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

OAE

To avoid confusion, official problems about triple-overlapping groups typically avoid the use of both, either or neither.
I believe that the following reflects the intent of the problem posted above:

There are a total of 400 students at Happy High School, which offers exactly three after-school activities: chorus, baseball, and Italian. This year, 120 students participate in chorus, 40 students participate in chorus and Italian, 45 students participate in chorus and baseball, and 15 students participate in all three activities. If 220 students participate in at least two of the three activities, how many students participate in none of the three activities?

A)40
B)60
C)70
D)100
E)130

Use a VENN DIAGRAM to organize the data.
Work from the INSIDE OUT.

There are a total of 400 students at Happy High School.
120 students participate in chorus.
15 students participate in all three activities.
The following Venn diagram is yielded:


40 students participate in chorus and Italian.
45 students participate in chorus and baseball.
Since a total of 40 students participate in chorus and Italian, and 15 students participate in all 3 activities, the number who participate only in chorus and Italian = 40-15 = 25.
Since a total of 45 students participate in chorus and baseball, and 15 students participate in all 3 activities, the number who participate only in chorus and baseball = 45-15 = 30.
Since a total of 120 students participate in chorus, and 25+30+15=70 students participate in chorus and at least one other activity, the number who participate only in chorus = 120-70 = 50.
The following Venn diagram is yielded:


220 students participate in at least two of the three activities.
Since 25 students participate only in chorus and Italian, 30 students participate only in chorus and baseball, and 15 students participate in all 3 activities, the number who participate only in Italian and baseball = 220-25-30-15 = 150.
The following Venn diagram is yielded:


How many students participate in none of the three activities?
Subtracting the values in the Venn diagram from 400 -- the total number of students -- we get:
N = 400 - (50+25+30+15+150) = 130.

The correct answer is E.