A spherical balloon has its volume increased by 700%. By what percentage does its surface area increase?

Introduction / Context:
This aptitude question tests how the volume and surface area of a sphere change when the radius is scaled. It is a classic example of geometric similarity, where solid figures with proportional dimensions have related ratios for volume and surface area. Understanding this relationship is useful in many real life situations, such as inflating balloons, designing spherical tanks, or scaling models. The key idea is that volume of a sphere depends on the cube of the radius, while surface area depends on the square of the radius. So, a percentage change in volume does not translate directly into the same percentage change in surface area, and we must carefully work with ratios and powers of the scale factor.

Given Data / Assumptions:
• Initial volume of the spherical balloon is V₁ (unknown but not needed numerically).
• Final volume V₂ is obtained by increasing the volume by 700%.
• A 700% increase means the new volume is 8 times the old volume.
• Let the initial radius be r₁ and the final radius be r₂.
• Surface area is proportional to the square of the radius for a sphere.

Concept / Approach:
For a sphere, volume is proportional to r³ and surface area is proportional to r². If we know how many times the volume has increased, we can find the scale factor of the radius using the cube root. Once the new radius factor is known, we square that factor to obtain the factor by which surface area changes. Finally, we convert this factor into a percentage increase relative to the original surface area. The percentage increase is given by (new area - old area) / old area multiplied by 100.

Step-by-Step Solution:
Step 1: A 700% increase in volume means the new volume is V₂ = V₁ + 7V₁ = 8V₁. Step 2: Since volume of a sphere is proportional to r³, we have V ∝ r³, so V₂ / V₁ = (r₂³) / (r₁³). Step 3: Substitute the volume ratio: 8 = (r₂³) / (r₁³), so r₂³ = 8 r₁³. Step 4: Take the cube root of both sides: r₂ = 2 r₁, so the radius doubles. Step 5: Surface area of a sphere is proportional to r², so the area ratio is A₂ / A₁ = (r₂²) / (r₁²) = (2²) / 1² = 4. Step 6: A factor of 4 means the surface area becomes four times the original, which is a 300% increase, because (4 - 1) * 100 = 300.

Verification / Alternative check:
We can verify with simple numbers. Assume the initial radius r₁ is 1 unit. Then the initial volume is proportional to 1³ = 1. If the volume becomes 8, the new radius r₂ must be 2, because 2³ = 8. For surface area, initially it is proportional to 1² = 1, and after the change it is proportional to 2² = 4. So the surface area has become four times the original. Converting this factor to a percentage increase gives (4 - 1) / 1 * 100 = 300%. This confirms the earlier reasoning and shows that the exact initial numerical values do not matter, only the ratios and the power relationships are important.

Why Other Options Are Wrong:
100% is wrong because it would mean the surface area only doubled, which would correspond to a radius multiplied by sqrt(2), not a factor of 2 in radius.
200% is incorrect because it suggests the area becomes three times the original, while the radius doubling makes area four times the original.
400% and 500% are too large, implying area factors of 5 and 6. These would require radius scale factors greater than 2, which contradicts the volume relationship that gave r₂ = 2 r₁.
Therefore, only 300% matches the correct geometric scaling.

Common Pitfalls:
A common mistake is to assume that if volume increases by 700%, surface area must also increase by 700%, which is not true because they depend on different powers of the radius. Another error is to treat percentage increase as a simple ratio without converting 700% into a factor of 8. Some learners also confuse area and volume formulas or forget that volume scales with the cube of the linear dimension and area with the square. Careful attention to the exponents and the meaning of percentage increase avoids these issues.

Final Answer:
The percentage increase in the surface area of the spherical balloon is 300%.