In this number analogy, 27 is related to 125 and 64 is related to which number so that the analogy 27 : 125 :: 64 : ? is correctly completed?

Introduction / Context:
This question involves a numerical analogy based on powers of integers, particularly perfect cubes. Many competitive exams include patterns involving squares and cubes because they are standard mathematical concepts. Recognising that 27 and 125 are both perfect cubes is the key to solving this question efficiently.

Given Data / Assumptions:
First pair: 27 : 125. Second pair: 64 : ?. All numbers are positive integers. The pattern is likely to involve powers such as squares or cubes.

Concept / Approach:
We start by factorising 27 and 125. Quickly noticing that 27 is 3^3 and 125 is 5^3 reveals that the pair links cubes of different integers. This suggests that 64, which is 4^3, should be connected to another cube, and the relationship may involve increasing the base by 2 in each case. Identifying this pattern allows us to determine the correct number to pair with 64.

Step-by-Step Solution:
Step 1: Express 27 and 125 as cubes. 27 = 3^3. 125 = 5^3. Step 2: Observe the bases. The base increases from 3 to 5, which is an increase of 2. Step 3: Express 64 as a cube. 64 = 4^3. Step 4: Apply the same base increase of 2. 4 + 2 = 6. Step 5: Compute the corresponding cube. 6^3 = 6 * 6 * 6 = 216. Therefore, 64 should be paired with 216.

Verification / Alternative check:
We can phrase the pattern as follows: first term is n^3, second term is (n + 2)^3. For the first pair n = 3, so (3 + 2)^3 = 5^3 = 125, which matches. For the second pair n = 4, so (4 + 2)^3 = 6^3 = 216, matching our derived value. This confirms that 216 maintains exactly the same relation with 64 that 125 has with 27.

Why Other Options Are Wrong:
517 is not a perfect cube and does not correspond to any simple base related to 4. 162 is not equal to 6^3 and does not match a neat power rule for this analogy. 273 is not a perfect cube of a small integer and does not follow the n^3 to (n + 2)^3 pattern. 343 is 7^3, which would correspond to an increase of 3 in the base from 4, not an increase of 2 as seen in the first pair.

Common Pitfalls:
Some learners may look for additive or multiplicative relationships directly between 27 and 125, such as differences or ratios, and overlook the fact that both are perfect cubes. A solid habit in such questions is to immediately check for squares and cubes when small integers are involved. Recognising standard powers saves time and leads to a clear and consistent pattern that can then be applied to the second pair without confusion.

Final Answer:
The number that correctly completes the analogy is 216.