Select the pair in which the numbers are similarly related as in the given below :
9 : 27 : : ………. : ………

Options

A. `5 : 125`
B. `8 : 64`
C. `15 : 135`
D. `81 : 729`

Answer

To solve the problem, we need to identify the relationship between the numbers in the given pair (9 : 27) and find another pair that has a similar relationship.

### Step-by-Step Solution:

1. **Identify the relationship in the given pair (9 : 27)**:
- The first number is 9, which is \(3^2\) (a perfect square).
- The second number is 27, which is \(3^3\) (a perfect cube).
- Thus, we see that the first number is the square of 3, and the second number is the cube of 3.

2. **Formulate the required relationship**:
- We need to find a pair of numbers where the first number is a perfect square of some integer \(n\) (i.e., \(n^2\)) and the second number is the perfect cube of the same integer \(n\) (i.e., \(n^3\)).

3. **Check the options**:
- We need to check each option to see if it fits the criteria of \(n^2\) and \(n^3\) for the same integer \(n\).

- **Option A**: 5 and 125
- 5 is not a perfect square.
- 125 is \(5^3\).
- This does not fit the criteria.

- **Option B**: 8 and 512
- 8 is \(2^3\) (not a perfect square).
- 512 is \(8^3\).
- This does not fit the criteria.

- **Option C**: 15 and 225
- 15 is not a perfect square.
- 225 is \(15^2\).
- This does not fit the criteria.

- **Option D**: 81 and 729
- 81 is \(9^2\) (a perfect square).
- 729 is \(9^3\) (a perfect cube).
- This fits the criteria perfectly.

4. **Conclusion**:
- The correct answer is **Option D: 81 : 729**.