SOLUTION: a radio station surveyed 190 audience to determine the types of music they liked.the survey revealed that 114 liked rock music,50 liked folk music ,41 liked classical music and 11

Click here to see ALL problems on sets-and-operations

Question 874971: a radio station surveyed 190 audience to determine the types of music they liked.the survey revealed that 114 liked rock music,50 liked folk music ,41 liked classical music and 11 liked classical music and folk music.5 liked all the three types of music. find (1) how many did not like any of the 3 types? (2) how many liked any two types only (3) how many liked folk music but not rock music
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
there were 114 people who liked rock.
of those 5 liked all three.
that leave 109 rock only.

there were 50 people who liked folk.
of those, 11 liked classical and folk and 5 liked all three.
that leaves 34 that liked folk only.

there were 41 people who liked classical.
of those, 11 liked classical and folk and 5 liked all 3.
that leaves 25 that liked classical only.

you have:

25 liked classical only.
34 liked folk only.
109 liked rock only.
11 liked classical and folk only.
5 liked all 3.

add those up and you get 184 total out of 190.
the remainde must not have liked any of the 3.
that makes a remainder of 190 - 184 = 6

6 people didn't like any of the 3.

the key is to place people in categories that are unique.
that eliminates any double counting.

people who liked all 3 had to have their own categories so they had to be subtracted from the categories they were part of (classical and folk and rock)

likewise people who liked 2 had to have their own categories so they had to be subtracted from the categories they were part of (classical and folk).

once you have unique categories so that none of the people in any category belong to any other category you can then just add them up,