In this numerical analogy question, select the related number from the given alternatives to complete the proportion: 0.04 : 0.2 :: 0.09 : ?

Introduction / Context:
This numerical analogy question uses decimal numbers and a square root relationship. The pair 0.04 and 0.2 is given, and you must determine which number completes the analogy 0.04 : 0.2 :: 0.09 : ?. Recognising square roots and squares of small decimal values is important in many quantitative reasoning questions, especially when dealing with proportions or analogy patterns.

Given Data / Assumptions:
The first pair is 0.04 and 0.2.
The second pair starts with 0.09, and we must find the related number.
The answer options are 0.3, 0.6, 0.9, 2, and 0.04.
We assume the same numerical relationship that links 0.04 to 0.2 must link 0.09 to the correct answer.

Concept / Approach:
Note that 0.2 * 0.2 = 0.04, which means 0.2 is the square root of 0.04 and 0.04 is the square of 0.2. This suggests a clear square root or squaring relationship between the terms of the pair. To maintain the analogy, we must apply the same type of relationship between 0.09 and the unknown number. Since 0.3 * 0.3 = 0.09, it follows that the square root of 0.09 is 0.3. Therefore, the pattern likely maps each decimal to its square root or vice versa in a consistent direction.

Step-by-Step Solution:
Step 1: Express the first pair using squares. 0.04 can be written as 0.2^2.Step 2: This means that 0.2 is the square root of 0.04, so the mapping is 0.04 mapped to sqrt(0.04) which equals 0.2, or equivalently 0.2 mapped to its square 0.04 depending on direction.Step 3: For consistency with the order 0.04 : 0.2, we see that the first term is the square of the second term.Step 4: Apply the same idea to the second pair. We must find a number whose square equals 0.09.Step 5: We know that 0.3 * 0.3 = 0.09, so 0.09 is 0.3^2, and thus 0.3 is the square root of 0.09.Step 6: Therefore, to preserve the pattern, 0.09 must be paired with 0.3, just as 0.04 is paired with 0.2.

Verification / Alternative check:
We can verify the pattern by comparing both pairs in terms of square roots. In the first pair, sqrt(0.04) = 0.2. In the second pair, sqrt(0.09) must be equal to the unknown number. Since sqrt(0.09) = 0.3, the second pair becomes 0.09 : 0.3, which mirrors the structure 0.04 : 0.2. None of the other options squared equals 0.09, confirming that 0.3 is uniquely correct. For instance, 0.6^2 = 0.36, 0.9^2 = 0.81, and 2^2 = 4, all of which are inconsistent with 0.09.

Why Other Options Are Wrong:
Option 0.6 has a square of 0.36, not 0.09, so it breaks the square relationship used in the first pair.
Option 0.9 squared gives 0.81, which is far from 0.09 and clearly does not mirror the original pattern.
Option 2 squared is 4, which is completely unrelated to the small decimal value 0.09 in the context of this analogy.
Option 0.04, if paired with 0.09, would create a pattern based on different squared values and does not serve as the square root of 0.09.

Common Pitfalls:
Some candidates may attempt to treat the relationship as simple multiplication or division by a factor of 5 or 2 because the numbers are decimals, but quick checks show these factors do not hold consistently for both pairs. Another mistake is to confuse 0.3 with 0.03, leading to incorrect squares. To avoid these issues, it is helpful to remember squares of small decimals like 0.1, 0.2, and 0.3, or to rewrite the decimals as fractions and square them to verify the pattern accurately.

Final Answer:
Following the same square and square root relationship as 0.04 : 0.2, the correct completion of the analogy 0.04 : 0.2 :: 0.09 : ? is 0.3.