Introduction:
The problem states that the radius of a spherical balloon increases from 7cm to 14cm as air is being pumped into it. We are required to find the ratio of the surface area of the balloon in the two cases.

Formula for Surface Area of a Sphere:
The formula for the surface area of a sphere is given by:
Surface Area = 4πr^2
where r is the radius of the sphere.

Calculating the Surface Area of the Balloon:
1. Initial Surface Area:
- Given that the initial radius of the balloon is 7cm.
- Using the formula for the surface area of a sphere, we can calculate the initial surface area as follows:
Initial Surface Area = 4π(7^2) = 4π(49) = 196π cm^2

2. Final Surface Area:
- Given that the final radius of the balloon is 14cm.
- Using the formula for the surface area of a sphere, we can calculate the final surface area as follows:
Final Surface Area = 4π(14^2) = 4π(196) = 784π cm^2

Calculating the Ratio:
To find the ratio of the surface area of the balloon in the two cases, we divide the final surface area by the initial surface area:
Ratio = Final Surface Area / Initial Surface Area

Substituting the values we calculated earlier:
Ratio = (784π cm^2) / (196π cm^2)

Simplifying, we can cancel out the common factor of π:
Ratio = 784 / 196

Final Answer:
Ratio = 4

Explanation:
The ratio of the surface area of the balloon when the radius increases from 7cm to 14cm is 4. This means that the surface area of the balloon increases by a factor of 4 when the radius doubles. The increase in surface area is due to the additional air being pumped into the balloon, which causes it to expand and stretch. As the radius increases, the surface area increases exponentially, resulting in a larger ratio.