The only library of a business school has total 91 books that are shared among all the students in such a way that every two students share a book on Organizational Behaviour, every three students share a book on Financial Management and every four students share a book on Operations Research. The number of students in the school is :

Options

A. 81
B. 84
C. 48
D. can't be determined

Answer

To solve the problem, we need to establish a relationship between the number of students and the books they share. Let's denote the number of students as \( n \).

1. **Understanding the sharing of books**:
- Every 2 students share 1 book on Organizational Behaviour. Therefore, the number of books on Organizational Behaviour is \( \frac{n}{2} \).
- Every 3 students share 1 book on Financial Management. Therefore, the number of books on Financial Management is \( \frac{n}{3} \).
- Every 4 students share 1 book on Operations Research. Therefore, the number of books on Operations Research is \( \frac{n}{4} \).

2. **Setting up the equation**:
- The total number of books is given as 91. Therefore, we can set up the equation:
\[ \frac{n}{2} + \frac{n}{3} + \frac{n}{4} = 91 \]

3. **Finding a common denominator**:
- The least common multiple (LCM) of 2, 3, and 4 is 12. We will use this to combine the fractions:
\[ \frac{n}{2} = \frac{6n}{12}, \quad \frac{n}{3} = \frac{4n}{12}, \quad \frac{n}{4} = \frac{3n}{12} \]
- Therefore, we can rewrite the equation as:
\[ \frac{6n + 4n + 3n}{12} = 91 \]
- Simplifying this gives:
\[ \frac{13n}{12} = 91 \]

4. **Solving for \( n \)**:
- To eliminate the fraction, multiply both sides by 12:
\[ 13n = 91 \times 12 \]
- Calculate \( 91 \times 12 \):
\[ 91 \times 12 = 1092 \]
- Now, divide both sides by 13:
\[ n = \frac{1092}{13} \]
- Calculate \( \frac{1092}{13} \):
\[ n = 84 \]

5. **Conclusion**:
- The number of students in the school is \( n = 84 \).