Question

In a group of $15$ students, $7$ have studied German, $8$ have studied French, and $3$ have not studied either. How many students have studied both German and French?
A. $0$
B. $3$
C. $4$
D. $5$

Answer 1

Hint: So, here in this question we can find the no. of students who studied both the subjects that are German and French, by subtracting the total no. of students who studied even one subject from the sum of students who studied German and the no. of students who studied French.

Step by Step solution:
In this question first of all we will start solving this question by listing the given quantities.
So, the given information in this question is
Total no. of students $ = 15 = n\left( U \right)$
No. of students who studied German $ = 7 = n\left( G \right)$
No. of students who studied French $ = 8 = n\left( F \right)$
No. of students who have not studied either $ = 3 = n\left( x \right)$
So, from the above given information we can conclude that the no. of students who studied even one subject would be equal to the subtraction of no. of students who have not studied from Total no. of students.
That is,
No. of students who studied even one subject $ = $ Total no. of students $ - $ no. of students who have not studied
$ \Rightarrow n\left( {G \cup F} \right) = n\left( U \right) - n\left( x \right)$
$
   \Rightarrow n\left( {G \cup F} \right) = 15 - 3 \\
   \Rightarrow n\left( {G \cup F} \right) = 12 \\
 $
So, the no. of students who studied even one subject$ = 12$
Now, from here we can conclude that
No. of students who studied even one subject$ = $ (No. of students who studied German) $ + $ (No. of students who studied French$ - $ (No. of students who studied both French and German)
$
   \Rightarrow n\left( {G \cup F} \right) = n\left( G \right) + n\left( F \right) - n\left( {G \cap F} \right) \\
   \Rightarrow 12 = 7 + 8 - n\left( {G \cap F} \right) \\
   \Rightarrow n\left( {G \cap F} \right) = 15 - 12 \\
 $
$ \Rightarrow n\left( {G \cap F} \right) = 3$
That means the No. of students who studied both French and German are equal to $3$.
$ \Rightarrow $Option B is correct.

Note: In these types of questions we have an alternative method to understand the given statements easily, which is by drawing Venn Diagrams.
So, by making Venn diagrams of the given information we can solve these types of questions in competitive exams.