A solid cube is cut into 27 identical smaller cubes. By what percentage does the total surface area increase as a result of this cutting?

Introduction / Context:
This question explores how the total surface area changes when a solid cube is cut into many identical smaller cubes. It is a classic aptitude problem that tests spatial reasoning and understanding of how surface area depends on side length and the number of exposed faces after cutting.

Given Data / Assumptions:

• We start with one solid cube of side length a centimetres.
• The cube is cut into 27 identical smaller cubes.
• 27 is 3^3, so each edge of the original cube is divided into 3 equal parts.
• We need the percentage increase in total surface area from the original cube to all 27 smaller cubes combined.

Concept / Approach:
Surface area of a cube with side a is 6 * a^2. When the big cube is cut into 27 identical smaller cubes, each smaller cube has side length a / 3. We compute the surface area of one smaller cube, multiply by 27, and compare this new total with the original surface area. The percentage increase is (increase / original) * 100. The value of a will cancel, so the final result is independent of the actual size of the original cube.

Step-by-Step Solution:
Step 1: Let the original cube have side length a. Its surface area S₁ = 6 * a^2. Step 2: Cutting into 27 smaller cubes means each side is divided into 3 equal segments, so each small cube has side a / 3. Step 3: Surface area of one smaller cube is S_small = 6 * (a / 3)^2 = 6 * a^2 / 9 = (2/3) * a^2. Step 4: There are 27 such smaller cubes, so total surface area after cutting is S₂ = 27 * S_small = 27 * (2/3) * a^2 = 18 * a^2. Step 5: Original surface area S₁ = 6 * a^2, new surface area S₂ = 18 * a^2. Step 6: Increase in surface area = S₂ − S₁ = 18 * a^2 − 6 * a^2 = 12 * a^2. Step 7: Percentage increase = (increase / original) * 100 = (12 * a^2 / 6 * a^2) * 100 = 2 * 100 = 200 %.

Verification / Alternative check:
To verify numerically, assume a simple value such as a = 3 cm. Then the original cube has side 3 cm and surface area 6 * 9 = 54 square centimetres. Each smaller cube then has side 1 cm, surface area 6 square centimetres, and 27 cubes together have area 27 * 6 = 162 square centimetres. The increase is 162 − 54 = 108, and (108 / 54) * 100 = 200 %, confirming our algebraic result.

Why Other Options Are Wrong:
150 %, 250 %, and 300 % correspond to different ratios of new surface area to old surface area. For example, a 150 % increase would mean the new area is 2.5 times the old area, and a 300 % increase would mean it is 4 times the old area. Our calculation shows the new area is exactly 3 times the original (18 a^2 compared to 6 a^2), which corresponds to a 200 % increase, not any of the other listed values.

Common Pitfalls:
A typical mistake is to assume that the surface area stays the same or only increases slightly because the volume is unchanged. Another error is to misinterpret “27 identical cubes” and not correctly deduce that each edge is divided into 3 equal segments. Finally, some learners forget to square the side length when computing surface area, which leads to incorrect comparisons. Using a variable for side length and carefully applying the cube surface area formula prevents these errors.

Final Answer:
The total surface area increases by 200 % when the cube is cut into 27 identical cubes.