Solving the Number Analogy: $3 : 24 :: 5 : ?$

This is a number analogy problem. We need to find the relationship between the first pair of numbers ($3$ and $24$) and apply the same relationship to the second number ($5$) to find the missing number.

Identifying the Pattern

Let the first number be $N$. We examine the relationship between $N=3$ and the second number, $24$.

One possible relationship is a cubic pattern:

• Calculate $N^3 - N$.
• For $N=3$: $3^3 - 3 = 27 - 3 = 24$. This matches the first pair.

Another possible relationship is:

• Calculate $N \times (N^2 - 1)$.
• For $N=3$: $3 \times (3^2 - 1) = 3 \times (9 - 1) = 3 \times 8 = 24$. This also matches.

Both patterns confirm the relationship derived from the first pair.

Applying the Pattern

Now, we apply the identified pattern ($N^3 - N$ or $N \times (N^2 - 1)$) to the second number, $N=5$.

• Using the $N^3 - N$ pattern: $5^3 - 5 = 125 - 5 = 120$.
• Using the $N \times (N^2 - 1)$ pattern: $5 \times (5^2 - 1) = 5 \times (25 - 1) = 5 \times 24 = 120$.

Both methods yield the same result.

Conclusion

The missing number is $120$. This corresponds to Option B.