A right square pyramid has a square base of side 4 cm and a height of 9 cm. The pyramid is cut into three parts of equal height by two planes parallel to the base. What is the ratio of the volumes of the top, middle and bottom parts respectively?

Introduction / Context:
This question tests the concept of how the volume of a pyramid changes when it is cut by planes parallel to the base. Because similar pyramids have volumes proportional to the cube of their corresponding linear dimensions, cutting a right square pyramid into parts of equal height does not give equal volumes. Understanding this idea is very important for many aptitude and geometry problems involving frustums and similar solids.

Given Data / Assumptions:

The solid is a right square pyramid.
Side of the square base = 4 cm.
Height of the pyramid = 9 cm.
Two planes parallel to the base cut the pyramid into three parts of equal height, so each part has height 3 cm along the axis.
We assume ideal cuts and no loss of material.

Concept / Approach:
For similar pyramids, volume is proportional to the cube of the height. A plane parallel to the base creates a smaller pyramid at the top that is similar to the original one. The volumes of the three parts can be obtained by subtracting the volumes of nested similar pyramids. We first compute the relative volumes of the pyramids formed up to heights 3 cm, 6 cm and 9 cm from the apex and then take differences to find the volumes of top, middle and bottom parts.

Step-by-Step Solution:
Let k be a proportionality constant so that volume V = k * h^3 for a pyramid of height h. For the full pyramid, height h = 9 cm, so V_total = k * 9^3 = k * 729. For the small top pyramid of height 3 cm, V_top = k * 3^3 = k * 27. For the pyramid formed up to height 6 cm from the apex, V_up_to_6 = k * 6^3 = k * 216. The middle part is the frustum between heights 3 cm and 6 cm, so V_middle = V_up_to_6 - V_top = k * 216 - k * 27 = k * 189. The bottom part is the frustum between heights 6 cm and 9 cm, so V_bottom = V_total - V_up_to_6 = k * 729 - k * 216 = k * 513. Therefore the ratio of the volumes is V_top : V_middle : V_bottom = 27k : 189k : 513k. Divide all three by 27 to simplify: 27k : 189k : 513k = 1 : 7 : 19.

Verification / Alternative check:
An alternative check is to note that if the heights are 3, 6 and 9, the corresponding volume factors are 3^3, 6^3 and 9^3. Using cumulative volumes and differences is a standard method for solid segments created by parallel planes. The simplified ratio 1 : 7 : 19 is unique among the options and is consistent with the h^3 proportionality rule for similar pyramids.

Why Other Options Are Wrong:
1 : 8 : 27 assumes that volumes directly follow the simple cube of 1, 2 and 3 without subtracting overlapping regions, which is not correct for segments of a single pyramid.
1 : 8 : 20 does not correspond to any consistent application of the h^3 law and fails when you reconstruct cumulative volumes.
1 : 3 : 5 assumes linear or nearly linear scaling of volume with height, which contradicts the cubic relationship for similar solids.
1 : 9 : 25 is another arbitrary ratio that does not arise from any correct geometric reasoning with heights 3, 6 and 9 cm.

Common Pitfalls:
A common mistake is to think that equal heights mean equal volumes, which is true for prisms but not for pyramids or cones. Another error is to treat each part as independent and directly cube the segment heights without considering that each frustum is the difference between two similar pyramids. Some learners also confuse area scaling (which is proportional to the square of the linear dimension) with volume scaling, which is proportional to the cube of the linear dimension.

Final Answer:
The ratio of the volumes of the top, middle and bottom parts of the right square pyramid is 1 : 7 : 19.