Ratio update after adding a constant: Two numbers are in the ratio 2 : 3. After adding 9 to each number, the new ratio becomes 3 : 4. Compute the product of the original two numbers.

Introduction / Context:
Like other ratio-shift problems, we parameterize the original numbers, apply the given increment, and enforce the new ratio to solve for the scaling factor. Once the originals are known, compute their product directly.

Given Data / Assumptions:

• Original numbers = 2x and 3x.
• Add 9 to each ⇒ (2x + 9), (3x + 9).
• New ratio = 3 : 4.

Concept / Approach:
Set (2x + 9)/(3x + 9) = 3/4. Solve for x, then compute (2x)*(3x) as the product of the original pair.

Step-by-Step Solution:

Verification / Alternative check:
After adding 9 → 27 and 36; ratio 27 : 36 = 3 : 4, confirming the setup.

Why Other Options Are Wrong:
360, 432, 480, 512 are products derived from incorrect x or arithmetic errors when multiplying.

Common Pitfalls:
Forgetting to add 9 to both numbers or cross-multiplying incorrectly in the proportion.

Final Answer:
486