The work done by `(x - 3)` men in `(2x + 1)` days and the work done by `(2x + 1)` men in `(x + 4)` days are in the ratio `3 : 10`. Find the value of x.

Answer

To solve the problem, we need to find the value of \( x \) given that the work done by \( (x - 3) \) men in \( (2x + 1) \) days and the work done by \( (2x + 1) \) men in \( (x + 4) \) days are in the ratio \( 3 : 10 \).

### Step-by-Step Solution:

1. **Define the Work Done**:
- Let the work done by \( (x - 3) \) men in \( (2x + 1) \) days be denoted as \( W_1 \).
- The work done can be expressed as:
\[ W_1 = (x - 3) \times (2x + 1) \]

2. **Define the Second Work Done**:
- Let the work done by \( (2x + 1) \) men in \( (x + 4) \) days be denoted as \( W_2 \).
- The work done can be expressed as:
\[ W_2 = (2x + 1) \times (x + 4) \]

3. **Set Up the Ratio**:
- According to the problem, the ratio of \( W_1 \) to \( W_2 \) is given as:
\[ \frac{W_1}{W_2} = \frac{3}{10} \]
- Substituting the expressions for \( W_1 \) and \( W_2 \):
\[ \frac{(x - 3)(2x + 1)}{(2x + 1)(x + 4)} = \frac{3}{10} \]

4. **Simplify the Equation**:
- We can cancel \( (2x + 1) \) from both sides (assuming \( 2x + 1 \neq 0 \)):
\[ \frac{x - 3}{x + 4} = \frac{3}{10} \]

5. **Cross Multiply**:
- Cross multiplying gives us:
\[ 10(x - 3) = 3(x + 4) \]

6. **Expand Both Sides**:
- Expanding both sides:
\[ 10x - 30 = 3x + 12 \]

7. **Rearrange the Equation**:
- Bringing all terms involving \( x \) to one side and constant terms to the other:
\[ 10x - 3x = 12 + 30 \]
\[ 7x = 42 \]

8. **Solve for \( x \)**:
- Dividing both sides by 7:
\[ x = \frac{42}{7} = 6 \]

### Final Answer:
The value of \( x \) is \( 6 \).