A certain sum triples in 4 years at compound interest, interest being compounded annually. In how many years would it become 27 times itself ?

Options

A. 9
B. 10
C. 12
D. 16

Answer

To solve the problem step by step, we will first determine the rate of interest and then use that to find out how long it will take for the sum to become 27 times itself.

### Step 1: Understand the problem
We know that a certain sum of money triples in 4 years at compound interest. We need to find out how many years it will take for this sum to become 27 times itself.

### Step 2: Set up the equation for tripling
Let the principal amount be \( P \). According to the problem, the amount after 4 years is:
\[ A = 3P \]
Using the formula for compound interest:
\[ A = P(1 + r)^t \]
Where:
- \( A \) is the amount after time \( t \)
- \( P \) is the principal
- \( r \) is the rate of interest
- \( t \) is the time in years

Substituting the values we have:
\[ 3P = P(1 + r)^4 \]
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 3 = (1 + r)^4 \]

### Step 3: Solve for \( 1 + r \)
Now, we will take the fourth root of both sides:
\[ 1 + r = 3^{1/4} \]

### Step 4: Set up the equation for becoming 27 times
Next, we want to find out how long it will take for the sum to become 27 times itself:
\[ A = 27P \]
Using the compound interest formula again:
\[ 27P = P(1 + r)^t \]
Dividing both sides by \( P \):
\[ 27 = (1 + r)^t \]

### Step 5: Substitute \( 1 + r \) into the equation
Now, we can substitute \( 1 + r \) from Step 3 into this equation:
\[ 27 = (3^{1/4})^t \]
This simplifies to:
\[ 27 = 3^{t/4} \]

### Step 6: Express 27 as a power of 3
We know that \( 27 = 3^3 \), so we can rewrite the equation:
\[ 3^3 = 3^{t/4} \]

### Step 7: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal:
\[ 3 = \frac{t}{4} \]

### Step 8: Solve for \( t \)
Now, we can solve for \( t \):
\[ t = 3 \times 4 = 12 \]

### Conclusion
Thus, it will take **12 years** for the sum to become 27 times itself.

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