11 : 1210 :: ?

Options

A. `6 : 216`
B. `7 : 1029`
C. `8 : 448`
D. `9 : 729`

Explanation

To solve the analogy problem given as 11 : 1210 :: ? : ?, we need to identify the relationship between the first pair (11 and 1210) and apply the same logic to find the second pair.

### Step-by-Step Solution:

1. **Identify the relationship in the first pair (11 : 1210)**:
- We can observe that 11 multiplied by 110 gives us 1210.
- This can be expressed as:
\[ 11 \times 110 = 1210 \]

2. **Understanding the multiplier**:
- The multiplier 110 can be broken down as \(11 \times 10\).
- Therefore, we can express the relationship as:
\[ 11 \times (11 \times 10) = 1210 \]

3. **Finding the second pair**:
- Let's denote the unknown number as \(x\).
- If we follow the same logic, we need to find a number \(x\) such that:
\[ x \times (x \times 10) = ? \]
- This means we need to find \(x\) such that the result is a specific number.

4. **Testing possible values for \(x\)**:
- Let's try \(x = 6\):
\[ 6 \times (6 \times 10) = 6 \times 60 = 360 \]
- Now let's try \(x = 7\):
\[ 7 \times (7 \times 10) = 7 \times 70 = 490 \]
- Next, try \(x = 8\):
\[ 8 \times (8 \times 10) = 8 \times 80 = 640 \]
- Finally, try \(x = 9\):
\[ 9 \times (9 \times 10) = 9 \times 90 = 810 \]

5. **Finding the correct multiplier**:
- We can see that the pattern is consistent with the first pair. The second number in the second pair should be \(x \times 10\).
- After testing, we find that \(x = 8\) gives us \(640\), which follows the same multiplication pattern as \(11 : 1210\).

6. **Final answer**:
- Therefore, the answer to the analogy is:
\[ 8 : 640 \]