If $x = \frac{1}{3}y$ and $y = \frac{1}{2}z$, then $x : y : z$ is equal to

### Concept & Compounding Ratios To find a combined ratio $x : y : z$ from two separate ratios ($x:y$ and $y:z$), establish a common value for the shared variable ($y$) in both ratios. ### Step-by-Step Solution * **Given equations:** $x = \frac{1}{3}y$ and $y = \frac{1}{2}z$. * **Convert to individual ratios:** From $x = \frac{1}{3}y$, we get $\frac{x}{y} = \frac{1}{3} \implies x : y = 1 : 3$. From $y = \frac{1}{2}z$, we get $\frac{y}{z} = \frac{1}{2} \implies y : z = 1 : 2$. * **Align the ratios:** Look at the value of the common term, $y$, in both ratios. In $x : y$, $y$ is $3$. In $y : z$, $y$ is $1$. * **Make the common term equal:** Multiply the second ratio by $3$ so that $y$ matches in both. $y : z = (1 \times 3) : (2 \times 3) = 3 : 6$. * **Combine the ratios:** Now that $y$ is $3$ in both cases, you can link them directly. $x : y : z = 1 : 3 : 6$. ### Exam Strategy & Shortcut You can substitute directly without finding common terms. Let $z = 6$ (a convenient multiple). Since $y = \frac{1}{2}z$, $y = \frac{1}{2}(6) = 3$. Since $x = \frac{1}{3}y$, $x = \frac{1}{3}(3) = 1$. The values are $x=1, y=3, z=6$. The ratio is directly $1 : 3 : 6$. ### Common Pitfall Trying to merge the ratios without equalizing the bridging variable ($y$), resulting in an invalid sequence. ### Final Answer Therefore, the correct answer is **$1 : 3 : 6$**.