A certain sum becomes k times in 5 years at compound interest. In 30 years it will become how many times?

Options

A. `5k `
B. `6k`
C. `k^6`
D. `k^5`

Answer

To solve the problem step by step, we need to understand the concept of compound interest and how the amount grows over time.

### Step-by-Step Solution:

1. **Understanding the Initial Condition**:
- We are given that a certain sum becomes \( K \) times in 5 years at compound interest.
- This means if we start with a principal amount \( P \), after 5 years, the amount \( A \) can be expressed as:
\[ A = P \times K \]

2. **Using the Compound Interest Formula**:
- The formula for compound interest is:
\[ A = P \times \left(1 + \frac{R}{100}\right)^T \]
- For our case, after 5 years, we can write:
\[ P \times K = P \times \left(1 + \frac{R}{100}\right)^5 \]

3. **Cancelling the Principal**:
- We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ K = \left(1 + \frac{R}{100}\right)^5 \]

4. **Finding the Amount After 30 Years**:
- Now, we need to find out how many times the amount will become after 30 years.
- Using the same compound interest formula for 30 years:
\[ A_{30} = P \times \left(1 + \frac{R}{100}\right)^{30} \]

5. **Expressing in Terms of K**:
- We can express \( \left(1 + \frac{R}{100}\right)^{30} \) in terms of \( K \):
\[ \left(1 + \frac{R}{100}\right)^{30} = \left(\left(1 + \frac{R}{100}\right)^5\right)^6 = K^6 \]
- Therefore, the amount after 30 years becomes:
\[ A_{30} = P \times K^6 \]

6. **Final Expression**:
- Again, cancelling \( P \):
\[ \text{Amount after 30 years} = K^6 \]
- Thus, the sum becomes \( K^6 \) times the original principal after 30 years.

### Conclusion:
The final answer is that the sum will become \( K^6 \) times after 30 years.