In the following number analogy, 3 is related to 27 in a certain way. Using the same numerical pattern, 4 is related to which of the following numbers?

Introduction / Context:
This is a basic number analogy that tests knowledge of cubes. The pair 3 and 27 is a strong hint because 27 is a well known cube. The task is to recognise the cube relationship and then apply it to the number 4. Such questions are common in quantitative aptitude sections and help learners become quicker at identifying powers of small integers.

Given Data / Assumptions:

• First pair: 3 : 27
• Second first term: 4
• We must find a number that stands in the same relation to 4 as 27 does to 3.
• We work with standard integer powers.

Concept / Approach:
The key observation is that 27 equals 3 raised to the power 3, that is 3^3. In many reasoning questions the mapping from one number to another involves squaring or cubing. If 27 is 3^3, then we suspect that the second term is the cube of the first term. We will apply this idea to 4 and check which option matches 4^3.

Step-by-Step Solution:
Step 1: Compute the cube of 3. We have 3 * 3 * 3 = 27. Step 2: Observe that the second term 27 is exactly 3^3. This suggests the rule: second number equals the cube of the first number. Step 3: Apply the same rule to the number 4. Compute 4 * 4 * 4. Step 4: 4 * 4 = 16 and 16 * 4 = 64, so 4^3 equals 64. Step 5: Identify the option that equals 64. This is option 64.

Verification / Alternative check:
Verify that no other common power pattern fits as neatly. 3 squared is 9, which would not produce 27 by any simple one step operation. However 3 cubed is 27 exactly. Applying the same idea to 4 gives 4 cubed equal to 64, which appears in the options. There is no need for any additional constants or extra arithmetic. The clean match of the cube pattern confirms that the correct related number is 64.

Why Other Options Are Wrong:
Option 63 is near 64 but does not represent a simple cube of an integer, so it fails the analogy. Option 15 would correspond to 3 multiplied by 5, which is unrelated to the original pair. Option 65 is also close numerically but would not arise from cubing 4 or from any symmetric cube based rule for 3 and 4 together. Therefore all alternatives except 64 break the simple pattern of n^3.

Common Pitfalls:
Sometimes students are tempted by numbers that are numerically close to the correct answer, such as 63 or 65 here. This usually happens when they guess based on rough estimation instead of recognising the exact operation. To avoid this, always compute small integer squares and cubes explicitly and compare them with the option list. Over time, memorising cubes up to at least 10 helps in quickly solving such analogy questions.

Final Answer:
The number that completes the analogy 3 : 27 :: 4 : ? is 64.