A sum of money double itself in 5 years com pounded annually. It will become 4 times itself in

Options

A. 10 years
B. 12 years
C. 15 years
D. 20 years

Answer

To solve the problem, we need to determine how long it will take for a sum of money to quadruple (become 4 times itself) if it doubles in 5 years under compound interest.

### Step-by-Step Solution:

1. **Understanding the Doubling Time**:
- The problem states that a sum of money doubles itself in 5 years. This means if we start with a principal amount \( P \), after 5 years, it will be \( 2P \).

2. **Identifying the Growth Factor**:
- Since the money doubles in 5 years, we can express this as:
\[ A = P(1 + r)^5 = 2P \]
- Here, \( A \) is the amount after 5 years, \( r \) is the rate of interest, and \( P \) is the principal amount.

3. **Finding the Rate of Interest**:
- Dividing both sides by \( P \):
\[ (1 + r)^5 = 2 \]
- To find \( r \), we take the fifth root of both sides:
\[ 1 + r = 2^{1/5} \]
- Therefore, \( r = 2^{1/5} - 1 \).

4. **Calculating the Time to Quadruple**:
- We want to find out how long it takes for the principal to become \( 4P \):
\[ A = P(1 + r)^t = 4P \]
- Dividing both sides by \( P \):
\[ (1 + r)^t = 4 \]

5. **Relating Quadrupling to Doubling**:
- Since \( 4 = 2^2 \), we can express this as:
\[ (1 + r)^t = (1 + r)^{10} \]
- This implies that \( t = 10 \) years because \( (1 + r)^{10} \) is the same growth factor as \( (1 + r)^5 \) doubling twice.

6. **Conclusion**:
- Therefore, it will take **10 years** for the sum of money to become 4 times itself.

### Final Answer:
The sum of money will become 4 times itself in **10 years**.

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