If a sum of money on compound interest becomes three times in 4 years, then at the same interest rate, the sum will become 27 times in `:`

Options

A. a) 15 years
B. b) 12 years
C. c) 10 years
D. d) 16 years

Answer

To solve the problem, we need to determine how long it will take for a sum of money to become 27 times its original amount at the same compound interest rate, given that it becomes 3 times in 4 years.

### Step-by-Step Solution:

1. **Understanding 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 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).
- \( t \) is the time the money is invested or borrowed for, in years.

2. **Setting Up the First Condition**:
We know that the amount becomes 3 times the principal in 4 years. Thus, we can write:
\[ 3P = P \left(1 + \frac{r}{100}\right)^4 \]
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 3 = \left(1 + \frac{r}{100}\right)^4 \]
This is our **Equation 1**.

3. **Setting Up the Second Condition**:
Now, we want to find the time \( t \) when the amount becomes 27 times the principal:
\[ 27P = P \left(1 + \frac{r}{100}\right)^t \]
Again, dividing both sides by \( P \):
\[ 27 = \left(1 + \frac{r}{100}\right)^t \]
This is our **Equation 2**.

4. **Relating the Two Equations**:
From **Equation 1**, we have:
\[ 3 = \left(1 + \frac{r}{100}\right)^4 \]
We can express \( \left(1 + \frac{r}{100}\right) \) in terms of 3:
\[ 1 + \frac{r}{100} = 3^{1/4} \]
Now substituting this into **Equation 2**:
\[ 27 = \left(3^{1/4}\right)^t \]
We know that \( 27 = 3^3 \), so we can write:
\[ 3^3 = \left(3^{1/4}\right)^t \]
This simplifies to:
\[ 3^3 = 3^{t/4} \]

5. **Equating the Exponents**:
Since the bases are the same, we can equate the exponents:
\[ 3 = \frac{t}{4} \]
Multiplying both sides by 4 gives:
\[ t = 12 \]

### Final Answer:
The sum will become 27 times in **12 years**.