Select the related number from the given alternatives to complete the analogy: 1 : 8 :: ? : 64

Introduction / Context:
This is a number analogy question where we need to identify the pattern linking the first pair of numbers and then apply the same pattern to find the missing number in the second pair. Such problems test quick observation of numerical relationships, such as squares, cubes or other simple operations.

Given Data / Assumptions:

• The given analogy is 1 : 8 :: ? : 64.
• We must choose the correct number from the options that maintains the same relationship between the pair as in the first pair.
• All numbers are positive integers.

Concept / Approach:
We first analyse the relationship between 1 and 8. Notice that 1 = 1^3 and 8 = 2^3. The sequence 1, 8, 27, 64, 125, and so on are cubes of natural numbers 1, 2, 3, 4, 5, etc. It is reasonable to suspect that the pattern involves consecutive cubes. If 1 corresponds to 8 (1^3 and 2^3), then the missing number should correspond to 64, which is 4^3. The natural candidate is 27, which is 3^3, forming a smooth progression 1^3, 2^3, 3^3, 4^3.

Step-by-Step Solution:
Observe that 1 = 1^3 and 8 = 2^3. So the first pair can be seen as (1^3, 2^3). We are given 1 : 8 :: ? : 64. Here 64 is 4^3. To maintain a pattern of consecutive cubes, the missing number should be 3^3, which is 27. Then we have a chain: 1^3, 2^3, 3^3, 4^3 → 1, 8, 27, 64. Thus the correct completion is 1 : 8 :: 27 : 64.

Verification / Alternative check:
We can check if any simple multiplication or ratio pattern also works. The ratio 8 / 1 = 8 and 64 / 27 is not 8, so the pair is not directly in the same ratio. Instead, the pattern of cubes is consistent and natural for both pairs. No other simple operation, such as squaring or adding a fixed value, arranges the numbers as neatly as consecutive cubes. Therefore the cube based pattern is the most reasonable and mathematically sound explanation.

Why Other Options Are Wrong:
25 is 5^2, 36 is 6^2, 30 is not a perfect square or cube and 49 is 7^2; none of these fit a consecutive cube pattern with 64 as 4^3. Only 27 is a perfect cube (3^3) and fits between 8 (2^3) and 64 (4^3) in the natural cube sequence.

Common Pitfalls:
A common mistake is to look for constant differences (like 8 − 1 = 7) and then try to apply the same difference to 64, which would suggest 57, a number not present in the options. Another error is to assume a direct ratio, such as 1 : 8 = x : 64, which would give x = 8. Missing the fact that 8 and 64 are cubes prevents recognition of the intended pattern. Always check for powers (squares, cubes) in analogy questions involving small integers.

Final Answer:
The missing number in the analogy 1 : 8 :: ? : 64 is 27.