In the following number analogy, 9 is related to 25 in a particular way. Using the same pattern, 49 is related to which of the following numbers?

Introduction / Context:
This analogy uses perfect squares of odd numbers. The pair 9 and 25 are themselves squares, and they correspond to consecutive odd integers. We must see how 49 fits into the same pattern and select the correct related number from the choices. Recognising square numbers and relationships between them is a fundamental skill for numerical reasoning.

Given Data / Assumptions:

• First pair: 9 : 25
• Second first term: 49
• We must find the number related to 49 in the same way that 25 is related to 9.
• All numbers are non negative integers.

Concept / Approach:
First observe that 9 is 3^2 and 25 is 5^2. Here 3 and 5 are consecutive odd numbers. Another way to see the relation is that between 3 and 5 the difference is 2, and we are mapping the square of a smaller odd number to the square of the next odd number. Therefore the rule may be described as: square of one odd number is related to square of the next odd number in the sequence. We will find which odd number squared gives 49 and then map to the next odd number.

Step-by-Step Solution:
Step 1: Express 9 as a square. 9 = 3^2. Step 2: Express 25 as a square. 25 = 5^2. Step 3: Note that 3 and 5 are consecutive odd numbers with difference 2. Step 4: Express 49 as a square. 49 = 7^2, where 7 is also an odd number. Step 5: The next odd number after 7 is 9. Its square is 9^2 = 81. Step 6: Conclude that 49 should be related to 81 to match the pattern.

Verification / Alternative check:
We can summarise the pattern as "square of an odd number n is related to square of the odd number n plus 2". In the first pair, n is 3, so 3^2 : 5^2 becomes 9 : 25. In the second pair, n is 7, so 7^2 : 9^2 becomes 49 : 81. No other simple interpretation of the relation between 9 and 25 fits as smoothly with 49 and one of the options. Therefore 81 is the only consistent and correct answer.

Why Other Options Are Wrong:
Numbers 36 and 64 are squares of even integers (6^2 and 8^2 respectively), and they do not preserve the odd square to next odd square pattern. Number 54 is not even a perfect square, so it does not match the structure of the first pair. Since the relation clearly involves consecutive odd squares, selecting any of these alternatives would break the analogy.

Common Pitfalls:
A frequent mistake is to focus only on numerical closeness, picking a number that "looks near" 49 without checking the exact role of squares. Some learners may also confuse arithmetic progressions with squares and try to apply linear differences. To handle such questions efficiently, it is useful to memorise squares of small integers and pay attention to whether given numbers are perfect squares or cubes.

Final Answer:
The number that completes the analogy 9 : 25 :: 49 : ? is 81.