Question

Out of 100 students, 50 fail in English and 30 in Mathematics. If 12 students fail in both English and Mathematics, the number of students passing both these subjects are:
(a). 8
(b). 20
(c). 32
(d). 50

Answer 1

Hint: Using Set theory find the total number of students failing that are failing either Mathematics or English i.e. find \[n\left( A\cup B \right)\]. Subtract this value from the total number of students to get the number of students who passed both subjects.

Complete step-by-step answer:

Here the total number of students = 100.
Let the number of students who fail in English be taken as n (A).
The number of students who fail in English = n (A) = 50.
Similarly, the number of students who fail in mathematics = n (B) = 30.
Thus the total number of students who fail in both English and Mathematics = 12.
It can be represented as A intersection B.
\[n\left( A\cap B \right)=12\]
Thus the total number of students who fail in both subjects will be \[n\left( A\cup B \right)\].
\[\begin{align}
  & n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)=50+30-12 \\
 & n\left( A\cup B \right)=68 \\
\end{align}\]
Thus the total number of students who failed are 68.
Thus we need to find the number of students who passed both subjects.
= Total students – The number of students who failed.
= \[100-n\left( A\cup B \right)\]
\[\begin{align}
  & =100-68 \\
 & =32 \\
\end{align}\]
\[\therefore \] The total number of students who passed both subjects = 32.
\[\therefore \] Option (c) is correct.

Note: We can also find it without the use of set theory.
Number of students fail in English = 50 – 12 = 38.
Similarly, the number of students who fail in mathematics = 30 – 12 = 18.
Number of students who failed in both = 12.
\[\therefore \] Total failing students = 38 + 18 + 12 = 68 students.
Thus the number of students who passed in both = 100 – 68 = 32 students.