Choosing a number related to a given number in the same manner as the two numbers of another given pair are related to each other
48 : 122 :: 168 : ?

Options

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

Explanation

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

### Step 1: Identify the relationship between 48 and 122
First, we can express 48 and 122 in terms of squares:
- \(48 = 7^2 - 1\) (since \(7^2 = 49\) and \(49 - 1 = 48\))
- \(122 = 11^2 + 1\) (since \(11^2 = 121\) and \(121 + 1 = 122\))

### Step 2: Analyze the relationship between the squares
Now, we can observe the numbers \(7\) and \(11\):
- The difference between \(11\) and \(7\) is \(11 - 7 = 4\).

### Step 3: Apply the same relationship to 168
Next, we need to find a number \(x\) such that:
- \(168 = x^2 - 1\)

To find \(x\):
- Rearranging gives \(x^2 = 168 + 1 = 169\).
- Taking the square root, we find \(x = 13\) (since \(13^2 = 169\)).

### Step 4: Determine the next number in the sequence
Now, we need to find the next number using the same relationship:
- The next number should be \(x + 4 = 13 + 4 = 17\).
- We will calculate \(17^2 + 1\):
- \(17^2 = 289\)
- Therefore, \(17^2 + 1 = 289 + 1 = 290\).

### Conclusion
The unknown number that corresponds to 168 in the analogy is \(290\).

Thus, the answer is:
\[ \text{?} = 290 \]