In this number analogy involving positive and negative signs, 5 is related to -10. Using the same pattern, which number should complete the analogy 5 : -10 :: -8 : ?

Introduction / Context:
This numerical analogy tests understanding of sign changes and multiplication by a constant. The given pair 5 : -10 suggests that the transformation involves both magnitude and sign. Once the exact operation is identified, it must be applied to -8 to determine the missing number. Such problems are basic but important for reinforcing comfort with negative numbers.

Given Data / Assumptions:
First pair: 5 : -10. Second pair: -8 : ?. Positive and negative signs are meaningful. The same arithmetic operation should be applied consistently.

Concept / Approach:
We look for a simple arithmetic relation between 5 and -10. The most natural interpretation is multiplication by a negative constant. Observing that -10 equals 5 multiplied by -2 indicates that the mapping uses -2 as the factor. Therefore, to find the number related to -8, we multiply -8 by the same factor. Understanding multiplication of signed numbers is crucial here.

Step-by-Step Solution:
Step 1: Express the relation between 5 and -10. 5 * -2 = -10. Step 2: Interpret the rule. The second number is the first number multiplied by -2. Step 3: Apply this rule to -8. Compute -8 * -2. A negative times a negative gives a positive result. -8 * -2 = 16. Step 4: Match 16 with the options. 16 is present as option A and fits the rule exactly.

Verification / Alternative check:
We can test whether -16 or other options can fit any similar pattern. If we tried to use -16, the required factor would have to be 2 (since -8 * 2 = -16), but this would conflict with the factor -2 used earlier, where 5 multiplied by 2 gives 10, not -10. Since the same factor must hold for both pairs, only multiplication by -2 is acceptable. Thus, 16 is uniquely consistent with the relationship seen in the first pair.

Why Other Options Are Wrong:
-16 requires multiplying -8 by 2, which breaks the pattern of multiplying by -2. 13 would need a non integer factor from -8 and does not match the pattern of the first pair. -13 again breaks the constant factor rule and would not come from multiplying -8 by -2. 8 is just the opposite sign of -8 and does not reflect any multiplication by -2 relationship.

Common Pitfalls:
Learners sometimes forget the rule that the product of two negative numbers is positive, or they attempt to match only the magnitude without considering the sign. In analogy questions involving signed numbers, carefully track both magnitude and sign, and always confirm that the same factor works for all given pairs. This avoids errors and builds confidence in handling negative values.

Final Answer:
The number that correctly completes the analogy is 16.