If $x$ is inversely proportional to $y$ and $y$ is proportional to $z$, then $xz$

### Concept & Variation Chaining When variables vary with each other, we can express their relationships using proportionality constants. By expressing all variables in terms of a single linking variable, we can determine the composite relationship. ### Step-by-Step Solution 1. Given that $x$ is inversely proportional to $y$, we can write: $$ x \propto \frac{1}{y} \Rightarrow x = \frac{k_1}{y} $$ where $k_1$ is a constant of proportionality. 2. Given that $y$ is directly proportional to $z$, we can write: $$ y \propto z \Rightarrow y = k_2z $$ where $k_2$ is a constant of proportionality. 3. We need to find the nature of the product $xz$. First, express $z$ in terms of $y$: $$ z = \frac{y}{k_2} $$ 4. Now, multiply the expressions for $x$ and $z$: $$ xz = \left( \frac{k_1}{y} \right) \times \left( \frac{y}{k_2} \right) $$ 5. The variable $y$ cancels out from the numerator and denominator: $$ xz = \frac{k_1}{k_2} $$ 6. Since $k_1$ and $k_2$ are both constants, their ratio $\frac{k_1}{k_2}$ is also a constant. 7. Therefore, the product $xz$ is a constant. ### Exam Strategy & Shortcut **Number Plugging:** Let $x = 2$, $y = 3$, $z = 6$ (so $y$ is half of $z$). If $y$ doubles to $6$, $z$ must double to $12$ (direct proportion). If $y$ doubles to $6$, $x$ must halve to $1$ (inverse proportion). Test $xz$ in scenario 1: $2 \times 6 = 12$. Test $xz$ in scenario 2: $1 \times 12 = 12$. Since $xz$ remains exactly the same regardless of the changes, it is a constant. ### Common Pitfall A common mistake is substituting $y = k_2z$ into the first equation to get $x = \frac{k_1}{k_2z}$, which shows $x$ is inversely proportional to $z$, and stopping there instead of analyzing the specific expression asked for ($xz$). ### Final Answer Therefore, the correct answer is **is a constant**.