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. 15 years
B. 12 years
C. 16 years
D. 10 years

Answer

To solve the problem step by step, we will use the formula for compound interest.

### Step 1: Understand the Problem
We know that a sum of money becomes three times in 4 years under compound interest. We need to find out how many years it will take for the same sum to become 27 times.

### Step 2: Set Up the Equation
Let the principal amount be \( P \).

According to the problem, after 4 years, the amount becomes:
\[ A = 3P \]

Using the compound interest formula:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where \( n \) is the number of years and \( r \) is the rate of interest.

Substituting the known values:
\[ 3P = P \left(1 + \frac{r}{100}\right)^4 \]

### Step 3: Simplify the Equation
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 3 = \left(1 + \frac{r}{100}\right)^4 \]

### Step 4: Solve for \( 1 + \frac{r}{100} \)
Taking the fourth root of both sides:
\[ 1 + \frac{r}{100} = 3^{1/4} \]

### Step 5: Find the Value of \( n \) for 27 Times
Now we need to find out how many years it will take for the amount to become 27 times:
\[ A = 27P \]
Using the same compound interest formula:
\[ 27P = P \left(1 + \frac{r}{100}\right)^n \]

Dividing both sides by \( P \):
\[ 27 = \left(1 + \frac{r}{100}\right)^n \]

### Step 6: Substitute the Value of \( 1 + \frac{r}{100} \)
Substituting \( 1 + \frac{r}{100} = 3^{1/4} \):
\[ 27 = \left(3^{1/4}\right)^n \]

### Step 7: Rewrite 27 in Terms of Powers of 3
Since \( 27 = 3^3 \), we can write:
\[ 3^3 = \left(3^{1/4}\right)^n \]

### Step 8: Equate the Exponents
Since the bases are the same, we can equate the exponents:
\[ 3 = \frac{n}{4} \]

### Step 9: Solve for \( n \)
Multiplying both sides by 4:
\[ n = 12 \]

### Conclusion
Thus, the sum will become 27 times in **12 years**.

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