In a class of MBA students `16.66%` students are from Science background and `12.5%` students are from commerce background and `6.66%` students from arts background and rest are from Engineering background. The minimum possible student of engineering background are:

Options

A. 45
B. 77
C. 100
D. 120

Answer

To solve the problem, we need to find the minimum possible number of students from the engineering background in a class of MBA students based on the given percentages of students from different backgrounds.

### Step-by-Step Solution:

1. **Identify the Percentages**:
- Science background: 16.66%
- Commerce background: 12.5%
- Arts background: 6.66%
- Engineering background: Rest of the students

2. **Convert Percentages to Fractions**:
- Science: 16.66% = 16.66/100 = 1/6 (approximately)
- Commerce: 12.5% = 12.5/100 = 1/8
- Arts: 6.66% = 6.66/100 = 1/15 (approximately)

3. **Calculate Total Percentage of Students from Science, Commerce, and Arts**:
- Total percentage from Science, Commerce, and Arts = 16.66% + 12.5% + 6.66%
- Total = 16.66 + 12.5 + 6.66 = 35.82%

4. **Calculate the Percentage of Students from Engineering Background**:
- Percentage of Engineering students = 100% - Total percentage from other backgrounds
- Percentage of Engineering students = 100% - 35.82% = 64.18%

5. **Determine the Minimum Number of Students**:
- Let the total number of students in the class be \( N \).
- The number of Engineering students = 64.18% of \( N \) = 0.6418 * \( N \).
- To find the minimum number of Engineering students, \( N \) must be a multiple of the denominators of the fractions used in the percentages. The least common multiple (LCM) of 6, 8, and 15 is 120.
- Therefore, if \( N = 120 \), the number of Engineering students = 0.6418 * 120 = 77.

6. **Conclusion**:
- The minimum possible number of students from the engineering background is 77.