Question

at a community college, a survey was taken to determine where students study on campus. of the 270 students surveyed, it was determined that 178 studied in the library, 135 studied in the cafeteria, 84 studied in both the library and the cafeteria. complete parts a) through c). a) of those students surveyed, how many studied only in the library? (type a whole number.) b) of those students surveyed, how many studied only in the cafeteria? (type a whole number.) c) of those students surveyed, how many did not study in either of these places? (type a whole number.)

Explanation:

Step1: Use set - theory principle

Let \(L\) be the set of students who studied in the library, \(C\) be the set of students who studied in the cafeteria. We know \(n(L)=178\), \(n(C)=135\), \(n(L\cap C)=84\), and the total number of students surveyed \(n = 270\).

Step2: Find students who studied only in the library

The number of students who studied only in the library is \(n(L)-n(L\cap C)\). So \(178 - 84=94\).

Step3: Find students who studied only in the cafeteria

The number of students who studied only in the cafeteria is \(n(C)-n(L\cap C)\). So \(135 - 84 = 51\).

Step4: Find students who did not study in either place

First, find the number of students who studied in at least one place using the formula \(n(L\cup C)=n(L)+n(C)-n(L\cap C)\). \(n(L\cup C)=178 + 135-84=229\). Then the number of students who did not study in either place is \(270 - 229 = 41\).

Answer:

a) 94
b) 51
c) 41