Question 809820: At a certain summer camp the participants were polled to find out what type of
programs they like to watch. The results were: 160 liked comedies. 160 liked
to watch documentaries, 190 liked to watch dramas, 20 campers like all three
programs, 50 watched comedies and documentaries, but not dramas, and 70 liked to
watch documentaries and drama, but not comedies. 130 watched dramas, but not
comedies. 130 watched dramas, but not comedies. Assuming all campers
participated in the survey how many campers were there total?
Note: It has been a while since I have done a Venn Diagram. Help?
Answer by Edwin McCravy(20059)   (Show Source):
You can put this solution on YOUR website!
Since it makes things easier to use the first letter of what
each set is composed of, and since Documents and Dramas both
start with D, I changed Dramas to Musicals. That way we can
use set D for documentaries, C for comedies, and M for musicals,
with no conflicts of letters.
CLUES
------------------------------------------------------------
1. 160 liked Comedies. (1 set is mentioned)
2. 160 liked to watch Documentaries, (1 set mentioned)
3 190 liked to watch Musicals, (1 set mentioned)
4. 20 campers like all three programs, (3 sets mentioned)
5. 50 watched Comedies and Documentaries, but not Musicals, (3 sets mentioned)
6. 70 liked to watch Documentaries and Musical, but not Comedies. (3 sets mentioned)
7. 130 watched Musicals, but not Comedies. (2 sets mentioned)
The little letters will represent how many are
in each of the 8 regions.
Region They
with watch do not
this these watch
many programs these
campers programs
------------------------------
a C D, M
b C,D M
e D C,M
f C,M D
g C,D,M ----
h D,M C
i M C,D
j ---- C,D,M
Since all participated in the survey, and none
did not like to watch any of the three kinds of
programs, so we know that j=0.
We begin with the clues that mention the most
number of sets.
Clue 4 tells us that g=20.
Clue 5 tells us that b=50.
Clue 6 tells us that h=70.
Now we can find e from clue 2
b+g+h+e = 120 and b+g+h=50+20+70=140,
so e=140-120=20.
We can find i from clue 7.
h+i = 130, and since h=70, i=130-70=60,
so now we have
Now we can find f from clue 3.
f+g+h+i=190, and g+h+i=20+70+60=150,
so f = 190-150=40. So we have
Finally we can find a from clue 1.
a+b+f+g=160, and b+f+g=50+40+20=110
so a=160-110=50. So now our Venn
diagram is complete:
Now to find the total number of campers we add
a+b+e+f+g+h+i+j =
50+50+20+40+20+70+60+0 = 310 campers.
Edwin
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