Introduction:
In order to find the ratio of the volumes of two spheres, we need to understand the relationship between the radius and volume of a sphere. The volume of a sphere is directly proportional to the cube of its radius. Therefore, if the radius of a sphere is doubled, the volume will be increased by a factor of 2^3 = 8.

Given information:
The radius of the balloon increases from 4 cm to 8 cm.

Calculating the volumes:
The formula for the volume of a sphere is V = (4/3)πr^3, where V is the volume and r is the radius.

1. Initial volume (V1):
- Radius (r1) = 4 cm
- V1 = (4/3)π(4^3) = (4/3)π(64) = 268.08 cm^3

2. Final volume (V2):
- Radius (r2) = 8 cm
- V2 = (4/3)π(8^3) = (4/3)π(512) = 2144.66 cm^3

Calculating the ratio:
To find the ratio of the volumes, we divide the final volume by the initial volume.

Ratio of volumes = V2/V1 = 2144.66 cm^3 / 268.08 cm^3 ≈ 8

Explanation:
The ratio of the volumes of the two spheres is approximately 8. This means that the final volume of the balloon, when the radius is 8 cm, is eight times greater than its initial volume when the radius is 4 cm.

Conclusion:
The ratio of the volumes of the two spheres is 8. As the radius of the balloon doubles, the volume increases by a factor of 8.