In the analogy 3 : 12 :: 5 : ?, the same numerical pattern that converts 3 to 12 is applied to 5. What is the missing number that completes this proportion correctly?

Introduction / Context:
Analogy questions of the form a : b :: c : ? appear frequently in competitive exams. They check whether the candidate can identify a hidden pattern between the first pair and then apply the same pattern to the second pair. In this problem, we are given 3 : 12 as the first pair and must determine the number that corresponds to 5 in the same way that 12 corresponds to 3.

Given Data / Assumptions:
• First pair: 3 and 12. • Second pair: 5 and an unknown number. • Relationship between the first pair must be consistently applied to the second pair. • Options given: 17, 30, 26, 32.

Concept / Approach:
We are looking for a simple arithmetic or algebraic relation that transforms 3 into 12. Common patterns include multiplication, addition, squares, or a combined operation. Once a consistent pattern is identified (for example, n * n + n), we apply it to the second number 5 and see which option matches. The key is to ensure that the rule is simple and uniquely fits the given pair.

Step-by-Step Solution:
Step 1: Try a pattern n * n + n for the first pair. Step 2: For n = 3, compute 3 * 3 + 3 = 9 + 3 = 12, which matches the given second term. Step 3: Apply the same pattern to n = 5. Step 4: Compute 5 * 5 + 5 = 25 + 5 = 30. Step 5: Check if 30 is one of the options; it is present as an option.

Verification / Alternative check:
Alternative simple patterns like 3 * 4 = 12 or 3 + 9 = 12 do not extend naturally to give any single valid option for 5. For instance, if we used multiplication by 4, 5 * 4 would be 20, which is not listed. The rule n * n + n fits perfectly and is simple, making 30 the consistent choice.

Why Other Options Are Wrong:
• 17, 26 and 32 do not arise from any straightforward extension of a rule that converts 3 to 12. • None of them satisfy the pattern n * n + n when n = 5, so they break the analogy.

Common Pitfalls:
A typical mistake is to assume 3 : 12 is a direct multiplication by 4 and then guess 5 * 4 = 20, which is not in the options. Another pitfall is to overcomplicate the pattern with unnecessary operations rather than checking simple square-plus-number patterns first. Always test whether your rule uniquely fits the given pair and then apply it consistently.

Final Answer:
The number that correctly completes the analogy is 30.