Select the related number that completes the analogy: 20 is related to 11 in the same proportional relationship as 102 is related to which of the following numbers?

Introduction / Context:
This question presents a numerical analogy. The pair 20 : 11 hides a particular arithmetic relationship between the two numbers. The task is to discover that pattern and then apply it to the second number 102 to select the correct answer from the options. Questions of this type often involve simple operations such as addition, subtraction, multiplication, division, or combinations of these operations. Recognising such patterns quickly is essential for quantitative aptitude tests.

Given Data / Assumptions:
- The first number pair is 20 and 11, written as 20 : 11.
- The second pair begins with 102 and needs to be completed as 102 : ?.
- Options for the missing number are 49, 52, 61, and 72.
- The same single rule must connect both pairs; the relationship cannot change between pairs.

Concept / Approach:
Start by testing simple relationships between 20 and 11. We can check sums of digits, squares, halves, and similar operations. One neat pattern is that 20 divided by 2 gives 10, and if we add 1, we reach 11. So 11 equals 20 divided by 2 plus 1. The pattern can be written as Result = (Number / 2) + 1. Once this rule is identified, it must be applied to 102 in the same way. If that result matches one of the given options, we can confidently choose it as the correct answer.

Step-by-Step Solution:
Step 1: Examine the pair 20 : 11. Compute 20 / 2 = 10. Step 2: Add 1 to this result to obtain 10 + 1 = 11. This matches the second number in the pair. Step 3: Assume the rule is Result = (Number / 2) + 1. Step 4: Apply this rule to 102. Compute 102 / 2 = 51. Step 5: Add 1 to get 51 + 1 = 52. Step 6: Check the options. The number 52 appears as option B, so 102 : 52 fits the same pattern as 20 : 11.

Verification / Alternative check:
To verify, we can test whether any other simple pattern connects 20 to 11 while also giving a different answer for 102 from the options. For example, adding or subtracting a fixed number would not work because 20 minus 9 equals 11, but 102 minus 9 equals 93, which is not in the list. Multiplying and then adding or subtracting small values also fails to match both pairs consistently. The division by 2 followed by adding 1 works cleanly and gives an answer that is present in the options, confirming the pattern.

Why Other Options Are Wrong:
Option A, 49, would require a different pattern such as subtracting 53 from 102, which does not correspond to any simple rule connecting 20 to 11. Option C, 61, and option D, 72, also cannot be obtained from 102 using the same operation that links 20 to 11. Because analogies require a single consistent rule, any choice that breaks that rule must be rejected even if it is numerically close.

Common Pitfalls:
Many learners attempt random or overly complex operations instead of checking the simplest possibilities first. Some may try to involve digit sums or squares without verifying that the pattern holds for both pairs. Another common error is accepting a pattern that works only for the first pair and then forcing a different operation on the second pair. In numerical analogy questions, the same transformation must apply to every pair, so always test the pattern systematically on all given numbers.

Final Answer:
Following the pattern Result = (Number / 2) + 1, 102 becomes 52, so the completed analogy is 20 : 11 :: 102 : 52, and option B is correct.