Sub-duplicate ratio: Find the sub-duplicate ratio (i.e., the ratio of square roots) of 81 : 64.

Introduction / Context:
Sub-duplicate ratio refers to the ratio of the square roots of the given numbers. So, for p : q, the sub-duplicate ratio is √p : √q. This is standard terminology in ratio and proportion problems.

Given Data / Assumptions:

• Given ratio = 81 : 64.
• We assume both are non-negative, perfect squares.

Concept / Approach:
Compute the square roots and express their ratio in the simplest whole-number form. For 81 and 64, the roots are integers, making the computation straightforward.

Step-by-Step Solution:
√81 = 9 and √64 = 8. Therefore, sub-duplicate ratio = 9 : 8. This is already simplified.

Verification / Alternative check:
Check by squaring the result: (9 : 8) duplicated (squared) gives 81 : 64, confirming that (9 : 8) is indeed the sub-duplicate ratio.

Why Other Options Are Wrong:

• 8 : 9 is the inverse order.
• 4 : 9 and 7 : 8 are unrelated to square roots of 81 and 64.
• 3 : 4 would be correct for 9 : 16, not for 81 : 64.

Common Pitfalls:
Inverting the ratio or using cube roots instead of square roots. Carefully match each term with its square root.

Final Answer:
9 : 8