Find the value of '?' in the following series.
`6:30 :: 7: ?`

Options

A. 56
B. 49
C. 35
D. 42

Answer

To find the value of '?' in the series `6:30 :: 7:?`, we can analyze the relationship between the numbers.

### Step-by-Step Solution:

1. **Identify the relationship in the first pair (6:30)**:
- We notice that if we square the first number (6), we get:
\[ 6^2 = 36 \]
- Then, we subtract the first number (6) from this square:
\[ 36 - 6 = 30 \]
- Thus, the relationship can be expressed as:
\[ x^2 - x = y \]
- Where \( x \) is the first number (6) and \( y \) is the second number (30).

2. **Apply the same relationship to the second pair (7:?)**:
- Now we will use the same formula for the number 7:
\[ 7^2 = 49 \]
- Next, we subtract the first number (7) from this square:
\[ 49 - 7 = 42 \]
- Therefore, the value of '?' is 42.

3. **Conclusion**:
- The value of '?' in the series `6:30 :: 7:?` is:
\[ ? = 42 \]

### Final Answer:
The value of '?' is **42**.