If `(x + 3)` : `(4x +1)` is the duplicate ratio of `3 : 5`, find the value of `x`.

Answer

To solve the problem, we need to find the value of \( x \) given that the ratio \( (x + 3) : (4x + 1) \) is the duplicate ratio of \( 3 : 5 \).

### Step-by-Step Solution:

1. **Understanding Duplicate Ratio**:
The duplicate ratio of \( a : b \) is given by \( a^2 : b^2 \). Therefore, the duplicate ratio of \( 3 : 5 \) is \( 3^2 : 5^2 \), which simplifies to \( 9 : 25 \).

2. **Setting Up the Equation**:
We can express the given ratio as:
\[ \frac{x + 3}{4x + 1} = \frac{9}{25} \]

3. **Cross Multiplying**:
To eliminate the fraction, we cross multiply:
\[ 25(x + 3) = 9(4x + 1) \]

4. **Expanding Both Sides**:
Now, expand both sides of the equation:
\[ 25x + 75 = 36x + 9 \]

5. **Rearranging the Equation**:
Next, we rearrange the equation to isolate \( x \):
\[ 25x + 75 - 9 = 36x \]
This simplifies to:
\[ 25x + 66 = 36x \]

6. **Combining Like Terms**:
Now, subtract \( 25x \) from both sides:
\[ 66 = 36x - 25x \]
This simplifies to:
\[ 66 = 11x \]

7. **Solving for \( x \)**:
Finally, divide both sides by \( 11 \):
\[ x = \frac{66}{11} = 6 \]

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

---