In the questions, there is a certain relationship between the two given numbers on one side of :: and one number is given on another side of :: . It is required to find another number from the given alternative having the same relationship with this number as the numbers of the given pair has :
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 the relationship between the numbers on the left side of the analogy and apply the same relationship to the number on the right side.

### Step-by-Step Solution:

1. **Identify the relationship between the first pair of numbers (48 and 122)**:
- We can examine the difference between the two numbers:
\[ 122 - 48 = 74 \]
- This suggests that the second number (122) is derived from the first number (48) by adding 74.

2. **Express the relationship mathematically**:
- We can express this as:
\[ 48 + 74 = 122 \]

3. **Apply the same relationship to the second pair (168 and ?)**:
- We need to find a number that, when added to 168, gives us the same difference of 74:
\[ ? = 168 + 74 \]

4. **Calculate the value of ?**:
- Performing the addition:
\[ 168 + 74 = 242 \]

5. **Check the options provided**:
- The options given are:
- Option 1: 215
- Option 2: 225
- Option 3: 290
- Option 4: 292
- Since 242 is not listed among the options, we need to re-evaluate the relationship.

6. **Reassess the relationship**:
- On closer inspection, we realize that the relationship might involve a different operation or pattern. Let's analyze the numbers more deeply.
- The first number (48) is related to (122) in a way that involves prime numbers.

7. **Identify prime numbers**:
- The prime number after 7 is 11, and we can see that:
\[ 7^2 - 1 = 49 - 1 = 48 \]
\[ 11^2 + 1 = 121 + 1 = 122 \]

8. **Apply the same logic to 168**:
- We need to find a prime number such that:
\[ p^2 - 1 = 168 \]
- The closest square to 168 is 169, which is \(13^2\):
\[ 13^2 - 1 = 169 - 1 = 168 \]

9. **Find the next prime number**:
- The next prime number after 13 is 17. We need to calculate:
\[ 17^2 + 1 = 289 + 1 = 290 \]

10. **Conclusion**:
- Therefore, the answer to the analogy is:
\[ 168 : 290 \]
- The correct option is **Option 3: 290**.