The radius of the earth is approximately 3,960 miles. find the approximate surface-area-to-volume ratio of the earth.

To put it another way, that ratio exists, no matter what distance units you use to express lengths, such as the radius of the Earth, but using different units will result in a different numerical part. 
You can't, for instance, say the the radius of the Earth is 3960, and leave it at that. 
If someone comes along who's measuring everything in km, he'll tell you that it's 6373. 
Or if he's using meters, he'll say it's 6,373,000. 
Or in yards, 6,969,600. 

So r = 3960 mi 
And as others have said, the area, A, and volume, V, of a sphere, in terms of its radius, are 
A = 4πr² 
V = 4πr³/3 
so that the area to volume ratio is 
A/V = 3/r 

So the answer is 
3/(3960 mi) = (1/1320) /mi.