Introduction / Context:
This question is from the number analogy type, where you must identify the pattern linking the first pair of numbers and then apply the same pattern to find the missing number in the second pair. Here, 8 is related to 27 and 64 is related to an unknown number. Many such patterns involve squares, cubes or simple arithmetic operations, and recognizing powers is a key skill.
Given Data / Assumptions:
- First pair: 8 and 27.
- Second pair: 64 and ?
- The relationship between the first two numbers should be applied consistently to the second pair.
Concept / Approach:
Observe that 8 = 2^3 and 27 = 3^3, so both numbers are consecutive cubes. This suggests that the pattern may involve cubes of consecutive integers. Check 64 to see whether it also fits a cube pattern. Indeed, 64 = 4^3. Therefore we expect the number paired with 64 to be the next cube, namely 5^3. This reasoning follows a simple power sequence of natural numbers.
Step-by-Step Solution:
1) Identify 8 in the first pair and express it as a power: 8 = 2^3.
2) Identify 27 in the first pair and express it as a power: 27 = 3^3.
3) Notice that the bases 2 and 3 are consecutive integers, and both numbers are cubes.
4) Now look at the second pair: 64 and ?.
5) Express 64 as a cube: 64 = 4^3.
6) Following the same pattern, the number paired with 64 should be the cube of the next integer after 4, that is 5^3.
7) Compute 5^3 = 5 * 5 * 5 = 125.
8) Therefore, the missing number in the analogy is 125.
Verification / Alternative check:
Another way to verify is to think in terms of pairs of cubes of consecutive numbers. The first pair (8, 27) is (2^3, 3^3). The next pair in the same sequence would be (3^3, 4^3) = (27, 64). Shifting the view slightly, the problem has (8, 27) and (64, ?), which can be interpreted as the pattern progressing in the base numbers. To keep the progression consistent from base 2 to base 5, after 4^3 the next term must be 5^3 = 125. This supports our earlier conclusion.
Why Other Options Are Wrong:
Options A (190), B (144), D (72) and E (216) do not fit the simple cube pattern of consecutive integers. For example, 144 is 12^2 and 216 is 6^3, neither of which align with a cube of a number that makes the sequence 2^3, 3^3, 4^3, 5^3. Only 125 matches 5^3 and preserves the clear relationship between the numbers in both pairs.
Common Pitfalls:
Some learners may search for complicated arithmetic operations such as adding or multiplying constants instead of checking simple powers. Others might focus on differences like 27 - 8 = 19 and try to apply a similar difference to 64, which does not produce an integer from the options. Always first test common patterns like squares and cubes in number analogy questions, because they are frequently used and easy to verify.
Final Answer:
The number that should replace the question mark is
125.
Discussion & Comments