In this number analogy, “36 is to 25 as 100 is to ______”. Select the number that completes the analogy by following the same pattern of moving from one perfect square to the square of the next lower integer.

Introduction / Context:
This analogy connects pairs of numbers through perfect square relationships. The given pair 36 : 25 is not simply a case of subtraction or division; instead, both 36 and 25 are squares of consecutive integers. The question is asking you to apply the same pattern to 100 in order to find its related number among the options. Knowledge of squares of integers and spotting patterns among them is critical for such problems.

Given Data / Assumptions:

• First pair: 36 and 25. • Second pair: 100 and an unknown number. • Options: 81, 30, 35, 40. • Standard squares: 5² = 25, 6² = 36, 9² = 81, 10² = 100.

Concept / Approach:
Observe that 36 is 6² and 25 is 5². Thus, the first number is the square of some integer n, and the second number is the square of n − 1. In other words, the pair moves from the square of an integer to the square of the previous integer. To maintain the same relationship, when the first number is 100, which is 10², the related second number should be the square of 9, which is 9² = 81. We then match this logic to the options.

Step-by-Step Solution:
Step 1: Express 36 and 25 as squares. 36 = 6 * 6 = 6². 25 = 5 * 5 = 5². Step 2: Identify the pattern. The second number is the square of one less than the square root of the first number: from 6² to 5². Step 3: Express 100 as a square. 100 = 10². Step 4: Apply the pattern. One less than 10 is 9, so the related number should be 9². 9² = 81. Step 5: Compare with the options. Only 81 matches the required pattern.

Verification / Alternative check:
Verify that alternative simple relationships such as subtracting a fixed value from 36 do not generalise. While 36 − 11 = 25, applying minus 11 to 100 gives 89, which is not among the choices. The square to previous square pattern is more natural and uses standard mathematical structures. Additionally, checking 81 as 9² reinforces that the relationship is about consecutive squares.

Why Other Options Are Wrong:
• 30 and 35: These numbers are not perfect squares and therefore do not fit a pattern of consecutive squares. • 40: Also not a perfect square and has no simple square based link to 100 analogous to 36 and 25.

Common Pitfalls:
Test takers may perform quick subtraction and assume that 36 and 25 are connected by subtracting 11 and then search for a similar difference with 100. This fails to capture the deeper square structure. To avoid such mistakes, always check whether the numbers in analogies are squares or cubes, especially when they look familiar like 25, 36, 81, and 100.

Final Answer:
The number that correctly completes the analogy is 81.