Question

In a class of 100 students, 55 students have passed in physics and 67 students have passed in mathematics. Find the number of students passed in physics only.

Answer 1

Hint: This question can be solved using set theory formula. Generally in these types of questions one should interpret the information properly. Generally ‘OR’ is used for union and ‘AND’ is used for intersection. Union means individually + common parts of two or more regions. Intersection only includes the common region.

Complete step-by-step solution:
Let A be the number of students passed in mathematics and B be the number of students passed in physics. So in the above question.
$n\left( A \right) = 67$ $n\left( B \right) = 55$
Total students i.e. $\left( {A \cup B} \right)$ who has passed = $100$
Using set theory formula i.e.
$n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)$
$
  100 = 67 + 55 - n\left( {A \cap B} \right) \\
  122 - 100 = n\left( {A \cap B} \right) \\
  22 = n\left( {A \cap B} \right) \\
$
 $n\left( {A \cap B} \right)$ = number of students passed in both subjects i.e. mathematics and physics
So if we subtract the number of students who have passed both from the number of students who have passed physics then we will get the number of students who have passed only physics.
So, $n\left( B \right) - n\left( {A \cap B} \right) = 55 - 22 = 33$
So there are $33$ students who have passed only physics.
Additional information: we can use other method to solve this type of questions i.e. Venn diagram. In that diagram assumptions can be made easily and interpretation of information from a venn diagram is easy. The calculations may be similar but steps can be lower in number.

Note: One can make a mistake in intersection or union and in interpreting ‘OR’ and ‘AND’ in this type of question. It’s a type of logical reasoning question also. So we should use our assumption skills properly.