A sum of money double itself at compound intrest I 15 years.In how many years it will become eight times?

Options

A. 30
B. 45
C. 50
D. 60

Answer

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

### Step 1: Understand the Problem
We know that a sum of money doubles itself in 15 years under compound interest. We need to find out how many years it will take for the same sum to become eight times.

### Step 2: Use the Compound Interest Formula
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial sum of money).
- \( r \) is the annual interest rate (in percentage).
- \( n \) is the number of years the money is invested or borrowed.

### Step 3: Set Up the Equation for Doubling
Since the sum doubles in 15 years, we can set up the equation:
\[ 2P = P \left(1 + \frac{r}{100}\right)^{15} \]

### Step 4: Simplify the Equation
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 2 = \left(1 + \frac{r}{100}\right)^{15} \]

### Step 5: Take the 15th Root
To isolate \( r \), we take the 15th root of both sides:
\[ 1 + \frac{r}{100} = 2^{\frac{1}{15}} \]

### Step 6: Solve for \( r \)
Subtract 1 from both sides:
\[ \frac{r}{100} = 2^{\frac{1}{15}} - 1 \]
Multiply both sides by 100:
\[ r = 100 \left(2^{\frac{1}{15}} - 1\right) \]

### Step 7: Set Up the Equation for Eight Times
Now we need to find out how long it will take for the amount to become eight times:
\[ 8P = P \left(1 + \frac{r}{100}\right)^n \]

### Step 8: Simplify the Equation
Dividing both sides by \( P \):
\[ 8 = \left(1 + \frac{r}{100}\right)^n \]

### Step 9: Substitute the Value of \( r \)
From our previous calculation, we have:
\[ 1 + \frac{r}{100} = 2^{\frac{1}{15}} \]
Thus, we can substitute:
\[ 8 = \left(2^{\frac{1}{15}}\right)^n \]

### Step 10: Rewrite 8 as a Power of 2
Since \( 8 = 2^3 \), we can rewrite the equation:
\[ 2^3 = \left(2^{\frac{1}{15}}\right)^n \]

### Step 11: Set the Exponents Equal
Since the bases are the same, we can set the exponents equal to each other:
\[ 3 = \frac{n}{15} \]

### Step 12: Solve for \( n \)
Multiply both sides by 15:
\[ n = 3 \times 15 = 45 \]

### Conclusion
Thus, it will take **45 years** for the sum of money to become eight times itself at the same compound interest rate.