In the following number analogy, 48 is related to 63 in a certain way. Using the same pattern, 80 is related to which of the following numbers?

Introduction / Context:
This number analogy involves recognising that both numbers in each pair are one less than perfect squares of consecutive integers. The pair 48 and 63 is very close to 49 and 64, which are 7 squared and 8 squared respectively. Using this idea, we can determine the related number for 80 by expressing it as one less than a square and then mapping to the next integer in the sequence.

Given Data / Assumptions:

• First pair: 48 : 63
• Second first term: 80
• Options: 97, 98, 99, 101.
• We assume standard integer squares.

Concept / Approach:
Start by checking if 48 and 63 can be written in the form n^2 minus 1. Compute the squares of nearby integers and compare. If a pattern emerges such as n^2 minus 1 for the first number and (n plus 1)^2 minus 1 for the second number, we can reapply it to 80. These kinds of patterns featuring squares offset by one are quite common in reasoning sets.

Step-by-Step Solution:
Step 1: Check 48. We know that 7^2 is 49. So 48 = 49 - 1 = 7^2 - 1. Step 2: Check 63. We know that 8^2 is 64. So 63 = 64 - 1 = 8^2 - 1. Step 3: Conclude the pattern for the first pair: 48 = 7^2 - 1 and 63 = 8^2 - 1, so the pair uses two consecutive integers 7 and 8. Step 4: Express 80 similarly. We know 9^2 = 81, hence 80 = 81 - 1 = 9^2 - 1. Step 5: The next integer after 9 is 10. So we follow the pattern and take 10^2 - 1. Step 6: Compute 10^2 - 1 = 100 - 1 = 99.

Verification / Alternative check:
Summarise the rule: if the first number is n^2 - 1, then the second is (n + 1)^2 - 1. For 48, n is 7, and the partner is 63, which is 8^2 - 1. For 80, n is 9, and the partner should be 10^2 - 1 equal to 99. Among the options provided, only 99 matches, confirming it as the correct answer.

Why Other Options Are Wrong:
Number 97 is not one less than a perfect square of an integer near 10, as 10^2 is 100 and 9^2 is 81. Number 98 equals 49 times 2, not of the same form as 10^2 - 1. Number 101 is one more than 100, not one less. Therefore these alternatives do not continue the pattern of consecutive squares minus one and must be rejected.

Common Pitfalls:
A common error is to try simple linear differences, such as 63 minus 48 equal to 15, and then to apply that difference to 80 to get 95. However 95 is not listed and also does not reveal a deeper pattern. When numbers are close to known squares like 49, 64, 81, and 100, it is usually productive to check square based relationships before looking at simple differences.

Final Answer:
The number that correctly completes the analogy 48 : 63 :: 80 : ? is 99.