Question

If 9000 students appeared in two papers in mathematics at a certain examination. Exactly 7400 and 6600 students passed in paper A and B respectively. Find the number of students who failed in both papers.

Answer 1

Hint: We draw Venn diagrams for the given situation and use set theory to find the required number of students. We find the number of students that passed in A or B by calculating union from the formula. Subtract the number of students passed in both papers from the total number of students to get the number of students who failed in both papers.
* If \[P(A)\] denotes the number of elements in A, \[P(B)\] denotes the number of elements in B then their intersection \[P(A \cap B)\] denotes the number of elements in both A and B. Then the number of elements in A or B is given by \[P(A \cup B)\]. Then the formula for set theory is
\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]

Complete step-by-step solution:
We are given that
Total number of students that appeared in two papers \[ = 9000\]
\[U = 9000\]
Number of students who passed in paper A \[ = 7400\]
\[P(A) = 7400\]
Number of students who passed in paper B \[ = 6600\]
\[P(B) = 6600\]
Number of students who passed in both paper A and B \[ = 6400\]
\[P(A \cap B) = 6400\]
Then the number of students who passed in paper A or paper B is given by \[P(A \cup B)\]
Then we can draw a Venn diagram depicting the number of students passing the exam A, B and both A and B. Here U is the universal set which depicts the total number of students who appeared in the two papers.

We use the formula \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\] to find the value of union of A and B.
Substitute the value of \[P(A) = 7400\],\[P(B) = 6600\] and \[P(A \cap B) = 6400\] in the formula
\[ \Rightarrow P(A \cup B) = 7400 + 6600 - 6400\]
Add the terms in RHS
\[ \Rightarrow P(A \cup B) = 14000 - 6400\]
Calculate the difference
\[ \Rightarrow P(A \cup B) = 7600\]
Now we have the number of students which have passed either paper A or paper B or both A and B.
Therefore, the number of students who have not passed both the papers is given by subtracting the number of students which have passed A or B from the total number of students who appeared in two examinations.
\[ \Rightarrow \]Number of students who failed in both papers \[ = U - P(A \cup B)\]
Substituting the value of \[U = 9000,P(A \cup B) = 7600\]
\[ \Rightarrow \]Number of students who failed in both papers \[ = 9000 - 7600\]
\[ \Rightarrow \]Number of students who failed in both papers \[ = 1400\]

\[\therefore \]Number of students who failed in both papers is 1400.

Note: Students might make the mistake of assuming the union of the two elements as the total number of elements. Keep in mind union denoted the elements that are in A, B or in both A and B but we have elements that are neither in A, nor in B also. So, the universal set denotes the total number of elements available.