A sum of money becomes 8 times of itself in 12 years at C.I. In how many years it will become 16 times of itself.

Answer

To solve the problem step by step, we will follow the principles of compound interest.

### Step 1: Understand the Problem
We need to find out how many years it takes for a sum of money to become 16 times itself, given that it becomes 8 times itself in 12 years.

### Step 2: Set Up the Known Information
Let:
- Principal amount = \( P \)
- Rate of interest = \( R \)
- Time for 8 times = 12 years
- Amount after 12 years = \( 8P \)

### Step 3: Use the Compound Interest Formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- \( A \) is the amount after time \( T \)
- \( P \) is the principal
- \( R \) is the rate of interest
- \( T \) is the time in years

For our case, we can write:
\[ 8P = P \left(1 + \frac{R}{100}\right)^{12} \]

### Step 4: Simplify the Equation
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 8 = \left(1 + \frac{R}{100}\right)^{12} \]

### Step 5: Express 8 as a Power of 2
We know that:
\[ 8 = 2^3 \]
Thus, we can rewrite the equation as:
\[ 2^3 = \left(1 + \frac{R}{100}\right)^{12} \]

### Step 6: Take the Cube Root
To simplify, we take the cube root of both sides:
\[ 2 = \left(1 + \frac{R}{100}\right)^{4} \]

### Step 7: Rewrite the Equation
Now we can express \( 1 + \frac{R}{100} \):
\[ 1 + \frac{R}{100} = 2^{\frac{1}{4}} = \sqrt[4]{2} \]

### Step 8: Find the Time for 16 Times
Now we need to find \( T \) when the amount becomes \( 16P \):
\[ 16P = P \left(1 + \frac{R}{100}\right)^T \]
Dividing both sides by \( P \):
\[ 16 = \left(1 + \frac{R}{100}\right)^T \]
Since \( 16 = 2^4 \), we can write:
\[ 2^4 = \left(1 + \frac{R}{100}\right)^T \]

### Step 9: Substitute the Value of \( 1 + \frac{R}{100} \)
From Step 7, we know:
\[ 1 + \frac{R}{100} = 2^{\frac{1}{4}} \]
Substituting this into the equation:
\[ 2^4 = \left(2^{\frac{1}{4}}\right)^T \]

### Step 10: Equate the Exponents
This gives us:
\[ 4 = \frac{T}{4} \]
Multiplying both sides by 4:
\[ T = 16 \]

### Conclusion
Thus, it will take **16 years** for the sum of money to become 16 times itself.

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