In this numerical analogy question, 8 is related to 27, so you must select the number that shows the same relationship with 64 from the given alternatives.

Introduction / Context:

This question belongs to the category of numerical analogies where you must detect a mathematical relationship between two numbers and then apply the same rule to another given number. The pair "8 : 27" suggests a relationship involving powers or cubes because these numbers are familiar as 2 cube and 3 cube. You must check this idea and then apply it to 64 to find the missing number.

Given Data / Assumptions:

- First pair of numbers: 8 and 27.
- Second pair to be completed: 64 and ?.
- Options: 75, 65, 100, 125 and 216.
- We assume standard arithmetic operations: addition, subtraction, multiplication, division and powers such as squares and cubes.

Concept / Approach:

The key concept is recognising perfect cubes. In many reasoning questions, the numbers 8 and 27 immediately hint at cubes, since 8 = 2^3 and 27 = 3^3. Once we observe this, it is natural to check whether 64 is also a cube and then continue the pattern using the next integer. This approach avoids unnecessary complex operations and uses standard number properties that are frequently examined in aptitude tests.

Step-by-Step Solution:

Verification / Alternative check:

To double-check, there is no simple consistent rule such as multiplication by a constant or addition of a fixed number that can convert 8 into 27. However, the cube idea works elegantly and extends naturally to 64. Also, among the options, only 125 is a perfect cube, which further supports the cube pattern. The sequence 8, 27, 64, 125 can be recognised as consecutive cubes of 2, 3, 4 and 5.

Why Other Options Are Wrong:

75 is not a perfect cube and does not fit any natural extension of the pattern created by 8 and 27.

65 is close to 64 but again has no cube property or simple relation that mirrors the original pair.

100 is a square (10^2), not a cube, and does not follow the consecutive cube pattern.

216 is 6^3 and would correspond to continuing the cube sequence even further, but the immediate next cube after 64 is 125 (5^3), not 216.

Common Pitfalls:

Many students do not immediately recognise 8 and 27 as cubes and instead try to apply ad hoc operations such as adding 19 or multiplying by fractions. Another common mistake is to think that 64 should be doubled or increased by some fixed difference. In analogy questions, always check for familiar patterns like squares, cubes and factorials before resorting to complex calculations.

Final Answer:

The correct completion of the analogy is 64 : 125, following the pattern of consecutive cubes.