Let the number of people who own only television be B
Let the number of people who own only radio be D
Let the number of people who own only both be C
Let the number of people who neither be A

From the question stem,
The value of B+C = 75 - (1)
The value of D+C = 35 - (2)
The value of C+A = (1/2)*(B+D) -(3)
C:A = 5:6. If C = 5x, A = 6x. - (4)
We know that C+A = (1/2)*(B+D) From (3) and (4)
5x + 6x = (1/2)*(B+D)
22x = (B+D). - (5)

B+D+C+C=110 - Add (1) and (2)
22x + 10x = 110, x =110/32. C = 5x = 5*110/32, is not an integer(which shouldn't be the case). hmm, Interesting.

Are you sure the question is right?

abcgmat wrote:n Greensboro County, 75 people own a television and 35 people own a radio. The number of people who own both a television and a radio plus the number of people who own neither is half the amount of the people who own either. If the ratio of people who own both to the people who own neither is 5:6 how many people live in the county ?
A 55
B 85
C 105
D 110
E 115

Approaching a different way we also see that the question is defective.

Let's call those who own both "b" and those who own neither "n". So, we have:

(1) b + n = 1/2(75+35-b)

(we subtract the extra b because we want "the amount [which should be "the number" - bad grammar on a GMAT question is never a good sign!] who own either" and if we just add 75 and 35 we're ignoring the overlap between the two).

We also have:

(2) b/n = 5/6

Two equations, two unknowns, time to solve!

Rewriting (2):

n = 6/5(b)

Simplifying and subbing into (1):

b + n = 1/2(75+35-b)
b + 6/5(b) = 1/2(110 - b)
11/5(b) = 55 - 1/2(b)
11/5(b) - 1/2(b) = 55
22/10(b) - 5/10(b) = 55
17/10(b) = 55
17b = 550
b = a non-integer... yes indeedy, this question is defective.