In the following number analogy, 18 is related to 30 using a simple linear rule. Using the same rule, 36 is related to which of the following numbers?

Introduction / Context:
This number analogy asks you to identify a simple arithmetic rule that connects two numbers and then apply it to a new input. You are told that 18 relates to 30, and must determine the corresponding value for 36 under the same rule. Such patterns often involve multiplication, addition or a combination of the two, forming a linear relation.

Given Data / Assumptions:
- First pair: 18 and 30.
- Second given number: 36.
- Options: 78, 64, 66 and 68.
- We assume that a single arithmetic transformation is used consistently for both pairs.

Concept / Approach:
Test basic transformations. Doubling 18 gives 36, while 30 is slightly less. Specifically, 2 * 18 = 36 and 36 - 6 = 30. So a neat rule is result = 2 * number - 6. We then apply this rule to 36. This kind of rule appears frequently in reasoning sets because it is simple yet requires a bit of attention to notice the minus constant component.

Step-by-Step Solution:
Step 1: Check if a direct multiplication works for 18. Doubling gives 36, not 30, so we need an adjustment. Step 2: Compute 2 * 18 - 6 = 36 - 6 = 30. This exactly matches the given related number. Step 3: Conclude that the transformation rule is f(n) = 2 * n - 6. Step 4: Apply this rule to 36. Compute 2 * 36 = 72. Step 5: Subtract 6 from 72 to get 72 - 6 = 66. Step 6: Match 66 with the options and select option C.

Verification / Alternative check:
Check that alternative simple rules do not fit as well. For example, adding a fixed number to 18 would require 12 to reach 30. Adding 12 to 36 would give 48, which is not among the options. Squaring or more complex functions are unnecessary when a clean linear expression works. Because 2n - 6 satisfies the first pair and gives a value present in the options for the second, it is the best explanation.

Why Other Options Are Wrong:
Option A 78, option B 64 and option D 68 cannot be obtained using the rule 2n - 6 applied to 36. For instance, 2 * 36 - 6 equals 66 only, not 78 or 64 or 68. Choosing any of these would either require a different constant or a different operation, breaking the requirement that the same pattern works for both pairs.

Common Pitfalls:
Sometimes students try to relate digits instead of the whole numbers, for example using 1 + 8 or 3 + 6, which can lead to confusing or contrived rules. Others fail to test their guessed rule on both the sample and the new number. Always verify that your transformation works consistently before deciding on the final answer.

Final Answer:
Using the rule result = 2 * number - 6, 18 maps to 30 and 36 maps to 66.