In the following number analogy, select the related number so that the pair 7 : 48 is to 11 : ? in the same way according to a consistent rule.

Introduction / Context:
This question belongs to the category of number analogies in aptitude. We are given a pair of numbers 7 and 48, and we must find a number that relates to 11 in the same way that 48 relates to 7. The key is to identify a simple and consistent numerical pattern, often involving squares, cubes, or basic arithmetic operations. Such questions test comfort with numerical patterns and recognition of common transformations like n squared minus one or n cubed plus a constant.

Given Data / Assumptions:

• First pair: 7 : 48
• Second pair: 11 : ?
• We must choose one number from the options that fits the same rule.
• We assume ordinary integer arithmetic and typical exam level patterns such as squares or cubes.

Concept / Approach:
A good starting point is to check whether 48 can be expressed in terms of 7 using squares or cubes. The square of 7 is 49, which is very close to 48. This suggests the pattern 7^2 - 1 = 48. Once this rule is identified we can apply the same pattern to 11 and see which option matches. Square based analogies are very common in reasoning tests because they are easy to compute yet still require observation.

Step-by-Step Solution:
Step 1: Compute the square of 7. We have 7^2 = 49. Step 2: Compare 49 with the second number 48. We see that 48 is equal to 49 - 1. Step 3: Conclude that the rule used in the first pair is "second number equals square of the first number minus one". Symbolically, if the first number is n, then the second is n^2 - 1. Step 4: Apply the same rule to 11. Compute 11^2 = 121. Step 5: Subtract one. So the required number is 121 - 1 = 120.

Verification / Alternative check:
We can verify the rule for consistency. For n = 7, n^2 - 1 = 7^2 - 1 = 49 - 1 = 48, which matches the first pair exactly. For n = 11, n^2 - 1 = 11^2 - 1 = 121 - 1 = 120. None of the other common simple rules, such as 2n^2, n^3, or n(n - 1), produce 48 for n equal to 7, so n^2 - 1 is the cleanest pattern. Therefore, following the same pattern, the only correct second number for 11 is 120.

Why Other Options Are Wrong:
Option 131 would correspond to 11^2 + 10, which has no link to the original pair. Option 130 does not match any simple expression derived from 7 that gives 48, and it would require a different rule. Option 171 would be closer to 11^3 plus a constant, but then 7 would not produce 48 with the same constant. Because a single consistent rule must work for both pairs, these alternatives are invalid. Only 120 fits the pattern n^2 - 1 for both 7 and 11.

Common Pitfalls:
A common mistake is to try random operations like multiplying by a constant or adding arbitrary numbers without checking whether the same formula works for both pairs. Another pitfall is to overcomplicate the pattern with multiple steps when a simple square based rule is sufficient. When solving number analogies, always test candidate patterns on the given pair before applying them to the unknown pair. This disciplined approach avoids choosing an option that fits one side only by coincidence.

Final Answer:
The number that completes the analogy 7 : 48 :: 11 : ? is 120.