12 : 144: :?

Options

A. `10:40`
B. `15:125`
C. `20: 400`
D. `22:464`

Explanation

To solve the analogy problem 12 : 144 :: ? we need to identify the relationship between the two numbers in the first pair (12 and 144) and then apply the same relationship to find the missing number.

### Step-by-Step Solution:

1. **Identify the relationship between 12 and 144**:
- We can observe that 144 is the square of 12.
- Mathematically, this can be expressed as:
\[ 12^2 = 144 \]

2. **Apply the same relationship to find the missing number**:
- We need to find a number (let's call it x) such that when we square it, we get the same relationship as in the first pair.
- We will check the options given to find which number squared gives us a result that fits the pattern.

3. **Evaluate the options**:
- **Option A: 10**
\[ 10^2 = 100 \quad (\text{not a match}) \]
- **Option B: 40**
\[ 40^2 = 1600 \quad (\text{not a match}) \]
- **Option C: 15**
\[ 15^2 = 225 \quad (\text{not a match}) \]
- **Option D: 20**
\[ 20^2 = 400 \quad (\text{not a match}) \]
- **Option E: 22**
\[ 22^2 = 484 \quad (\text{not a match}) \]

4. **Conclusion**:
- None of the options provided (10, 40, 15, 20, 22) directly relate to the square of 12 in the same way that 12 relates to 144. However, the problem seems to imply that we are looking for a number that fits the pattern of squaring.
- The correct answer based on the relationship of squaring is not listed among the options provided.