Questions: At a community college, a survey was taken to determine where students study on campus. Of the 250 students surveyed, 176 studied in the library. 138 studied in the cafeteria. 81 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.)

At a community college, a survey was taken to determine where students study on campus. Of the 250 students surveyed, 176 studied in the library. 138 studied in the cafeteria. 81 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.)

Transcript text: At a community college, a survey was taken to determine where students study on campus. Of the 250 students surveyed, 176 studied in the library. 138 studied in the cafeteria. 81 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.)

Solution

Solution Steps

Solution Approach

a) To find the number of students who studied only in the library, subtract the number of students who studied in both places from the total number of students who studied in the library.

b) To find the number of students who studied only in the cafeteria, subtract the number of students who studied in both places from the total number of students who studied in the cafeteria.

c) To find the number of students who did not study in either place, subtract the number of students who studied in at least one of the places from the total number of students surveyed. Use the principle of inclusion-exclusion to find the number of students who studied in at least one place.

Step 1: Students Who Studied Only in the Library

To find the number of students who studied only in the library, we use the formula: \[ \text{Only Library} = \text{Library Students} - \text{Both Places Students} \] Substituting the values: \[ \text{Only Library} = 176 - 81 = 95 \]

Step 2: Students Who Studied Only in the Cafeteria

To find the number of students who studied only in the cafeteria, we use the formula: \[ \text{Only Cafeteria} = \text{Cafeteria Students} - \text{Both Places Students} \] Substituting the values: \[ \text{Only Cafeteria} = 138 - 81 = 57 \]

Step 3: Students Who Did Not Study in Either Place

To find the number of students who did not study in either the library or the cafeteria, we use the formula: \[ \text{Neither Place} = \text{Total Students} - \text{At Least One Place} \] Where: \[ \text{At Least One Place} = \text{Library Students} + \text{Cafeteria Students} - \text{Both Places Students} \] Calculating: \[ \text{At Least One Place} = 176 + 138 - 81 = 233 \] Then: \[ \text{Neither Place} = 250 - 233 = 17 \]