Question

Out of 500 first year students, 260 passed in the first semester and 210 passed in the second semester. If 170 did not pass in either semester, how many passed in both semester?
A. 30
B. 40
C. 70
D. 140

Answer 1

Hint: We first try to find the number of students who failed in the first and second semesters. We denote those sets separately and try to find the number of students that failed in at least 1 semester using the formula of $ n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right) $ . The complement set of the set $ \left( A\cup B \right) $ is the number of students who passed in both semesters.

Complete step by step answer:
There are 500 first-year students. If S denotes the event of all students then $ n\left( S \right)=500 $ .
Out of them 260 passed in the first semester.
So, the number of students failed in the first semester is $ 500-260=240 $ .
Also, 210 passed in the second semester.
So, the number of students failed in the second semester is $ 500-210=290 $ .
Let us denote the students failed in first semester as set A and the students who failed in the second semester as set B.
So, $ n\left( A \right)=240 $ and $ n\left( B \right)=290 $ . We also have that 170 students failed in both semesters.
This means $ n\left( A\cap B \right)=170 $ .
Now we need to find the number of students that failed in at least 1 semester which is denoted by $ n\left( A\cup B \right) $ .
We know the theorem $ n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right) $ and putting the values we get
 $ n\left( A\cup B \right)=240+290-170=360 $ .
So, out of 500 students, 360 failed in at least 1 semester. The complement set will be those students who passed in both semesters which is denoted by $ n{{\left( A\cup B \right)}^{c}} $ .
We know $ n{{\left( A\cup B \right)}^{c}}=n\left( S \right)-n\left( A\cup B \right)=500-360=140 $ .
The correct option is D.

Note:
We need to remember that just like failed numbers we could have taken those sets as the number of students passed. In that case, the $ 500-170=330 $ number of students would have passed in at least one of the semesters. From there we could have found the number of students who passed in only 1 semester separately. We subtract those two numbers from 330 to find the final answer.