If the work done by (x-1) men in (x + 1) days is to the work done by (x + 2) men in (x - 1) days are in the ratio 9:10, then the value of x is equal to :

Options

A. 5
B. 6
C. 7
D. 8

Answer

To solve the problem, we need to find the value of \( x \) given that the work done by \( (x - 1) \) men in \( (x + 1) \) days is to the work done by \( (x + 2) \) men in \( (x - 1) \) days is in the ratio \( 9:10 \).

### Step-by-step Solution:

1. **Define the Work Done**:
- Let \( W_1 \) be the work done by \( (x - 1) \) men in \( (x + 1) \) days.
- Let \( W_2 \) be the work done by \( (x + 2) \) men in \( (x - 1) \) days.

2. **Calculate Work Done**:
- The work done by \( (x - 1) \) men in \( (x + 1) \) days can be expressed as:
\[ W_1 = (x - 1) \times (x + 1) \]
- The work done by \( (x + 2) \) men in \( (x - 1) \) days can be expressed as:
\[ W_2 = (x + 2) \times (x - 1) \]

3. **Set Up the Ratio**:
- According to the problem, the ratio of \( W_1 \) to \( W_2 \) is \( 9:10 \):
\[ \frac{W_1}{W_2} = \frac{9}{10} \]

4. **Substitute the Work Expressions**:
- Substitute \( W_1 \) and \( W_2 \) into the ratio:
\[ \frac{(x - 1)(x + 1)}{(x + 2)(x - 1)} = \frac{9}{10} \]

5. **Simplify the Equation**:
- Cancel \( (x - 1) \) from both sides (assuming \( x \neq 1 \)):
\[ \frac{x + 1}{x + 2} = \frac{9}{10} \]

6. **Cross-Multiply**:
- Cross-multiplying gives:
\[ 10(x + 1) = 9(x + 2) \]

7. **Expand Both Sides**:
- Expanding both sides:
\[ 10x + 10 = 9x + 18 \]

8. **Rearrange the Equation**:
- Rearranging the equation to isolate \( x \):
\[ 10x - 9x = 18 - 10 \]
\[ x = 8 \]

9. **Conclusion**:
- The value of \( x \) is \( 8 \).