If x : y = 2 : 3 and 2 : x = 1 : 2, then find the value of y.

Options

A. 4
B. 6
C. `1/3`
D. `3/2`

Answer

To solve the problem, we need to work with the given ratios step by step.

### Step 1: Understand the Ratios
We have two ratios given:
1. \( x : y = 2 : 3 \)
2. \( 2 : x = 1 : 2 \)

### Step 2: Express x in terms of y
From the first ratio \( x : y = 2 : 3 \), we can express \( x \) in terms of \( y \):
\[ x = \frac{2}{3}y \]

### Step 3: Express x in terms of a constant
From the second ratio \( 2 : x = 1 : 2 \), we can cross-multiply to find \( x \):
\[ 2 \cdot 2 = 1 \cdot x \implies 4 = x \]

### Step 4: Substitute x in terms of y
Now we have \( x = 4 \) from the second ratio. We can substitute this value into the equation we found in Step 2:
\[ 4 = \frac{2}{3}y \]

### Step 5: Solve for y
To find \( y \), we can multiply both sides of the equation by \( \frac{3}{2} \):
\[ y = 4 \cdot \frac{3}{2} = 6 \]

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