In the questions, select the pair in which the numbers are similarly related as in the given pair.
8: 256 :: ?:?

Options

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

Explanation

To solve the analogy problem given in the question, we need to identify the relationship between the numbers in the first pair (8 and 256) and then apply the same relationship to find the correct pair from the options provided.

### Step-by-step Solution:

1. **Identify the relationship in the pair (8: 256)**:
- The first number is 8.
- The second number is 256.
- To find the relationship, we can check if the second number is derived from the first number using a mathematical operation.

2. **Calculate the cube of the first number**:
- \( 8^3 = 512 \)

3. **Divide the cube by 2**:
- \( \frac{512}{2} = 256 \)
- This confirms that the second number (256) is equal to the cube of the first number (8) divided by 2.

4. **Apply the same relationship to the options**:
- We need to check each option to see if it follows the same pattern: \( x^3 / 2 = y \), where \( x \) is the first number and \( y \) is the second number.

5. **Check Option 1: 7: 343**:
- Calculate \( 7^3 = 343 \).
- Now divide by 2: \( \frac{343}{2} = 171.5 \) (not equal to 343).
- So, Option 1 is eliminated.

6. **Check Option 2: 9: 243**:
- Calculate \( 9^3 = 729 \).
- Now divide by 2: \( \frac{729}{2} = 364.5 \) (not equal to 243).
- So, Option 2 is eliminated.

7. **Check Option 3: 10: 500**:
- Calculate \( 10^3 = 1000 \).
- Now divide by 2: \( \frac{1000}{2} = 500 \) (equal to 500).
- This option follows the same pattern, so it is a potential answer.

8. **Check Option 4: 5: 75**:
- Calculate \( 5^3 = 125 \).
- Now divide by 2: \( \frac{125}{2} = 62.5 \) (not equal to 75).
- So, Option 4 is eliminated.

9. **Conclusion**:
- The only option that follows the same pattern as the original pair (8: 256) is Option 3: 10: 500.

### Final Answer:
The correct answer is **10: 500**.