If a certain sum becomes 4 times in 4 years at compound Interest, then in how many years, it will become 64 times?

Options

A. 5
B. 12
C. 16
D. 24

Answer

To solve the problem step by step, we will use the concept of compound interest.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that a certain sum of money (let's call it P) becomes 4 times itself in 4 years at compound interest. We need to find out how many years it will take for the same sum to become 64 times itself.

2. **Setting Up the First Equation**:
From the information given, we can set up the equation for the first scenario:
\[ 4P = P \left(1 + \frac{R}{100}\right)^4 \]
Dividing both sides by P (assuming P ≠ 0), we get:
\[ 4 = \left(1 + \frac{R}{100}\right)^4 \]

3. **Setting Up the Second Equation**:
Now, we need to find out how long it will take for the sum to become 64 times itself:
\[ 64P = P \left(1 + \frac{R}{100}\right)^x \]
Again, dividing both sides by P, we have:
\[ 64 = \left(1 + \frac{R}{100}\right)^x \]

4. **Expressing 64 in Terms of 4**:
We know that \(64\) can be expressed as \(4^3\):
\[ 64 = 4^3 \]

5. **Substituting the First Equation into the Second**:
From our first equation, we have:
\[ 1 + \frac{R}{100} = 4^{1/4} \]
Therefore, we can substitute this into the second equation:
\[ 4^3 = \left(4^{1/4}\right)^x \]

6. **Using the Power of a Power Property**:
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we can rewrite the equation:
\[ 4^3 = 4^{x/4} \]

7. **Equating the Exponents**:
Since the bases are the same, we can equate the exponents:
\[ 3 = \frac{x}{4} \]

8. **Solving for x**:
To find x, multiply both sides by 4:
\[ x = 3 \times 4 = 12 \]

9. **Conclusion**:
Therefore, it will take **12 years** for the sum to become 64 times itself.

### Final Answer:
The answer is **12 years**.