In this numerical analogy question, 5 is related to 100, so you must select the number that stands in the same relation to 6 from the given alternatives.

Introduction / Context:

This question is a numerical analogy where a specific mathematical operation relates the first number to the second. The pair "5 : 100" suggests that the answer might involve powers or products rather than simple addition or subtraction. Your task is to detect the exact rule used to obtain 100 from 5 and then apply the same rule to the number 6 to find the missing term.

Given Data / Assumptions:

- First pair of numbers: 5 and 100.
- Second pair to be completed: 6 and ?.
- Options: 180, 150, 160, 170 and 144.
- Standard arithmetic operations and basic algebraic expressions such as n^2, n^3 or combinations are considered.

Concept / Approach:

We suspect that a combined power and product rule may link 5 and 100. Clearly, 5^2 = 25 and 5^3 = 125, neither equals 100 alone. However, if we multiply 5^2 by (5 minus 1), that is 25 * 4, we obtain 100. This leads to a compact expression n^3 - n^2 = n^2 * (n - 1). We can verify this expression for n = 5 and then apply the same for n = 6 to determine the second term.

Step-by-Step Solution:

Verification / Alternative check:

We can verify by checking whether any simpler rules fit equally well. Doubling 5 or using 5 * 20 gives 100, but applying the same fixed factor to 6 would give 120, which is not in the options. Similarly, 5^2 * 4 = 100 uses a factor (n - 1) when n = 5. For n = 6, this would be 6^2 * 5 = 36 * 5 = 180, which agrees with our n^3 - n^2 formula. Among the options provided, only 180 matches this result, confirming the correctness of the rule.

Why Other Options Are Wrong:

150 cannot be expressed as 6^3 - 6^2 or 6^2 * (6 - 1), so it does not follow the same pattern.

160 also fails to match the derived expression and thus breaks the analogy.

170 is an arbitrary nearby number with no clear relation to 6 via the rule n^3 - n^2.

144 might be attractive as 12^2, but the original mapping from 5 to 100 did not use a pure square alone, so 144 is inconsistent with the rule.

Common Pitfalls:

Some candidates may try simple multiplication or addition first and then pick an option that seems reasonably close. However, numerical analogies in exams often depend on structured expressions like n^2 * (n - 1) or combinations of powers. Another pitfall is stopping after testing only one plausible formula without confirming that it fits both the given pair and the options uniformly.

Final Answer:

Following the same rule that maps 5 to 100, the number 6 is mapped to 180.