Select the related number from the given alternatives. In a certain numeric pattern 534 is related to 645. Using the same rule, 381 will be related to which of the following numbers?

Introduction / Context:
This is a number analogy question where one three digit number is connected to another by a simple arithmetic pattern. Once you identify the rule that converts 534 to 645, you must apply that same rule to 381 to obtain the missing related number.

Given Data / Assumptions:

• Given pair: 534 : 645
• Target pair: 381 : ?
• Options are all three digit numbers, so the rule should keep the magnitude similar.
• We assume a straightforward arithmetic relation such as addition or subtraction rather than complex operations.

Concept / Approach:
For numeric analogies, it is natural to first check the difference between the numbers in the first pair. If the difference is simple and clean, that difference is often the rule to use for the second pair. Occasionally, operations on individual digits are involved, but here a direct difference works very well and keeps the question at an easy level.

Step-by-Step Solution:

Verification / Alternative check:
We can quickly test whether any of the other options give a constant difference when compared with 381. For example, 446 − 381 = 65, 486 − 381 = 105, and 412 − 381 = 31. None of these match 111. Only 492 − 381 = 111 which is exactly the difference found in the first pair, hence confirming the rule.

Why Other Options Are Wrong:

• 446 does not preserve the difference of 111 when subtracted from 381.
• 486 produces a different difference and suggests a different rule, so it cannot match the given first pair.
• 412 is too close to 381 and does not fit a clean pattern linked with 534 and 645.
• 502 also breaks the simple plus 111 pattern and would make the analogy inconsistent.

Common Pitfalls:
Some candidates overcomplicate such questions by trying to manipulate individual digits or by searching for squares or cubes. Others may spot an incorrect difference due to arithmetic mistakes and then carry that error into the second pair. Careful subtraction and choosing the simplest possible pattern usually leads to the correct answer efficiently.

Final Answer:
Using the same rule, 381 is related to 492.