If certain amount becomes 8 times of itself in 24 years at the certain rate of compound interest then in what time period that amount will become 4times of itself?

Options

A. 14 year
B. 15 year
C. 17 year
D. 16 year

Answer

To solve the problem, we need to determine the time period required for an amount to become 4 times itself, given that it becomes 8 times itself in 24 years at a certain rate of compound interest.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that the amount becomes 8 times itself in 24 years. Let's denote the principal amount as P. Therefore, after 24 years, the amount A is:
\[ A = 8P \]

2. **Using the Compound Interest Formula**:
The formula for compound interest is given by:
\[ A = P(1 + r)^t \]
where:
- A is the final amount,
- P is the principal amount,
- r is the rate of interest,
- t is the time in years.

For our case, substituting the values we have:
\[ 8P = P(1 + r)^{24} \]

3. **Simplifying the Equation**:
Dividing both sides by P (assuming P ≠ 0):
\[ 8 = (1 + r)^{24} \]

4. **Finding the Rate of Interest**:
To find \(1 + r\), we take the 24th root of both sides:
\[ 1 + r = 8^{1/24} \]

5. **Calculating \(8^{1/24}\)**:
We know that \(8 = 2^3\), so:
\[ 8^{1/24} = (2^3)^{1/24} = 2^{3/24} = 2^{1/8} \]
Thus, we have:
\[ 1 + r = 2^{1/8} \]

6. **Finding the Time for Amount to Become 4 Times**:
Now we need to find the time \(t\) when the amount becomes 4 times itself:
\[ A = 4P \]
Using the compound interest formula again:
\[ 4P = P(1 + r)^t \]
Dividing both sides by P:
\[ 4 = (1 + r)^t \]

7. **Substituting \(1 + r\)**:
We substitute \(1 + r = 2^{1/8}\):
\[ 4 = (2^{1/8})^t \]

8. **Simplifying the Equation**:
We know that \(4 = 2^2\), so we can write:
\[ 2^2 = 2^{t/8} \]

9. **Equating the Exponents**:
Since the bases are the same, we can equate the exponents:
\[ 2 = \frac{t}{8} \]

10. **Solving for \(t\)**:
Multiplying both sides by 8 gives:
\[ t = 16 \]

### Final Answer:
The time period for the amount to become 4 times itself is **16 years**.