If `(2x+ 3) : (5x- 38)` be the duplicate ratio of `sqrt(5): sqrt(6)`. Find the value of x.

Options

A. 8
B. 32
C. 16
D. 18

Answer

To solve the problem, we need to find the value of \( x \) given that the ratio \( (2x + 3) : (5x - 38) \) is the duplicate ratio of \( \sqrt{5} : \sqrt{6} \).

### Step-by-Step Solution:

1. **Understanding Duplicate Ratio**:
The duplicate ratio of \( a : b \) is given by \( \frac{a^2}{b^2} \). Therefore, the duplicate ratio of \( \sqrt{5} : \sqrt{6} \) can be calculated as:
\[ \left(\frac{\sqrt{5}}{\sqrt{6}}\right)^2 = \frac{5}{6} \]

2. **Setting Up the Equation**:
We can set up the equation based on the given ratio:
\[ \frac{2x + 3}{5x - 38} = \frac{5}{6} \]

3. **Cross-Multiplying**:
To eliminate the fraction, we cross-multiply:
\[ 6(2x + 3) = 5(5x - 38) \]

4. **Expanding Both Sides**:
Now, we expand both sides of the equation:
\[ 12x + 18 = 25x - 190 \]

5. **Rearranging the Equation**:
Next, we rearrange the equation to isolate \( x \):
\[ 12x - 25x = -190 - 18 \]
\[ -13x = -208 \]

6. **Solving for \( x \)**:
Dividing both sides by -13 gives:
\[ x = \frac{208}{13} \]
\[ x = 16 \]

### Final Answer:
The value of \( x \) is \( 16 \).

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