In this number analogy, “36 is to 6 as 441 is to ______”. Choose the number that completes the analogy by following the same square root relationship used in the first pair.

Introduction / Context:
This numerical analogy uses the concept of perfect squares and square roots. In the pair 36 : 6, the second number is the square root of the first number. The question asks you to apply the same relationship to 441 and find the correct number from the given options. This type of problem tests basic knowledge of squares and roots, which are very common in quantitative aptitude and number system questions.

Given Data / Assumptions:

• First pair: 36 and 6. • Second pair: 441 and an unknown number. • Options: 11, 21, 25, 32. • We assume standard definitions of perfect squares: if n² = m, then n is the square root of m.

Concept / Approach:
Recognise that 36 is a perfect square and 6 is its square root, since 6 * 6 = 36. Therefore, the relationship in the first pair is "first number is the square of the second", or equivalently "second number is the square root of the first". To complete the analogy, we must find the square root of 441 and match it with one of the options. Knowing standard squares up to at least 25² is very helpful in such questions.

Step-by-Step Solution:
Step 1: Confirm the relationship in the first pair. 36 = 6 * 6, so 6² = 36. Hence, second number = square root of first number. Step 2: Factorise 441 or recall known squares. 21 * 21 = 441, so 21² = 441. Step 3: Apply the same relationship. If 36 : 6 follows the pattern first number : square root, then 441 must correspond to its square root, which is 21. Step 4: Check the options. Among 11, 21, 25, 32, only 21 squared equals 441.

Verification / Alternative check:
We can verify by checking squares of the options. 11² = 121, 21² = 441, 25² = 625, and 32² = 1024. Only 21 yields 441, which confirms that the relationship is indeed based on square roots. No other simple rule such as division or addition fits both pairs as cleanly as this square root pattern.

Why Other Options Are Wrong:
• 11: Its square 121 does not match 441, so it does not satisfy the required relationship. • 25: Squares to 625, which is not equal to 441. • 32: Squares to 1024, again not equal to 441 and far from the required value.

Common Pitfalls:
A common error is to think in terms of division, for example 36 / 6 = 6 and then trying to find a similar ratio for 441. However, the positions in an analogy are directional: here, we move from the larger square to its root, not from the root to the square. Remember to check for perfect square relationships whenever you see familiar numbers like 36, 49, 64, 81, 100, 144, and 441.

Final Answer:
The number that correctly completes the analogy is 21.