A sum of money doubled itself at a certain rate of compound interest in 10 years. In how many years will it become eight times ?

Options

A. 15
B. 30
C. 45
D. 25

Answer

To solve the problem step by step, we need to determine how many years it will take for a sum of money to become eight times itself at the same rate of compound interest that allowed it to double in 10 years.

### Step 1: Understand the doubling time
We know that the sum of money doubles in 10 years. Let's denote the principal amount as \( P \). After 10 years, the amount \( A \) will be:
\[ A = 2P \]

### Step 2: Use the compound interest formula
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- \( A \) is the amount after time \( T \),
- \( P \) is the principal amount,
- \( R \) is the rate of interest,
- \( T \) is the time in years.

### Step 3: Set up the equation for doubling
Using the information that the amount doubles in 10 years, we can set up the equation:
\[ 2P = P \left(1 + \frac{R}{100}\right)^{10} \]
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 2 = \left(1 + \frac{R}{100}\right)^{10} \]

### Step 4: Solve for \( 1 + \frac{R}{100} \)
Taking the 10th root of both sides:
\[ 1 + \frac{R}{100} = 2^{\frac{1}{10}} \]

### Step 5: Set up the equation for becoming eight times
Now we want to find out how long it will take for the amount to become eight times the principal:
\[ 8P = P \left(1 + \frac{R}{100}\right)^T \]
Again, dividing both sides by \( P \):
\[ 8 = \left(1 + \frac{R}{100}\right)^T \]

### Step 6: Substitute the value of \( 1 + \frac{R}{100} \)
Substituting \( 1 + \frac{R}{100} = 2^{\frac{1}{10}} \) into the equation:
\[ 8 = \left(2^{\frac{1}{10}}\right)^T \]

### Step 7: Express 8 as a power of 2
We know that \( 8 = 2^3 \). Therefore, we can rewrite the equation as:
\[ 2^3 = \left(2^{\frac{1}{10}}\right)^T \]

### Step 8: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[ 3 = \frac{T}{10} \]

### Step 9: Solve for \( T \)
Multiplying both sides by 10 gives:
\[ T = 30 \]

### Conclusion
Thus, it will take **30 years** for the sum of money to become eight times itself at the same rate of compound interest.