Three solid cubes with edge lengths 3 cm, 4 cm and 5 cm are melted together and recast into one large solid cube. What is the ratio of the total surface area of the three original cubes to the surface area of the new large cube?

Introduction / Context:
This question combines volume conservation with surface area comparisons. Three small cubes are melted and formed into a single larger cube. Because melting does not change the total volume of material, the sum of the volumes of the small cubes equals the volume of the large cube. We can use this fact to determine the edge length of the large cube, then find its surface area and compare it to the total surface area of the original cubes. This kind of question is common in aptitude exams involving solids.

Given Data / Assumptions:

• Edge lengths of the three small cubes: 3 cm, 4 cm and 5 cm.
• The cubes are melted with no loss or gain of material.
• The resulting solid is a cube with edge length L.
• We need the ratio (total surface area of original cubes) : (surface area of new cube).

Concept / Approach:
For a cube of side a, the volume and surface area are:
Volume = a^3 Surface area = 6 * a^2 First, we compute the total volume of the three small cubes and set it equal to L^3 to find L. Then we compute the total surface area of the three small cubes and the surface area of the large cube, and finally form their ratio.

Step-by-Step Solution:
Volume of cube with side 3 cm = 3^3 = 27 cm^3. Volume of cube with side 4 cm = 4^3 = 64 cm^3. Volume of cube with side 5 cm = 5^3 = 125 cm^3. Total volume = 27 + 64 + 125 = 216 cm^3. Let L be the edge of the new cube, so L^3 = 216. Since 216 = 6^3, L = 6 cm. Surface area of cube with side 3 cm = 6 * 3^2 = 6 * 9 = 54 cm^2. Surface area of cube with side 4 cm = 6 * 4^2 = 6 * 16 = 96 cm^2. Surface area of cube with side 5 cm = 6 * 5^2 = 6 * 25 = 150 cm^2. Total surface area of small cubes = 54 + 96 + 150 = 300 cm^2. Surface area of large cube (side 6 cm) = 6 * 6^2 = 6 * 36 = 216 cm^2. Required ratio = 300 : 216. Divide numerator and denominator by 12: 300 / 12 = 25, 216 / 12 = 18. So ratio = 25 : 18.

Verification / Alternative check:
The volume check is straightforward: 6^3 is indeed 216, matching the combined volume of the small cubes, so the side of the large cube is correctly found as 6 cm. The arithmetic for the surface areas is simple, and the ratio simplifies neatly, suggesting that the computation is correct. No inconsistencies appear when rechecking steps.

Why Other Options Are Wrong:
Ratios like 2 : 1 or 3 : 2 do not reflect the specific side lengths and would produce incorrect total surface area values. Ratios 27 : 20 and 27 : 16 have numerators and denominators inconsistent with the specific surface areas computed above. Only 25 : 18 matches the exact numeric values 300 and 216.

Common Pitfalls:
One common mistake is to add edge lengths instead of volumes, leading to an incorrect new cube side. Others miscompute 5^3 or 6^3. Students may also neglect to use the factor 6 in the surface area formula for cubes, or incorrectly reduce the final ratio. Careful step-by-step work avoids these issues.

Final Answer:
The ratio of the total surface area of the three original cubes to the surface area of the new large cube is 25 : 18.