How many of the freshmen like both classical and rock, but don't like jazz?
Understand the Problem
The question requires constructing a Venn diagram based on the provided data about freshmen's music preferences. Then, using the Venn diagram, determine how many freshmen like both classical and rock, but not jazz. We need to use the inclusion-exclusion principle and provided numbers to calculate the values for each region of the Venn diagram, and then extract value for "classical and rock, but not jazz".
Answer
33
Answer for screen readers
33
Steps to Solve
- Determine the number of freshmen who like at least one of the three genres
Subtract the number of freshmen who don't like any of the three genres from the total number of freshmen surveyed.
$598 - 118 = 480$
- Find the number of freshmen who like both classical and rock and jazz
This information is directly provided: 56 freshmen like all three types of music. This is the intersection of all three circles.
- Calculate the number of freshmen who like classical and rock, but not jazz
We know that 89 freshmen like both rock and classical. Out of these 89, 56 also like jazz. Therefore, the number of freshmen who like both classical and rock, but not jazz, is $89 - 56 = 33$.
33
More Information
The inclusion-exclusion principle is helpful for solving Venn diagram problems like these. By carefully subtracting and adding the sizes of overlapping sets, one can determine the size of various regions within the Venn diagram.
Tips
A common mistake is to forget to subtract the number of people who like all three genres when finding the number of people that like only two genres. In this problem, one might initially think that 89 students like both rock and classical but not jazz. However, since 56 of them also like Jazz, then only $89 - 56 = 33$ like rock and classical but not jazz.
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