Select the related number from the given alternatives: 27 : 125 :: 64 : ?

Introduction / Context:
This is a number analogy involving cubes. The given pair 27 : 125 connects two numbers that can both be expressed as cubes of integers. You must identify the rule connecting 27 and 125, and then apply the same logic to 64 in order to select the correct answer from the options. Questions like this train your ability to recognize number patterns and power relationships.

Given Data / Assumptions:
- 27 and 125 are the first pair, 64 and an unknown number form the second pair.
- Both 27 and 125 can be written as perfect cubes of small integers.
- The link between the base and its cube must be consistent in both pairs.
- Only one option among 162, 216, 273, and 514 will follow the same pattern.

Concept / Approach:
We first factor the numbers as cubes. 27 is 3^3 and 125 is 5^3. The pair shows that the base has increased from 3 to 5, so the rule may involve using consecutive odd numbers or adding 2 to the base. To complete the analogy, we identify the base of 64 and then see how we can apply the same base transformation to get the second number in the pair.

Step-by-Step Solution:
Step 1: Express 27 as a cube: 27 = 3^3. Step 2: Express 125 as a cube: 125 = 5^3. Step 3: Observe that the base increased from 3 to 5, which is an increment of 2. So we can describe the pattern as: if the first number is a^3, the second number is (a + 2)^3. Step 4: Write 64 as a cube: 64 = 4^3. Step 5: Increase the base by 2: 4 + 2 = 6. Compute 6^3: 6 * 6 * 6 = 216. Step 6: Therefore the number that stands in the same relationship with 64 is 216.

Verification / Alternative check:
Check that the same rule holds consistently. If we start with 3^3 = 27 and apply the rule, we use (3 + 2)^3 = 5^3 = 125, which matches the given pair. Using the same idea for 4^3 = 64, we get (4 + 2)^3 = 6^3 = 216. None of the other options correspond to 6^3, which confirms that 216 is the only correct completion of the analogy.

Why Other Options Are Wrong:
Option A "162" is not a perfect cube of an integer and does not match the cube pattern.
Option C "273" does not equal any small integer cubed and has no simple cube relationship with 64.
Option D "514" also fails the cube check and appears unrelated to 64 through simple power operations.

Common Pitfalls:
Some students recognize 27 and 125 as cubes but then incorrectly assume that 64 must go to 4^3 again or confuse squares and cubes. Others attempt linear rules on the numbers themselves, ignoring the cube structure. When two given numbers are perfect powers of small integers, always check for patterns based on exponents and base changes first.

Final Answer:
The correct related number is 216.