If`x : y : : 2 : 3` and `2 : x : : 4 : 8` the value of y is

Options

A. 6
B. 8
C. 4
D. 12

Answer

To solve the problem, we need to find the value of \( y \) given the ratios \( x : y = 2 : 3 \) and \( 2 : x = 4 : 8 \).

### Step-by-step Solution:

1. **Set up the first ratio**:
From the first ratio \( x : y = 2 : 3 \), we can express \( x \) in terms of \( y \):
\[ \frac{x}{y} = \frac{2}{3} \]
This implies:
\[ x = \frac{2}{3}y \quad \text{(Equation 1)} \]

2. **Set up the second ratio**:
From the second ratio \( 2 : x = 4 : 8 \), we can express it as:
\[ \frac{2}{x} = \frac{4}{8} \]
Simplifying \( \frac{4}{8} \) gives:
\[ \frac{2}{x} = \frac{1}{2} \]
Cross-multiplying gives:
\[ 2 \cdot 2 = 1 \cdot x \implies x = 4 \quad \text{(Equation 2)} \]

3. **Substitute \( x \) into Equation 1**:
Now that we have \( x = 4 \) from Equation 2, we substitute this value into Equation 1:
\[ 4 = \frac{2}{3}y \]

4. **Solve for \( y \)**:
To find \( y \), we can rearrange the equation:
\[ y = 4 \cdot \frac{3}{2} = 6 \]

Thus, the value of \( y \) is \( 6 \).

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

---