A sum of money invested at compound interest triples itself in five years. In how many years will it become 27 times itself at the same rate of compound interest ?

Answer

To solve the problem, we need to determine how many years it will take for a sum of money to become 27 times itself at the same rate of compound interest, given that it triples in 5 years.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that the amount triples in 5 years. This means if we start with an amount \( P \), after 5 years, it becomes \( 3P \).

2. **Using the Compound Interest Formula**:
The formula for compound interest is given by:
\[ A = P(1 + r)^t \]
where \( A \) is the amount after time \( t \), \( P \) is the principal amount, \( r \) is the rate of interest, and \( t \) is the time in years.

3. **Setting Up the Equation for Tripling**:
From the information provided, we can set up the equation for tripling:
\[ 3P = P(1 + r)^5 \]
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 3 = (1 + r)^5 \]

4. **Finding the Rate of Interest**:
To find \( r \), we take the fifth root of both sides:
\[ 1 + r = 3^{1/5} \]
Therefore,
\[ r = 3^{1/5} - 1 \]

5. **Setting Up the Equation for Becoming 27 Times**:
We want to find the time \( t \) when the amount becomes 27 times itself:
\[ 27P = P(1 + r)^t \]
Again, dividing both sides by \( P \):
\[ 27 = (1 + r)^t \]

6. **Relating the Two Equations**:
We know from step 3 that \( 3 = (1 + r)^5 \). We can express 27 as:
\[ 27 = 3^3 \]
Therefore, we can rewrite the equation:
\[ 3^3 = (1 + r)^t \]

7. **Equating the Powers**:
Since \( 3 = (1 + r)^5 \), we can express \( (1 + r)^t \) in terms of \( (1 + r)^5 \):
\[ (1 + r)^t = ((1 + r)^5)^{t/5} = 3^{t/5} \]
Thus, we can equate:
\[ 3^3 = 3^{t/5} \]

8. **Solving for \( t \)**:
Since the bases are the same, we can set the exponents equal to each other:
\[ 3 = \frac{t}{5} \]
Multiplying both sides by 5 gives:
\[ t = 15 \]

### Final Answer:
The sum of money will become 27 times itself in **15 years**.