Triplicate ratio: Determine the triplicate ratio (i.e., the ratio of cubes) of 7 : 5.

Introduction / Context: The triplicate ratio of a : b is a^3 : b^3. This extends the idea of duplicate (squared) ratios to cubes and appears often in classical proportion problems and speed/volume scaling analogies.

Given Data / Assumptions:

• Base ratio = 7 : 5.
• Triplicate ratio means cube each term.

Concept / Approach: Compute 7^3 and 5^3 and write the ratio in simplest integer form, preserving the original order.

Step-by-Step Solution: 7^3 = 343 and 5^3 = 125. Triplicate ratio = 343 : 125. No further simplification is possible since 343 and 125 share no common factor.

Verification / Alternative check: If the duplicate (squared) ratio would be 49 : 25, the triplicate is indeed 343 : 125. Both follow the nth-power definition of these classical terms.

Why Other Options Are Wrong:

• 125 : 343 is the inverse order.
• 344 : 125 and 343 : 126 are numerically off and do not represent cubes of 7 and 5.
• 49 : 25 is the duplicate (squared) ratio, not the triplicate ratio.

Common Pitfalls: Inverting the order or mixing duplicate and triplicate definitions. Always cube each term and keep the original order.

Final Answer: 343 : 125