If $x$% of $y$ is 100 and $y$% of $z$ is 200, find a relation between $x$ and $z$.

### Concept & Formula The foundational concept here is translating percentage statements into algebraic equations. The phrase "$a$% of $b$" translates directly to the product of $a$ and $b$ divided by 100. Setting up a system of equations allows us to divide them and find the direct ratio between the target variables, bypassing the need to solve for the intermediate variable entirely. $$ a\% \text{ of } b = \frac{a \times b}{100} $$ ### Step-by-Step Solution * **Given:** * $x$% of $y = 100$ * $y$% of $z = 200$ * **Calculation:** Convert the first statement into an equation: $$ \frac{x}{100} \times y = 100 \implies xy = 10000 $$ Convert the second statement into an equation: $$ \frac{y}{100} \times z = 200 \implies yz = 20000 $$ To find the relation between $x$ and $z$, we need to eliminate $y$. The most efficient way to do this when variables are multiplied is to divide the two equations. Divide the second equation by the first: $$ \frac{yz}{xy} = \frac{20000}{10000} $$ The $y$ terms cancel out cleanly: $$ \frac{z}{x} = 2 $$ Multiply both sides by $x$ to isolate $z$: $$ z = 2x $$ ### Exam Strategy & Shortcut Notice that the right side of the equations (200) is exactly double the other (100). Because both expressions take a percentage involving a common variable $y$ (i.e., $x \times y$ and $y \times z$), their products scale linearly. You can visually inspect this: $(y \times z)$ is double $(x \times y)$. The $y$ cancels immediately in your head, meaning $z$ must simply be double $x$. You can mark $z = 2x$ in under 5 seconds without writing anything! ### Common Pitfall A major time-wasting pitfall is attempting to isolate $y$ in the first equation (e.g., $y = \frac{10000}{x}$) and substituting it into the second. While mathematically sound, it introduces messy fractions and slows down your solving speed significantly. Always look to divide equations when eliminating a multiplied variable. ### Final Answer Therefore, the correct answer is **$z = 2x$**.