If 6x4 = 12, 4x12 = 24 and 12x6 = 36 under a special operation x, then what is the value of 6x9 under that same rule?

Introduction / Context:
Here the symbol x does not stand for usual multiplication. We must infer the special operation from three examples and then compute 6x9 using the same rule. Notice that the outputs 12, 24, and 36 relate closely to the ordinary product of the two numbers.

Given Data / Assumptions:

• 6x4 = 12.
• 4x12 = 24.
• 12x6 = 36.
• We must find 6x9.

Concept / Approach:
Compare each result with the regular product of the two numbers. The regular products are 6×4 = 24, 4×12 = 48, and 12×6 = 72. Each of these is exactly twice the coded result (12, 24, 36), suggesting that the operation x is defined as half the usual product.

Step-by-Step Solution:
Step 1: Test the rule a x b = (a × b) / 2. For 6x4: (6 × 4) / 2 = 24 / 2 = 12, which matches. For 4x12: (4 × 12) / 2 = 48 / 2 = 24, which matches. For 12x6: (12 × 6) / 2 = 72 / 2 = 36, which matches. Step 2: The pattern is confirmed: x means \"half the product\". Step 3: Apply this rule to 6x9. Compute the normal product: 6 × 9 = 54. Now take half: 54 / 2 = 27.

Verification / Alternative check:
No other simple function of 6 and 4 (such as sum, difference, or average) will produce all given examples correctly. The half-product interpretation works cleanly for all three, so it is the intended rule.

Why Other Options Are Wrong:
Values like 35, 24, and 31 arise from incorrect operations, such as adding extra constants or misapplying multiplication. Only 27 equals half of 54, which is 6 × 9.

Common Pitfalls:
Many learners initially assume x is normal multiplication and are confused when 6×4 ≠ 12. The key is always to compare the given result with normal arithmetic to discover the hidden operation.

Final Answer:
Under this operation, 6x9 evaluates to 27.