42: 56:: 110:?

Options

A. 118
B. 132
C. 136
D. 140

Explanation

To solve the analogy problem \( 42 : 56 :: 110 : ? \), we can follow these steps:

### Step 1: Analyze the first pair (42 and 56)
We notice that:
- \( 42 \) can be expressed as \( 7^2 - 7 = 49 - 7 = 42 \)
- \( 56 \) can be expressed as \( 8^2 - 8 = 64 - 8 = 56 \)

### Step 2: Identify the pattern
From the first pair, we can see that:
- The first number \( 42 \) corresponds to \( 7 \) (which is \( 7^2 - 7 \)).
- The second number \( 56 \) corresponds to \( 8 \) (which is \( 8^2 - 8 \)).
- The numbers \( 7 \) and \( 8 \) are consecutive integers.

### Step 3: Apply the same pattern to the second pair
Now, we need to find the number that corresponds to \( 110 \):
- We can express \( 110 \) as \( 11^2 - 11 = 121 - 11 = 110 \).

### Step 4: Find the next integer
Following the pattern of consecutive integers:
- After \( 11 \), the next integer is \( 12 \).

### Step 5: Calculate the value for the question mark
Now, we apply the same formula for \( 12 \):
- The value will be \( 12^2 - 12 = 144 - 12 = 132 \).

### Conclusion
Thus, the answer to the analogy \( 42 : 56 :: 110 : ? \) is \( 132 \).