Composite ratios with division: Given P : Q : R = 2 : 3 : 4, compute P/Q : Q/R : R/P in simplest integer terms.

Introduction / Context:
Transforming a three-term ratio into derived ratios like P/Q, Q/R, and R/P requires dividing corresponding terms, then simplifying to a clean integer ratio. This builds fluency with fractional manipulation in proportional reasoning.

Given Data / Assumptions:
P : Q : R = 2 : 3 : 4 with all positive terms.

Concept / Approach:
Compute each fraction: P/Q = 2/3, Q/R = 3/4, and R/P = 4/2 = 2. Then clear denominators by multiplying by a common multiple to get an integer ratio.

Step-by-Step Solution:
P/Q = 2/3; Q/R = 3/4; R/P = 2. Choose 12 as a convenient common multiple. (2/3)*12 = 8; (3/4)*12 = 9; (2)*12 = 24. Therefore, P/Q : Q/R : R/P = 8 : 9 : 24.

Verification / Alternative check:
Converting back to fractions: 8/12 = 2/3, 9/12 = 3/4, 24/12 = 2, matching the original computed values.

Why Other Options Are Wrong:
Options that permute or mis-scale these values do not correspond to the defined fractions; only 8 : 9 : 24 preserves all three relationships simultaneously.

Common Pitfalls:
Mixing up the order (e.g., writing Q/P instead of P/Q) or failing to multiply by a common factor to get integers.

Final Answer:
8 : 9 : 24