There are a total of 400 students at a school, which offers a chorus, baseball, and Italian. This year, 120 students are in the chorus, 40 students in both chorus & Italian, 45 students in both chorus & baseball, and 15 students do all three activities. If 220 students are in either Italian or baseball, then how many student are in none of the three activities?
(A) 40
(B) 60
(C) 70
(D) 100
(E) 130
For a discussion of set problem such as this, as well as the OA & OE of this particular question, please see:
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Mike
There are a total of 400 students at a school, which offers
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The school offers a chorus, baseball, and Italian, and 15 students do all three activities[email protected] wrote: ↑Mon Oct 20, 2014 10:45 amThere are a total of 400 students at a school, which offers a chorus, baseball, and Italian. This year, 120 students are in the chorus, 40 students in both chorus & Italian, 45 students in both chorus & baseball, and 15 students do all three activities. If 220 students are in either Italian or baseball, then how many student are in none of the three activities?
(A) 40
(B) 60
(C) 70
(D) 100
(E) 130
Draw 3 overlapping circles and start at the MIDDLE
Now we'll work from the middle to the outside.
40 students are in both chorus and Italian
So, 25 students must be in chorus and Italian, but not in baseball
45 students in both chorus & baseball
So, 30 students must be in chorus and baseball, but not in Italian
120 students are in the chorus
30 + 15 + 25 = 70
So, we've already accounted for 70 students in chorus.
So, the remaining 50 students must be in chorus ONLY
220 students are in either Italian or baseball
This is the trickiest part of the question.
This tells us that there are 220 students inside the two DARKENED circles below
As you can see, we've already accounted for 70 students inside the two DARKENED circles
So, the remaining 150 students are somewhere else inside the two DARKENED circles.
ASIDE: The precise location of those 150 doesn't matter, since the question doesn't specifically ask about this.
How many student are in none of the three activities?
At this point, the number of students we've accounted for = 50 + 30 + 15 + 25 + 150 = 270
There are 400 students in total.
So, the remaining 130 students must be OUTSIDE the circles.
On other words, 130 student are in none of the three activities
Answer: E
Cheers,
Brent
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We can use the formula:[email protected] wrote: ↑Mon Oct 20, 2014 10:45 amThere are a total of 400 students at a school, which offers a chorus, baseball, and Italian. This year, 120 students are in the chorus, 40 students in both chorus & Italian, 45 students in both chorus & baseball, and 15 students do all three activities. If 220 students are in either Italian or baseball, then how many student are in none of the three activities?
(A) 40
(B) 60
(C) 70
(D) 100
(E) 130
For a discussion of set problem such as this, as well as the OA & OE of this particular question, please see:
https://magoosh.com/gmat/2014/gmatprac ... lemssets/
Mike
Total = n(C) + n(B) + n(I) – n(C and B) – n(C and I) – n(B and I) + n(C and B and I) + n(No Set)
Here we are given that Total = 400, n(C) = 120, n(C and I) = 40, n(C and B) = 45, n(C and B and I) = 15. We need to find n(No Set). We are not given n(B), n(I) and n(B and I). However, we are given that n(B or I) = 220. Recall that n(B or I) = n(B) + n(I)  n(B and I), so n(B) + n(I)  n(B and I) = 220.
We can rearrange the terms in our formula, substitute the numbers, and solve for n(No Set):
Total = n(C) – n(C and B) – n(C and I) + [n(B) + n(I) – n(B and I)] + n(C and B and I) + n(No Set)
400 = 120  45  40 + 220 + 15 + n(No Set)
400 = 270 + n(No Set)
130 = n(No Set)
Answer: E
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