If `W_(1) :W_(2) = 2 : 3 ` and ` W_(1) : W_(3) = 1 : 2` then `W_(2) : W_(3)` is

Options

A. `3 : 4`
B. `4 : 3`
C. `2 : 3`
D. `4 : 5`

Answer

To solve the problem, we need to find the ratio \( W_2 : W_3 \) given the ratios \( W_1 : W_2 = 2 : 3 \) and \( W_1 : W_3 = 1 : 2 \).

### Step-by-Step Solution:

1. **Express the Ratios in Terms of a Common Variable:**
- From the ratio \( W_1 : W_2 = 2 : 3 \), we can express:
\[ W_1 = 2x \quad \text{and} \quad W_2 = 3x \]
- Here, \( x \) is a common variable.

2. **Express \( W_3 \) in Terms of \( W_1 \):**
- From the ratio \( W_1 : W_3 = 1 : 2 \), we can express:
\[ W_1 = 1y \quad \text{and} \quad W_3 = 2y \]
- Here, \( y \) is another common variable.

3. **Equate the Expressions for \( W_1 \):**
- Since both expressions represent \( W_1 \), we can set them equal to each other:
\[ 2x = 1y \quad \Rightarrow \quad y = 2x \]

4. **Substitute \( y \) back to find \( W_3 \):**
- Now substitute \( y \) in the expression for \( W_3 \):
\[ W_3 = 2y = 2(2x) = 4x \]

5. **Now we have all weights in terms of \( x \):**
- \( W_1 = 2x \)
- \( W_2 = 3x \)
- \( W_3 = 4x \)

6. **Find the Ratio \( W_2 : W_3 \):**
- Now we can find the ratio:
\[ W_2 : W_3 = 3x : 4x = 3 : 4 \]

### Final Answer:
Thus, the ratio \( W_2 : W_3 \) is \( 3 : 4 \).