Problem Solving

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:
https://magoosh.com/gmat/2014/gmat-prac ... lems-sets/

Mike

Mike@Magoosh wrote: ↑

Mon Oct 20, 2014 10:45 am

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

The school offers a chorus, baseball, and Italian, and 15 students do all three activities
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

Mike@Magoosh wrote: ↑

Mon Oct 20, 2014 10:45 am

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:
https://magoosh.com/gmat/2014/gmat-prac ... lems-sets/

Mike

We can use the formula:

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