If $W_1 : W_2 = 2 : 3$ and $W_1 : W_3 = 1 : 2$, then $W_2 : W_3$ is

### Concept & Linking Ratios When two ratios share a common term (in this case, $W_1$), we can combine them to find the ratio between the other two terms by manipulating the fractions. $$ \frac{W_2}{W_3} = \frac{W_2}{W_1} \times \frac{W_1}{W_3} $$ ### Step-by-Step Solution 1. We are given the following ratios: $\frac{W_1}{W_2} = \frac{2}{3}$ $\frac{W_1}{W_3} = \frac{1}{2}$ 2. We need to find the ratio $W_2 : W_3$, which is the fraction $\frac{W_2}{W_3}$. 3. First, take the reciprocal of the first ratio to get $\frac{W_2}{W_1}$: $\frac{W_2}{W_1} = \frac{3}{2}$ 4. Now, multiply this inverted ratio by the second ratio to cancel out $W_1$: $\frac{W_2}{W_3} = \left(\frac{W_2}{W_1}\right) \times \left(\frac{W_1}{W_3}\right)$ $\frac{W_2}{W_3} = \left(\frac{3}{2}\right) \times \left(\frac{1}{2}\right)$ 5. Multiply the numerators and denominators: $\frac{W_2}{W_3} = \frac{3 \times 1}{2 \times 2} = \frac{3}{4}$ 6. Express this fraction as a ratio: $W_2 : W_3 = 3 : 4$ ### Exam Strategy & Shortcut Alternatively, equalize the common variable $W_1$. $W_1 : W_2 = 2 : 3$ $W_1 : W_3 = 1 : 2 = 2 : 4$ (multiplying by $2$ to match $W_1=2$) Now that $W_1$ is $2$ in both, we can clearly see $W_2 = 3$ and $W_3 = 4$. Thus, $W_2 : W_3 = 3 : 4$. This visual scaling is incredibly fast. ### Common Pitfall Failing to notice that $W_1$ is the first term in *both* given ratios. If you blindly multiply $\frac{2}{3} \times \frac{1}{2}$, you calculate $\frac{(W_1)^2}{W_2 W_3}$, which is incorrect. ### Final Answer Therefore, the correct answer is **$3 : 4$**.