Understanding the Number Analogy Problem

The question asks us to find a missing number in a sequence based on a consistent relationship between pairs of numbers. The sequence is given as $9:56::12:?::5:12$.

We need to determine the relationship between the first number and the second number in each pair and apply it to find the missing term.

Identifying the Pattern

Let's analyze the given pairs:

• Pair 1: $9 \rightarrow 56$
• Pair 3: $5 \rightarrow 12$

We look for a mathematical operation that links the first number to the second.

1. Analyzing Pair 1 ($9:56$):
   • Try multiplying 9 by a number and adding a constant.
   • $9 \times 6 = 54$. Adding 2 gives $54 + 2 = 56$.
   • This suggests a potential relationship: $B = (A \times 6) + 2$.
2. Analyzing Pair 3 ($5:12$):
   • Let's test if a similar structure applies, using the form $B = (A \times k) + 2$.
   • For $A=5$, we need $B=12$. So, $12 = (5 \times k) + 2$.
   • Solving for $k$: $10 = 5 \times k \implies k=2$.
   • The relationship for this pair is $B = (A \times 2) + 2$.
3. Finding the Rule for the Multiplier:
   • We have multipliers $k=6$ for the first number $A=9$, and $k=2$ for $A=5$.
   • Let's find the relationship between $A$ and $k$.
   • For $A=9$, $k=6$. We see that $6 = 9 - 3$.
   • For $A=5$, $k=2$. We see that $2 = 5 - 3$.
   • This suggests the general rule for the multiplier is $k = A - 3$.
   • Therefore, the complete relationship between the first number ($A$) and the second number ($B$) is:
     $B = (A \times (A-3)) + 2$

Applying the Pattern to Find the Missing Number

Now, we apply the derived rule $B = (A \times (A-3)) + 2$ to the middle pair ($12:?$).

1. The first number in this pair is $A = 12$.
2. Calculate the multiplier using the rule $k = A - 3$: $k = 12 - 3 = 9$
3. Calculate the second number ($B$) using the formula $B = (A \times k) + 2$: $B = (12 \times 9) + 2$ $B = 108 + 2$ $B = 110$

The missing number is 110.

Conclusion

The number that fits the relationship in the sequence $9:56::12:?::5:12$ is 110.