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 semesters $?$

Answer 1

Determine the student passes in both the semester:

Here out of $500$ first-year students ,$260$ passed in the first semester and $210$ passed in the second semester whereas, $170$ students did not pass in either semester.

Assume $n\left(A\right)$ represents the number of students passed in the first semester and $n\left(B\right)$ represents the number of students passed in the second semester.

Such that $n\left(A\right)=260$,$n\left(B\right)=210$, $n\left(A\cup B\right)\text{'}=170$

Therefore, the number of students who passed in both semesters:

$n(A\cap B)=n\left(A\right)+n\left(B\right)-n(A\cup B)$

Apply respective values in the above equation to find the number of students who passed in both semesters:

$\begin{array}{c}n(A\cap B)=260+210-(500-170)\left[\because n(A\cup B)=500-n(A\cup B)\text{'}\right]\\ =470-330\\ =140\end{array}$

Hence, the number of students who passed in both semesters is $140$.