A sum of money doubles itself in 4 years at compound Interest. It will amount to 8 times itself at the same rate of interest in :

Options

A. 18 years
B. 12 years
C. 16 years
D. 24 years

Answer

To solve the problem, we need to determine how long it will take for a sum of money to amount to 8 times itself at the same rate of compound interest, given that it doubles in 4 years.

### Step-by-Step Solution:

1. **Understanding Compound Interest**:
- When a sum of money doubles in a certain period at compound interest, it means that the final amount is twice the principal amount.

2. **Given Information**:
- Let the principal amount be \( P \).
- After 4 years, the amount becomes \( 2P \).

3. **Using the Compound Interest Formula**:
- The formula for compound interest is given by:
\[ A = P(1 + r)^n \]
- Where:
- \( A \) is the amount after \( n \) years,
- \( P \) is the principal amount,
- \( r \) is the rate of interest,
- \( n \) is the number of years.

4. **Setting Up the Equation**:
- From the given information, after 4 years:
\[ 2P = P(1 + r)^4 \]
- Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 2 = (1 + r)^4 \]

5. **Finding the Rate of Interest**:
- To find \( r \), we take the fourth root of both sides:
\[ 1 + r = 2^{1/4} \]
- Thus:
\[ r = 2^{1/4} - 1 \]

6. **Finding the Time to Amount to 8 Times**:
- We want to find \( n \) such that:
\[ 8P = P(1 + r)^n \]
- Dividing both sides by \( P \):
\[ 8 = (1 + r)^n \]

7. **Substituting \( 1 + r \)**:
- From our earlier calculation, we know \( 1 + r = 2^{1/4} \):
\[ 8 = (2^{1/4})^n \]
- This simplifies to:
\[ 8 = 2^{n/4} \]

8. **Equating the Exponents**:
- Since \( 8 = 2^3 \), we can equate the exponents:
\[ 3 = \frac{n}{4} \]
- Multiplying both sides by 4 gives:
\[ n = 12 \]

### Final Answer:
It will take **12 years** for the sum of money to amount to 8 times itself at the same rate of compound interest.