In this number analogy, 4 is related to 12. Using the same pattern, which number should complete the analogy 4 : 12 :: 5 : ?

Introduction / Context:
This question involves a simple arithmetic relation between two numbers and asks you to extend that relation to another number. Recognising standard patterns such as multiplication by a nearby value, or n multiplied by (n minus 1), is very important for quickly solving such numerical analogy problems in competitive exams.

Given Data / Assumptions:
First pair: 4 : 12. Second pair: 5 : ?. All numbers are small positive integers. The transformation is likely to be a simple arithmetic rule.

Concept / Approach:
We start by looking for a straightforward operation that converts 4 into 12. While 4 times 3 gives 12, we must consider whether there is a more general rule that can apply to 5 in a similar way. One common pattern is n multiplied by (n minus 1). For 4, this is 4 * 3, which equals 12. Applying the same n * (n minus 1) rule to 5 gives a unique and consistent result.

Step-by-Step Solution:
Step 1: Express the transformation from 4 to 12. 4 * 3 = 12. Step 2: Rewrite the multiplier in terms of the original number. 3 is equal to 4 minus 1. So the rule can be written as n * (n - 1). Step 3: Apply this rule to the second number 5. Compute 5 * (5 - 1) = 5 * 4. 5 * 4 = 20. Step 4: Match with the options. 20 appears in the options and fits the pattern exactly.

Verification / Alternative check:
If we instead used a simple doubling or tripling rule, the result for 5 would be 10 or 15, which do not match the strong pattern we identified based on n * (n - 1). The n times (n minus 1) rule works perfectly for the first pair and produces a unique corresponding value for the second pair. Therefore, 20 is the only number that preserves the analogy in a systematic way.

Why Other Options Are Wrong:
25 is equal to 5^2 and does not reflect the relationship used for 4 and 12. 22 is not obtainable from 5 using the n * (n - 1) rule and lacks a clear connection to the transformation of 4 into 12. 30 would require multiplying 5 by 6, which does not match 4 multiplied by 3. 15 would correspond to 5 * 3, which is not in line with 4 * 3 because here the multiplier does not follow the n minus 1 pattern.

Common Pitfalls:
A frequent error is to guess transformations based on the first number alone without rewriting them in a consistent algebraic form. For instance, noticing 4 multiplied by 3 equals 12 is fine, but unless we identify 3 as (4 minus 1), we cannot reliably extend the pattern to other numbers like 5. Translating the relationship into a general expression such as n * (n - 1) makes it much easier to test and apply the rule to the next term in the analogy.

Final Answer:
The number that correctly completes the analogy is 20.