A sum of money becomes eight times in 3 years if the rate is compounded annually. In how much time, the same amount at the same compound interest rate will become sixteen times?

Options

A. 6 years
B. 4 years
C. 8 years
D. 5 years

Explanation

To solve the problem step by step, we will use the formula for compound interest and analyze the given information.

### Step 1: Understand the first condition
We are given that a sum of money becomes 8 times in 3 years with compound interest. Let's denote the principal amount as \( P \). Therefore, the amount after 3 years is:

\[ A = 8P \]

### Step 2: Write 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 after time \( n \),
- \( P \) is the principal amount,
- \( r \) is the rate of interest,
- \( n \) is the time in years.

### Step 3: Substitute the known values into the formula
From the first condition, we substitute \( A \) and \( n \):

\[ 8P = P \left(1 + \frac{r}{100}\right)^3 \]

### Step 4: Simplify the equation
We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):

\[ 8 = \left(1 + \frac{r}{100}\right)^3 \]

### Step 5: Express 8 as a power of 2
We can express 8 as \( 2^3 \):

\[ 8 = 2^3 \]

### Step 6: Take the cube root
Taking the cube root of both sides gives:

\[ 1 + \frac{r}{100} = 2 \]

### Step 7: Solve for \( r \)
Subtracting 1 from both sides:

\[ \frac{r}{100} = 1 \]

Multiplying both sides by 100:

\[ r = 100\% \]

### Step 8: Analyze the second condition
Now we need to find out how long it will take for the same principal \( P \) to become 16 times with the same interest rate. So we set up the equation:

\[ A = 16P \]

### Step 9: Substitute into the compound interest formula
Using the compound interest formula again:

\[ 16P = P \left(1 + \frac{100}{100}\right)^n \]

### Step 10: Simplify the equation
Cancel \( P \):

\[ 16 = (2)^n \]

### Step 11: Express 16 as a power of 2
We can express 16 as \( 2^4 \):

\[ 16 = 2^4 \]

### Step 12: Equate the exponents
Since the bases are the same, we can equate the exponents:

\[ n = 4 \]

### Conclusion
Thus, the time taken for the amount to become 16 times is:

\[ \text{Time} = 4 \text{ years} \]