48: 122 :: 168 : ?

Options

A. 215
B. 225
C. 290
D. 292

Explanation

To solve the analogy problem \( 48 : 122 :: 168 : ? \), we need to identify the relationship between the first pair of numbers (48 and 122) and then apply the same relationship to the second pair (168 and the unknown).

### Step 1: Analyze the first pair (48 and 122)

1. **Identify a pattern for 48**:
- We can express 48 as \( 7^2 - 1 \).
- Calculation: \( 7^2 = 49 \), so \( 49 - 1 = 48 \).

2. **Identify a pattern for 122**:
- We can express 122 as \( 11^2 + 1 \).
- Calculation: \( 11^2 = 121 \), so \( 121 + 1 = 122 \).

### Step 2: Establish the relationship

From the above, we see that:
- The first number (48) can be represented as \( n^2 - 1 \) where \( n = 7 \).
- The second number (122) can be represented as \( (n + 4)^2 + 1 \) where \( n + 4 = 11 \).

### Step 3: Apply the same relationship to the second pair (168 and ?)

1. **Identify a pattern for 168**:
- We can express 168 as \( 13^2 - 1 \).
- Calculation: \( 13^2 = 169 \), so \( 169 - 1 = 168 \).

2. **Find the corresponding second number**:
- We need to find \( (n + 4)^2 + 1 \) where \( n = 13 \).
- Calculation: \( n + 4 = 13 + 4 = 17 \).
- Now calculate \( 17^2 + 1 \):
- \( 17^2 = 289 \)
- So, \( 289 + 1 = 290 \).

### Conclusion

Thus, the unknown number is \( 290 \).

### Final Answer:
The answer is \( 290 \).

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