Help with a word problem?

Question

100 students were polled on whether they like country or rock music.
7 said they liked neither.
90 said they liked rock.
57 said they liked country.
How many liked both?

Answer 1

Draw a little diagram. You have a total of 100 students. Then draw two overlapping circles labelled rock(90) and country (57).

The area outside the middle circles is 7 (like neither), so the two regions must add up to 93:

So if you add 90 and 57 you get 147. But you need it to be 93, so that means 147 - 93 = 54 students like both.

Summarizing:
36 like *just* rock *
54 like both <----
3 like *just* country *
7 like neither
----
100 = total students polled

*Note: If you add 36+54 you get 90 students that like rock (with or without country). And if you add 54+3 you get 57 students that like country (with our without rock)

So the answer is 54 students liked both.

Answer 2

Alright,
Since 7 like neither we take that away from the 100
100-7=93
Now we add the amount who liked rock and the amount who said they liked country and get the total of 147. We now subtract 93 from 147 which brings 54. Thus 54 must have voted for both country and rock. To check we can add (90-54)+(57-54)+54+7
which comes out to 100
54 students voted for both

MATHEMATICAL:
100-7 = 93 total voted for at least one
90+57 = 147 voted
147 - 93 = 54 voted for both

Answer 3

N=100
B'=7
R=90
C=57

U=UNION
n=Intersection


R U C = N-B' =100 - 7 = 93 ( if we UNION we get 93)

R' = (R U C) - R = 93 -90 =3
( its mean 3 said liked country and doesn't liked rock)

C' = ( R U C) - C = 93 - 57 = 36
( its mean 36 said liked rock and doesn't liked country)

to know how many liked both
so the answer is = 54 said liked both

R n C = (R U C) - R' - C' = 93 -3 -36 =90 -36 =54

Answer 4

COOL QUESTION...

Okay 100 were asked

7 said they didn't like either one...
So the question pool dropped to 93.

Now you had 90 Votes Rock
And you had 57 Votes Country
or a Total of 147 Votes.

Only 93 were voting.

So 147 - 93 = 54 who liked both (voted for both).

Check: 93 + 54 = 147 Correct

Answer 5

100 students, 7 liked neither so 93 liked at least 1. 90 like rock so there are 3 who only like country. 57 like country so 93-57=36 omly like rock & 54 like both

Answer 6

None, according to your definition: The either liked rock, or country, or neither. In any case, the numbers don't add up. If 90 liked rock, and 57 liked country, and 7 liked neither, that adds up to 154 polled.

Answer 7

R is the students who like rock only, C is the students who like country only, B is the students who like both

R + B = 90

C + B = 57

R + C + B = 100 - 7 = 93

That's three equations, with three unknowns; you can solve from here with basic algebra.

You should solve this on you own for practice, but the answer is 54.

Answer 8

54 liked both

Answer 9

Why those who liked both, did not say they liked both ?

Answer 10

it's either 54 or 47...(I think)