The ratio of the total surface area to the volume of a solid sphere is 1 : 7. This sphere is melted and recast into many smaller solid spheres of equal size. The radius of each small sphere is 1/6 of the radius of the original large sphere. What is the sum, in square centimetres (cm^2), of the curved surface areas of all the small spheres together?

Introduction / Context:
This question combines sphere geometry with volume conservation. A single large sphere is melted and recast into many smaller identical spheres. The ratio of total surface area to volume for the large sphere is given, and the radius of each small sphere is a fraction of the large radius. Using this information, we find the radius of the large sphere, determine how many small spheres are formed, and finally compute the total curved surface area of all the small spheres, which is numerically the same as total surface area for spheres.

Given Data / Assumptions:

• The ratio of total surface area to volume of the large sphere is 1 : 7.
• Radius of the large sphere is R centimetres.
• Radius of each small sphere is R / 6.
• The large sphere is completely melted and recast into small spheres, so total volume is conserved.
• Curved surface area of a sphere equals its total surface area, that is 4 * pi * r^2.
• pi is taken as 22 / 7 for numerical evaluation.

Concept / Approach:
The total surface area of a sphere of radius R is 4 * pi * R^2 and its volume is (4 / 3) * pi * R^3. The given ratio of surface area to volume 1 : 7 allows us to solve for R. Once R is known, we use volume conservation: volume of the large sphere equals the total volume of all small spheres. From this we find the number of small spheres. Finally, the total curved surface area is the number of small spheres multiplied by the surface area of one small sphere of radius R / 6.

Step-by-Step Solution:
Step 1: For the large sphere, total surface area = 4 * pi * R^2 and volume = (4 / 3) * pi * R^3. Step 2: Given ratio (surface area) : (volume) = 1 : 7, so (4 * pi * R^2) / ((4 / 3) * pi * R^3) = 1 / 7. Step 3: Simplify the left side: (4 * pi * R^2) * (3 / (4 * pi * R^3)) = 3 / R. Step 4: So 3 / R = 1 / 7, therefore R = 21 cm. Step 5: Radius of each small sphere r_s = R / 6 = 21 / 6 = 3.5 cm. Step 6: Using volume conservation, number of small spheres n satisfies (4 / 3) * pi * R^3 = n * (4 / 3) * pi * r_s^3. Step 7: Cancel the common factor (4 / 3) * pi, giving R^3 = n * r_s^3, so n = (R / r_s)^3 = 6^3 = 216. Step 8: Surface area of one small sphere = 4 * pi * r_s^2 = 4 * pi * 3.5^2 = 4 * pi * 12.25 = 49 * pi. Step 9: Total curved surface area of all small spheres = n * 49 * pi = 216 * 49 * pi = 10584 * pi. Step 10: Using pi = 22 / 7, total area = 10584 * (22 / 7) = 1512 * 22 = 33264 cm^2.

Verification / Alternative check:
We can confirm the radius of the large sphere by substituting R = 21 back into the ratio. Surface area is 4 * pi * 21^2 = 1764 * pi and volume is (4 / 3) * pi * 21^3 = (4 / 3) * pi * 9261 = 12348 * pi. The ratio (1764 * pi) : (12348 * pi) simplifies to 1 : 7. Also, 21 / 3.5 = 6, so the cubic ratio gives 6^3 = 216 small spheres, confirming the count. Multiplying 216 by 49 * pi and using pi = 22 / 7 again gives 33264, confirming the total curved surface area.

Why Other Options Are Wrong:
31276, 36194 and 25182 are obtained from incorrect manipulation of the ratio, wrong radius, or a mistake in cube or square calculations. 29400 is a plausible looking distractor but does not match the precise product 10584 * pi with pi = 22 / 7. Only 33264 is consistent with all correct intermediate values and volume conservation.

Common Pitfalls:
Learners often confuse the ratio of surface area to volume and mistakenly treat it as R : something, instead of substituting the actual formulas. Another common error is forgetting that radius is raised to the power 3 in the volume formula and to the power 2 in the area formula, which affects how the ratio simplifies. Some also mix up the factor 1/6 between radius and diameter when recasting the sphere. Working carefully through algebraic simplification and using volume conservation systematically helps avoid these slips.

Final Answer:
The sum of the curved surface areas of all the small spheres is 33264 cm^2.