Question

a college radio station surveyed 299 incoming freshmen to see how many like jazz music and how many like classical music.
the venn diagram below shows the results. (each number gives the number of freshmen who fall into that venn diagram category.)
all freshman in the survey
(a) how many of the freshmen do not like jazz music?
(b) how many of the freshmen like jazz music or classical music (or both)?
(c) how many of the freshmen like jazz music but do not like classical music?

Explanation:

Step1: Find the number of freshmen who do not like jazz music

The total number of freshmen is \(299\). The number of freshmen who like jazz music is \(41 + 103=144\). So the number of freshmen who do not like jazz music is \(299-(41 + 103)\).

$$299-(41 + 103)=299 - 144=155$$

Step2: Find the number of freshmen who like jazz music or classical music

We use the formula \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\). Here \(n(A)\) (jazz) \(=41 + 103 = 144\), \(n(B)\) (classical) \(=103+33 = 136\), \(n(A\cap B)=103\). So \(n(A\cup B)=144+136 - 103\).

$$144+136-103=277 - 103=174$$

Step3: Find the number of freshmen who like jazz music but do not like classical music

The number of freshmen who like jazz music is \(41+103\), and the number of those who like both is \(103\). So it is \(41+103-103\).

$$41+103 - 103=41$$

Answer:

(a) \(155\)
(b) \(174\)
(c) \(41\)