8 : 256 :: ?

Options

A. `7 : 343`
B. `9: 243`
C. `10: 500`
D. `5: 75`

Explanation

To solve the analogy question "8 : 256 :: ?," we need to identify the relationship between the numbers 8 and 256 and then apply the same logic to find the missing pair.

### Step-by-Step Solution:

1. **Identify the relationship between 8 and 256:**
- Start by calculating the square of 8.
- \( 8^2 = 64 \)

2. **Determine the next operation:**
- The next step involves multiplying the square of 8 by \( \frac{8}{2} \).
- \( \frac{8}{2} = 4 \)

3. **Calculate the final result:**
- Now, multiply the square of 8 by 4.
- \( 64 \times 4 = 256 \)

4. **Formulate the logic:**
- The relationship can be summarized as:
- If \( x \) is the first number, then the second number is given by the formula:
- \( x^2 \times \frac{x}{2} \)

5. **Apply the same logic to find the missing pair:**
- We need to find a number \( y \) such that:
- \( y^2 \times \frac{y}{2} = ? \)

6. **Check the options:**
- Let's evaluate the options one by one to see which one follows the same logic.
- **Option A: 7**
- \( 7^2 = 49 \)
- \( 49 \times \frac{7}{2} = 49 \times 3.5 = 171.5 \) (not a match)
- **Option B: 9**
- \( 9^2 = 81 \)
- \( 81 \times \frac{9}{2} = 81 \times 4.5 = 364.5 \) (not a match)
- **Option C: 10**
- \( 10^2 = 100 \)
- \( 100 \times \frac{10}{2} = 100 \times 5 = 500 \) (this is a match)
- **Option D: 5**
- \( 5^2 = 25 \)
- \( 25 \times \frac{5}{2} = 25 \times 2.5 = 62.5 \) (not a match)

7. **Conclusion:**
- The only option that follows the same logic as 8 : 256 is **10 : 500**.

### Final Answer:
The answer to the analogy is **10 : 500**.

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