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Geometry

500 people are enrolled in at least two of these three classes: art, drama, and piano. 170 are enrolled in both art and drama, 150 are enrolled in both piano and drama, and 300 are enrolled in art and piano. How many of the 500 people are enrolled in all three?

A300
B330
C200
D120
E 90
F 60
Reply 1
Avatar for mqb2766
mqb2766
21
Original post by Phosphorescent
500 people are enrolled in at least two of these three classes: art, drama, and piano. 170 are enrolled in both art and drama, 150 are enrolled in both piano and drama, and 300 are enrolled in art and piano. How many of the 500 people are enrolled in all three?

A300
B330
C200
D120
E 90
F 60


There was some good advice given in
https://www.thestudentroom.co.uk/showthread.php?p=78929926&highlight=
What did you do, what problem did you have?
I used the venn diagram and got E 90 but after submitting my answer, the teacher told me its incorrect.
Original post by mqb2766
There was some good advice given in
https://www.thestudentroom.co.uk/showthread.php?p=78929926&highlight=
What did you do, what problem did you have?


I used the venn diagram and got E 90 but after submitting my answer, the teacher told me its incorrect.
Reply 4
Avatar for mqb2766
mqb2766
21
Original post by Phosphorescent
I used the venn diagram and got E 90 but after submitting my answer, the teacher told me its incorrect.


Can you upload your venn diagram (image)?
My Venn diagram is incorrect, I know the answer is F 60, but I don't know how to use the Venn diagram to get the answer.
Original post by mqb2766
Can you upload your venn diagram (image)?


My Venn diagram is incorrect, I know the answer is F 60, but I don't know how to use the Venn diagram to get the answer.
Original post by Phosphorescent
500 people are enrolled in at least two of these three classes: art, drama, and piano. 170 are enrolled in both art and drama, 150 are enrolled in both piano and drama, and 300 are enrolled in art and piano. How many of the 500 people are enrolled in all three?

A300
B330
C200
D120
E 90
F 60



tsrvennd.jpg
Reply 8
Avatar for math42
math42
20
Original post by Phosphorescent
My Venn diagram is incorrect, I know the answer is F 60, but I don't know how to use the Venn diagram to get the answer.


You want to draw the venn diagram with all three overlapping. You'll have four regions of interest: art and drama, art and piano, drama and piano, and all three. Since everyone is enrolled in at least two of the subjects, everyone is contained in these regions. Let's say there are "x" people doing all three. Then with the numbers given we have:
170 - x doing art, drama but not piano
150 - x doing piano, drama but not art.
300 - x doing piano, art but not drama.

Then, altogether, the number of people is
(170 - x) + (150 - x) + (300 - x) + x = 500

Then just solve for x.
Reply 9
Avatar for mqb2766
mqb2766
21
Original post by Phosphorescent
My Venn diagram is incorrect, I know the answer is F 60, but I don't know how to use the Venn diagram to get the answer.


Picture & formula for 3 overlapping sets is halfway down.
http://www.gmatfree.com/module-999/venn-diagrams-and-the-overlapping-set-equation/

|AuBuC| = |A| + |B| + |C| - |AnB| - |BnC| - |CnA| + |AnBnC|
or
|AnBnC| = |AuBuC| - |A| - |B| - |C| + |AnB| + |BnC| + |CnA|
...
You know |AuBuC|, and 3 bits of information like
|AuB| = |A| + |B| - |AnB|
what did you do?
Thank you, everyone, for helping. I now know how to solve using the equation and Venn diagram 😊 Thanks again

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