In a survey of 260 students in a school, 100 were listed as taking apple juice, 150 as taking orange juice, and 75 were listed as taking both apple as well as orange juice. Find how many students were taking neither apple juice nor orange juice.
85
To determine how many students were taking neither apple juice nor orange juice, we can use the principle of inclusion-exclusion in set theory. Let's define:
- Total number of students surveyed: \( N = 260 \)
- Students taking apple juice: \( A = 100 \)
- Students taking orange juice: \( O = 150 \)
- Students taking both apple and orange juice: \( A \cap O = 75 \)
The formula for finding the number of students taking either apple juice or orange juice (or both) is:
\[ |A \cup O| = |A| + |O| - |A \cap O| \]
Substituting the values, we get:
\[ |A \cup O| = 100 + 150 - 75 = 175 \]
This means 175 students were taking either apple juice or orange juice or both. To find the number of students taking neither, we subtract this from the total number of students:
\[ N - |A \cup O| = 260 - 175 = 85 \]
Therefore, 85 students were taking neither apple juice nor orange juice.
The correct answer is 85.
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