In a recent survey of 100 students, 34 said that they took a Math class as a freshman, 59 said that they took an English class as a freshman and 12 said they took both classes. Required:
How many students took neither class as a freshman?

Step-by-step explanation:

We will use the set notation to solve this question.

let n(U) be the total number of students surveyed = 100

n(M) be the number of student that took math = 34

n(E) be the number of student that took English = 59

n(M∩E) be the number of student that took both classes= 12

n(M∪E)' be the number of student that took neither class = ?

Using the formula n(U) = n(M∪E) + n(M∪E)'

n(M∪E)' = n(U)-n(M∪E)

Before we can get the number of student that took neither class i.e n(M∪E)'  we need to  get n(M∪E).

n(M∪E) = n(M)+n(E)- n(M∩E)

n(M∪E) = 34+59-12

n(M∪E)  = 81

Since n(M∪E)' = n(U)-n(M∪E);

n(M∪E)' = 100-81

n(M∪E)' = 19

Hence 19 students took neither class as a freshmen.