In Greensboro County, 75 people own a television and 35

That's because the question is flawed.

In 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

Given: "75 people own a television and 35 people own a radio", so we have that: total=75+35-{both}+{neither}, say {both}=x and {neither}=y --> total=75+35-x+y;

"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": x+y=(75+35-x)/2;

"The ratio of people who own both to the people who own neither is 5:6": x/y=5/6.

Now, if you solve x+y=(75+35-x)/2 and x/y=5/6 you won't get integer values for x and y, as you should (they represent # of people). So, there is something wrong with the question.

Based on the OA saying that total=115, Kaplan (if it's really their question) writes first equation as x+y=(75+35)/2, which is wrong because the # of people who own either a television or a radio is 75+35-{both}, we are subtracting {both} to avoid doublecounting of that group, since it's included both in 75 and in 35. But if you solve their version of equations: x+y=(75+35)/2 and x/y=5/6 you'll get x=25 and y=30 --> substitute it in initial equation: total=75+35-x+y=75+35-25+30=115, the answer they got as correct.

Hope it's clear.