Question

In a group of 15 students seven can read Hindi and 5 can read English and 6 can read neither. The number of students who can read Hindi and English both is _____.
(a) 1
(b) 2
(c) 3
(d) 12

Answer 1

Hint: Here, we will draw the Venn diagram for the given problem and then consider the number of elements in each of the regions obtained as x, y and z. The union set represents the total number of students in the group. The region of the union set which is outside the circles representing the number of students who can read Hindi or English represent those students who can read neither Hindi nor English.Therefore, we will calculate the value of x, y and z using the given conditions.

Complete step-by-step answer:
The Venn diagram for this problem can be drawn as:

Here, we can observe that:
Total number of students who can read Hindi = x+y
Total number of students who can read English = y+z
Total number of students who can read both Hindi and English = y
Therefore, from the given data, we can write:
 $ x+y=7..........\left( 1 \right) $
 $ y+z=5..........\left( 2 \right) $
Now, it is also given that 6 students can read neither English nor Hndi. So:
 ( Total number of students ) – ( x+y+z) = 6
 $ \Rightarrow 15-\left( x+y+z \right)=6 $
 $ \Rightarrow x+y+z=9.......\left( 3 \right) $
Now, we will find the values of y using equations (1), (2) and (3).
On subtracting equation (2) from equation (3), we get:
 $ \begin{align}
  & \left( x+y+z \right)-\left( y+z \right)=9-5 \\
 & \Rightarrow x+y+z-y-z=4 \\
 & \Rightarrow x=4 \\
\end{align} $
So, the value of x comes out to be equal to 4.
On putting x = 4 in equation (1), we get:
 $ \begin{align}
  & 4+y=7 \\
 & \Rightarrow y=7-4=3 \\
\end{align} $
So, the value of y comes out to be equal to 3 which means that 3 students can read both English and Hindi.
Hence, option (c) is the correct answer.

Note: Students should note here that the elements which are common to two sets come under their intersection. So, here the intersection of the two sets gives us the number of students who fall in both the categories. It should be kept in mind that the region of the union set outside the circles represent the number of students who cannot read any of the languages. So, we write that 15 – (x+y+z) is equal to 6. Some students directly write the answer by taking the values from the question as 15 - 7 - 5 = 3. This is not the right way to solve the question. Steps should be mentioned in detail to get full marks in the exams. It will also show the knowledge the student has in this topic.