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 Fraction Form:**
- From the ratio \( W_1 : W_2 = 2 : 3 \), we can express this as:
\[ \frac{W_1}{W_2} = \frac{2}{3} \]
- From the ratio \( W_1 : W_3 = 1 : 2 \), we can express this as:
\[ \frac{W_1}{W_3} = \frac{1}{2} \]

2. **Express \( W_2 \) and \( W_3 \) in terms of \( W_1 \):**
- From \( \frac{W_1}{W_2} = \frac{2}{3} \), we can rearrange it to find \( W_2 \):
\[ W_2 = \frac{3}{2} W_1 \]
- From \( \frac{W_1}{W_3} = \frac{1}{2} \), we can rearrange it to find \( W_3 \):
\[ W_3 = 2 W_1 \]

3. **Substitute \( W_1 \) in terms of \( W_2 \) or \( W_3 \):**
- We can express \( W_1 \) in terms of \( W_2 \):
\[ W_1 = \frac{2}{3} W_2 \]
- Now substitute \( W_1 \) into the equation for \( W_3 \):
\[ W_3 = 2 \left(\frac{2}{3} W_2\right) = \frac{4}{3} W_2 \]

4. **Find the Ratio \( W_2 : W_3 \):**
- Now we can find the ratio \( W_2 : W_3 \):
\[ W_2 : W_3 = W_2 : \frac{4}{3} W_2 \]
- This simplifies to:
\[ W_2 : W_3 = 1 : \frac{4}{3} \]
- To express this in a simpler form, we can multiply both sides by 3:
\[ W_2 : W_3 = 3 : 4 \]

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