Question

In an examination, $35\% $ of the total students failed in Hindi and $45\% $ failed in English and $20\% $ failed in both. The percentage of those who passed in both the subjects is:
A) $20$
B) $30$
C) $40$
D) $80$

Answer 1

Hint: Let the total number of students be \[100\]
Let A be the set of students who failed in Hindi
So $35\% {\text{ of 100}} = 35$
So $n(A) = 35$
Similarly let B be the set of students who failed in English
$n(B) = 45\% {\text{ of 100}} = 45$
And also we know that $20\% $ failed in both
So $n(A \cap B) = 20\% {\text{ of 100}} = 20$
We know that $n(A \cup B) = n(A) + n(B) - n(A \cap B)$

Complete step by step solution:
Here in this question, we are given that in an examination, $35\% $ of the total students failed in Hindi and $45\% $ failed in English and $20\% $ failed in both. So firstly we need to assume the total number of students in the examination. Let us assume that the total number of students be \[100\]
Total students who failed in Hindi$ = 35\% {\text{ of 100}} = \dfrac{{35}}{{100}} \times 100 = 35$
Now we also know that $45\% $ failed in English$ = 45\% {\text{ of 100}} = \dfrac{{45}}{{100}} \times 100 = 45$
And we know that $20\% $ of the students failed in both the subjects so number of students who failed in both the subjects$ = 20\% {\text{ of 100}} = 20$
Let A be the set of students who failed in Hindi
So $35\% {\text{ of 100}} = \dfrac{{35}}{{100}} \times 100 = 35$
So $n(A) = 35$
Similarly let B be the set of students who failed in English
$n(B) = 45\% {\text{ of 100}} = \dfrac{{45}}{{100}} \times 100 = 45$
Also let that $A \cap B$ be the total students who failed in both the subjects.
And also we know that $20\% $ failed in both
So $n(A \cap B) = 20\% {\text{ of 100}} = 20$
Now we need to find the percentage of the students who passed in both the subjects which is given by =${\text{total students }} - {\text{ n(A}} \cup {\text{B)}}$
Now as we know that
$n(A \cup B) = n(A) + n(B) - n(A \cap B)$
Putting the values of $n(A),n(B),n(A \cap B)$ we get
$n(A \cup B) = 35 + 45 - 20 = 60$
Here the $n(A \cup B)$ represents the total students who failed in Hindi and English.
And $n(A \cap B)$ represents the students who failed in both the subjects.
So the total students passing in Hindi and English$ = 100 - n(A \cup B)$
                                                                                         $ = 100 - 60 = 40$
Now for percentage, $\dfrac{{{\text{number of students}}}}{{{\text{total students}}}} \times 100$
$ = \dfrac{{40}}{{100}} \times 100 = 40\% $

Note: We can do it this way as it is given that $35\% $ of the total students failed in Hindi and $45\% $ failed in English and $20\% $ failed in both. It means that $(35 - 20)\% = 15\% $failed exclusively in Hindi and $(45 - 20)\% = 25\% $ exclusively in English. So total $\% $ failed in Hindi, English and Both are
$15\% + 25\% + 20\% = 60\% $
So the students passed in both subjects$ = (100 - 60)\% = 40\% $