In this numerical analogy question, select the related number from the given alternatives to complete the proportion: 64 : 8 :: 0.01 : ?

Introduction / Context:
This numerical analogy question asks you to observe how 64 is related to 8 and then apply the same mathematical relationship to 0.01. Such questions appear frequently in aptitude tests to check your familiarity with basic operations like taking square roots or powers, and your ability to apply them consistently to new numbers without getting confused by decimals.

Given Data / Assumptions:
The first pair of numbers is 64 and 8.
The second pair starts with 0.01, and we must find the related number.
The options provided are 0.1, 1, 10, 0.08, and 0.5.
We assume that the relationship between 64 and 8 is the same as the relationship between 0.01 and the correct answer.

Concept / Approach:
First, look for a simple and strong relationship between 64 and 8. A natural observation is that 8 is the square root of 64, since 8 * 8 = 64. This suggests that the mapping might be taking the square root of the first number to get the second number. If this is the rule, we just need to find the square root of 0.01 and then match it with one of the options. Square roots of decimals can be handled by remembering that 0.1 * 0.1 = 0.01.

Step-by-Step Solution:
Step 1: Express the relationship in the first pair. The square root of 64 is 8, because 8 * 8 = 64.Step 2: So we can write the mapping as 64 is mapped to sqrt(64) which equals 8.Step 3: Apply the same operation to the second pair. We must compute sqrt(0.01).Step 4: Recall that 0.1 * 0.1 = 0.01, so sqrt(0.01) = 0.1.Step 5: Check the options and see that 0.1 is explicitly listed, so it is the only number that matches the discovered pattern.

Verification / Alternative check:
We can check this relationship in another way by writing the numbers in power form. 64 can be written as 8^2, so taking the square root reverses the squaring and returns us to 8. Similarly, 0.01 can be expressed as 0.1^2, so taking the square root of 0.01 should give 0.1. If we tested any other option, squaring it would not produce 0.01, confirming that 0.1 is the only value that fits the square root mapping.

Why Other Options Are Wrong:
Option 1 squared gives 1, not 0.01, so it cannot be the square root of 0.01.
Option 10 squared gives 100, which is far from 0.01 and clearly incorrect.
Option 0.08 squared gives 0.0064, which again does not equal 0.01, so it does not satisfy the same relationship.
Option 0.5 squared gives 0.25, which is also not equal to 0.01, so it cannot mirror the mapping from 64 to 8.

Common Pitfalls:
One common mistake is to treat the relationship as simple division, for example thinking 64 divided by 8 equals 8, and then trying to divide 0.01 by some factor. However, this does not give a neat or consistent result with the given options. Another pitfall is to misremember square roots of decimals and choose 0.01 as the square of 0.01 itself, which is incorrect. Writing the numbers in exponent form or squaring the candidate answers is a reliable way to avoid such errors.

Final Answer:
Thus, using the same square root relationship as in 64 : 8, the correct completion of the analogy 64 : 8 :: 0.01 : ? is 0.1.