Difficulty: Easy
Correct Answer: 8
Explanation:
Introduction / Context:
This problem is a basic numerical analogy that checks whether you can identify an exponent based relationship between two numbers. You are given the pair 10 and 1000 and asked to determine which number should be related to 2 in the same way. Such questions are common in aptitude tests to quickly test your understanding of powers, especially cubes and squares, without involving lengthy calculations.
Given Data / Assumptions:
The first pair of numbers is 10 and 1000.
The second pair starts with the number 2, and we must find the related number.
The options provided are 4, 0.5, 20, 8, and 16.
We assume the same numerical rule that connects 10 to 1000 must also connect 2 to the correct answer.
Concept / Approach:
The most natural way to connect 10 to 1000 is to observe that 1000 is 10 raised to the power 3, that is, 10^3. This suggests a cube relation. In analogies, once we recognise that one term is the cube of the other, we simply apply the same exponent pattern to the second base number. Therefore, if 10 is mapped to 10^3, we expect 2 to be mapped to 2^3, which is 8.
Step-by-Step Solution:
Step 1: Express 1000 in terms of 10. We know 1000 = 10 * 10 * 10 = 10^3.Step 2: Interpret the analogy: 10 is mapped to its cube, 10^3.Step 3: Apply the same pattern to 2. The cube of 2 is 2 * 2 * 2 = 8, which is 2^3.Step 4: Check the options and confirm that 8 is listed among them.Step 5: Conclude that 8 is the only number that preserves the cube relationship between the elements of the pairs.
Verification / Alternative check:
We can also test whether any of the other options could form a similar exponent relationship. For example, 4 is 2^2, 0.5 is 2^-1, 20 is 10 times 2, and 16 is 2^4. None of these mirrors the pattern of mapping a number to its cube. Therefore, 8 is uniquely suitable, because it is exactly 2^3, just as 1000 is 10^3. This confirms that the underlying rule is consistent and correctly applied.
Why Other Options Are Wrong:
Option 4 equals 2^2, a square rather than a cube, so it does not parallel the mapping of 10 to 10^3.
Option 0.5 equals 2^-1, which involves reciprocals and is completely different from the positive cube relationship used in the first pair.
Option 20 equals 10 * 2, so it suggests multiplication by a constant factor rather than exponentiation to the third power.
Option 16 equals 2^4, so even though it is a power of 2, it corresponds to the fourth power rather than the cube, which breaks the exact analogy with 10 mapped to 10^3.
Common Pitfalls:
Candidates often jump to simple operations like multiplying by 10 or adding a constant, overlooking the more appropriate exponent pattern. Another typical error is to accept any power of the second number that appears in the options, even if the exponent does not match the first pair. To avoid these traps, always write the first pair explicitly using powers or factors and then replicate the exact same operation for the second pair.
Final Answer:
Thus, following the same cube relationship as 10 : 1000, the correct completion of the analogy 10 : 1000 :: 2 : ? is 8.
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