Question

In an examination, 80% of the students passed in English, 85% in Mathematics and 75% in both English and Mathematics. If 40 students failed in both the subjects, find the total number of students.
A. 350
B. 400
C. 450
D. 500

Answer 1

Hint: To solve this question, we will let the total number of students be x and we will use Venn diagram and the property ${\text{n(A}} \cup {\text{B) = n(A) + n(B) - n(A}} \cap {\text{B)}}$ to solve the given problem.

Complete step-by-step answer:
Let A be the set representing students passed in English and B be set representing students passed in Mathematics. Therefore, the Venn diagram is

Now, n(A) is equal to the elements in set A. Similarly, n(B) represents elements in set B.
So, we have n(A) = 80% of x = $\dfrac{{80}}{{100}} \times {\text{x}}$
n(B) = 85% of x = $\dfrac{{85}}{{100}} \times {\text{x}}$
Also, we are given that 75% students passed in both subjects. So, n (${\text{A}} \cap {\text{B}}$) = 75% of x = $\dfrac{{75}}{{100}} \times {\text{x}}$
Now, we will use the property ${\text{n(A}} \cup {\text{B) = n(A) + n(B) - n(A}} \cap {\text{B)}}$ to find the total students passed.
Therefore, n(${\text{A}} \cup {\text{B}}$) = $\dfrac{{80{\text{x}}}}{{100}}{\text{ + }}\dfrac{{85{\text{x}}}}{{100}}{\text{ - }}\dfrac{{75{\text{x}}}}{{100}}$ = $\dfrac{{90{\text{x}}}}{{100}}$ = $\dfrac{{9{\text{x}}}}{{10}}$
Total number of students passed = n (${\text{A}} \cup {\text{B}}$) = $\dfrac{{9{\text{x}}}}{{10}}$
So, students failed in both the subjects = total students – total number of students passed
Students failed = x - $\dfrac{{9{\text{x}}}}{{10}}$ = $\dfrac{{\text{x}}}{{10}}$
But, according to the question, students failed = 40
Therefore, $\dfrac{{\text{x}}}{{10}}$ = 40
$ \Rightarrow $ x = 400
Therefore, total number of students = x = 400
So, option (B) is correct.

Note: When we come up with such types of questions, we will first draw a Venn diagram to solve the problem. Venn diagrams help in better visualisation of questions and help in solving questions easily. Also, we will let the value asked in the question a variable and then apply the property ${\text{n(A}} \cup {\text{B) = n(A) + n(B) - n(A}} \cap {\text{B)}}$ to find the value of variable. Also, we will use the condition given in the question accordingly to solve the given problem.