forming a ratio, we can find the fraction of the cube volume that belongs to the pyramid.

Step-by-Step Solution:
Step 1: Let the side length of the cube be a units.Step 2: Volume of the cube V_cube = a^3.Step 3: Area of the square base of the cube = a^2.Step 4: The centre of the cube lies midway between its top and bottom faces, so the height of the pyramid h = a / 2.Step 5: Volume of the pyramid V_pyramid = (1 / 3) * base area * height = (1 / 3) * a^2 * (a / 2).Step 6: Simplify V_pyramid = (1 / 3) * (a^3 / 2) = a^3 / 6.Step 7: Fraction of the cube volume occupied by the pyramid = V_pyramid / V_cube = (a^3 / 6) / a^3 = 1 / 6.

Verification / Alternative check:
You can assign a simple numerical value to the cube side, for example a = 2 units. Then V_cube = 8 cubic units. Height of the pyramid is 1 unit, and its base area is 4 square units, so its volume is (1 / 3) * 4 * 1 = 4 / 3 cubic units. The ratio 4 / 3 divided by 8 equals (4 / 3) * (1 / 8) = 4 / 24 = 1 / 6, confirming the symbolic calculation.

Why Other Options Are Wrong:
Fractions like 1 / 3, 1 / 4, 1 / 5 and 1 / 8 would correspond to different combinations of base area and height. They might arise if someone mistakenly takes the height as the full side length of the cube, uses the wrong base area or forgets the factor 1 / 3 in the pyramid volume formula. None of these match the correct ratio derived from the proper formula and geometry of the situation.

Common Pitfalls:
Many students forget the factor 1 / 3 in the volume of a pyramid, incorrectly using base area * height instead of (1 / 3) * base area * height. Others may assume the height of the pyramid equals the full side of the cube rather than half, which happens because they misinterpret where the vertex lies. Carefully identifying base and height and recalling the correct formula avoids these mistakes.

Final Answer:
Thus, the pyramid occupies 1/6 of the volume of the cube.