$x, y, z, u$ are real numbers such that $x : y = y : z = z : u$ and $x : u = 64 : 27$. The value of $x : z$ is

### Concept & Continued Proportion Multiplication When variables are in a continued proportion (e.g., $\frac{x}{y} = \frac{y}{z} = \frac{z}{u} = k$), multiplying consecutive ratios together will eliminate intermediate variables and yield a relationship between the outer variables expressed as a power of the constant $k$. ### Step-by-Step Solution - **Given:** $\frac{x}{y} = \frac{y}{z} = \frac{z}{u} = k$ (where $k$ is a constant ratio). - **Relate Extremes:** Multiply the three individual ratios to find $\frac{x}{u}$. $\frac{x}{u} = \left(\frac{x}{y}\right) \times \left(\frac{y}{z}\right) \times \left(\frac{z}{u}\right)$ $\frac{x}{u} = k \times k \times k = k^3$ - **Solve for k:** We are given $\frac{x}{u} = \frac{64}{27}$. $k^3 = \frac{64}{27}$ Taking the cube root of both sides: $k = \sqrt[3]{\frac{64}{27}} = \frac{4}{3}$. - **Find Target Ratio:** We need to find $x : z$, which is $\frac{x}{z}$. Multiply the first two ratios: $\frac{x}{z} = \left(\frac{x}{y}\right) \times \left(\frac{y}{z}\right)$ $\frac{x}{z} = k \times k = k^2$ - **Calculate Final Value:** $k^2 = \left(\frac{4}{3}\right)^2 = \frac{16}{9}$. - **Result:** $x : z = 16 : 9$. ### Exam Strategy & Shortcut Recognize that moving from $x$ to $u$ involves $3$ steps ($x \to y$, $y \to z$, $z \to u$), so the ratio $x/u$ is the base ratio cubed. Thus, the base ratio is the cube root of $64/27$, which is $4/3$. Moving from $x$ to $z$ involves $2$ steps, so we square the base ratio: $(4/3)^2 = 16/9$. ### Common Pitfall A common pitfall is stopping after finding the base ratio $k = 4/3$ and selecting that as the answer, forgetting that the question specifically asks for $x : z$, which corresponds to $k^2$. ### Final Answer Therefore, the correct answer is **$16 : 9$**.