A sum becomes 8 times of itself in 7 years at the rate of compound interest (interest is compounded annually). In how many years will the sum becomes 4096 times of itself?

Options

A. 28
B. 24
C. 56
D. 16

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the Problem
We need to find out how many years it will take for a sum of money to become 4096 times itself, given that it becomes 8 times itself in 7 years at a certain rate of compound interest.

### Step 2: Set Up the Initial Equation
Let the principal amount be \( P \). According to the problem, the amount after 7 years is:
\[ A = 8P \]
Using the formula for compound interest:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Substituting the values we have:
\[ 8P = P \left(1 + \frac{R}{100}\right)^7 \]
By dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 8 = \left(1 + \frac{R}{100}\right)^7 \]

### Step 3: Solve for \( 1 + \frac{R}{100} \)
To isolate \( 1 + \frac{R}{100} \), we take the 7th root of both sides:
\[ 1 + \frac{R}{100} = 8^{\frac{1}{7}} \]

### Step 4: Calculate \( 8^{\frac{1}{7}} \)
We know that \( 8 = 2^3 \), therefore:
\[ 8^{\frac{1}{7}} = (2^3)^{\frac{1}{7}} = 2^{\frac{3}{7}} \]
Thus, we can write:
\[ 1 + \frac{R}{100} = 2^{\frac{3}{7}} \]

### Step 5: Set Up the Equation for 4096 Times
Now, we need to find out how many years \( T \) it will take for the amount to become 4096 times itself:
\[ A = 4096P \]
Using the compound interest formula again:
\[ 4096P = P \left(1 + \frac{R}{100}\right)^T \]
Dividing both sides by \( P \):
\[ 4096 = \left(1 + \frac{R}{100}\right)^T \]

### Step 6: Substitute \( 1 + \frac{R}{100} \)
Substituting \( 1 + \frac{R}{100} = 2^{\frac{3}{7}} \):
\[ 4096 = \left(2^{\frac{3}{7}}\right)^T \]

### Step 7: Express 4096 as a Power of 2
We know that \( 4096 = 2^{12} \), so we can write:
\[ 2^{12} = \left(2^{\frac{3}{7}}\right)^T \]

### Step 8: Set the Exponents Equal
Since the bases are the same, we can set the exponents equal:
\[ 12 = \frac{3}{7} T \]

### Step 9: Solve for \( T \)
To find \( T \), multiply both sides by \( \frac{7}{3} \):
\[ T = 12 \times \frac{7}{3} = 28 \]

### Conclusion
Thus, the sum will become 4096 times of itself in **28 years**.

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