In this number analogy question, select the related number from the given alternatives: 12 : 54 :: 15 : ?

Introduction / Context:
Number analogy questions require you to infer a rule that transforms one number into another and then apply the same rule to a different starting number. In this item, the pair 12 and 54 is given, and you must determine which number relates to 15 in the same way. Such problems strengthen your pattern recognition skills and your ability to work with simple formulas under time pressure.

Given Data / Assumptions:
- The first pair is 12 and 54, with 54 generated from 12 using some arithmetic operation.
- The second starting number is 15, and the correct related number must be selected from 56, 64, 67, and 69.
- We assume use of basic operations like multiplication and addition or subtraction rather than advanced or obscure operations.
- The rule should be reasonably simple and consistent when applied to both numbers.

Concept / Approach:
One standard method is to examine products of the number with small integers and see if adding or subtracting a small constant yields the given result. For 12 and 54, we look at multiples of 12: 12 * 3 = 36, 12 * 4 = 48, and 12 * 5 = 60. Only 12 * 5 - 6 equals 54. Once you find a neat rule like result = (number * 5) - 6, you apply it to 15 and then check against the answer choices.

Step-by-Step Solution:
Step 1: Compute 12 * 5 = 60 and then subtract 6 to get 54. This suggests the rule result = (number * 5) - 6.Step 2: Confirm there is no simpler rule such as number * 4 + 6, which would give 12 * 4 + 6 = 54 as well, but then check its effect on 15 carefully.Step 3: If we try result = (number * 4) + 6 on 15, we get 15 * 4 + 6 = 66, which is not among the options.Step 4: Now apply the chosen rule result = (number * 5) - 6 to 15: 15 * 5 = 75 and 75 - 6 = 69.Step 5: Among the options, only 69 matches this computation, so 69 is the correct answer.Step 6: The rule is consistent and uses small, natural integers, making it suitable for an exam analogy.

Verification / Alternative check:
Check whether any other simple pattern could map 12 to 54 and 15 to a different option. For instance, 12 * 3 + 18 also gives 54, but then 15 * 3 + 18 equals 63, which is not an option. Similarly, 12 * 4 + 6 works for 12 but fails for 15. The rule using multiplication by 5 and subtraction of 6 is both simple and consistent for the pair and produces one of the given options for 15, which makes it the most reasonable interpretation.

Why Other Options Are Wrong:
- 56: This would arise from 15 * 4 - 4, which does not mirror the transformation used on 12, so it lacks consistency.
- 64: No straightforward common rule applied to both 12 and 15 naturally results in 54 and 64 respectively.
- 67: Similarly, producing 67 from 15 requires an arbitrary combination of operations that does not fit the first pair cleanly.

Common Pitfalls:
Students often pick an option that seems arithmetically close to a rough estimate without verifying that the same pattern works with the given pair. Others may try too many complicated formulas and lose track of consistency. The best practice is to search for a rule that comfortably explains the first pair and yields exactly one of the offered choices for the second number. Always cross check your rule for both numbers before finalizing your answer.

Final Answer:
Using the pattern result = (number * 5) - 6, 12 maps to 54 and 15 maps to 69. Hence 69 is the correct answer.