A group of men decided to do a job in 8 days. But since 10 men dropped out every day, the job got completed at the end of the 12th day. How many men were there at the beginning?

Options

A. 165
B. 175
C. 80
D. 90

Answer

To solve the problem, we need to determine how many men were there at the beginning, given that they were supposed to complete a job in 8 days but finished it in 12 days due to 10 men dropping out every day.

### Step-by-Step Solution:

1. **Understand the Work Requirement**:
The total work is designed to be completed in 8 days. Let the total amount of work be represented as \( W \). If \( x \) is the number of men initially, then the work done per day by \( x \) men is \( x \) units of work per day. Therefore, the total work can be expressed as:
\[ W = 8x \]

2. **Calculate Work Done Over 12 Days**:
Since 10 men drop out every day, we need to calculate how many men are working each day for 12 days.
- On Day 1: \( x \) men work.
- On Day 2: \( x - 10 \) men work.
- On Day 3: \( x - 20 \) men work.
- On Day 4: \( x - 30 \) men work.
- On Day 5: \( x - 40 \) men work.
- On Day 6: \( x - 50 \) men work.
- On Day 7: \( x - 60 \) men work.
- On Day 8: \( x - 70 \) men work.
- On Day 9: \( x - 80 \) men work.
- On Day 10: \( x - 90 \) men work.
- On Day 11: \( x - 100 \) men work.
- On Day 12: \( x - 110 \) men work.

3. **Set Up the Equation for Total Work Done**:
The total work done over 12 days can be calculated as:
\[ W = x + (x - 10) + (x - 20) + (x - 30) + (x - 40) + (x - 50) + (x - 60) + (x - 70) + (x - 80) + (x - 90) + (x - 100) + (x - 110) \]
Simplifying this gives:
\[ W = 12x - (10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90 + 100 + 110) \]
The sum of the series \( 10 + 20 + 30 + ... + 110 \) is an arithmetic series with 11 terms:
\[ \text{Sum} = \frac{n}{2} \times (\text{first term} + \text{last term}) = \frac{11}{2} \times (10 + 110) = \frac{11}{2} \times 120 = 660 \]
Therefore, we have:
\[ W = 12x - 660 \]

4. **Equate the Two Expressions for Work**:
Since both expressions represent the total work \( W \), we can set them equal to each other:
\[ 8x = 12x - 660 \]

5. **Solve for \( x \)**:
Rearranging the equation gives:
\[ 660 = 12x - 8x \]
\[ 660 = 4x \]
\[ x = \frac{660}{4} = 165 \]

6. **Conclusion**:
Therefore, the number of men at the beginning was \( \boxed{165} \).