A sum of money becomes 3 times of itself in 6 years at compound interest. In how many years will it become 81 times?

Answer

To solve the problem, we need to determine how many years it will take for a sum of money to become 81 times its original amount at compound interest, given that it becomes 3 times its original amount in 6 years.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that a sum of money becomes 3 times itself in 6 years. This means if the principal amount is \( P \), then after 6 years, the amount \( A \) is:
\[ A = 3P \]

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

3. **Setting Up the Equation**:
From the information given, we can set up the equation for the first scenario (6 years):
\[ 3P = P(1 + r)^6 \]
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 3 = (1 + r)^6 \]

4. **Finding the Rate of Interest**:
To find \( r \), we take the sixth root of both sides:
\[ 1 + r = 3^{1/6} \]
Therefore,
\[ r = 3^{1/6} - 1 \]

5. **Finding the Time for 81 Times**:
Now, we want to find out how long it will take for the sum to become 81 times itself. So we set up the equation for this scenario:
\[ 81P = P(1 + r)^t \]
Dividing both sides by \( P \):
\[ 81 = (1 + r)^t \]

6. **Relating 81 to 3**:
Notice that \( 81 = 3^4 \). Therefore, we can rewrite the equation as:
\[ 3^4 = (1 + r)^t \]

7. **Using the Previous Equation**:
Since we know from step 4 that \( (1 + r)^6 = 3 \), we can express \( (1 + r)^t \) in terms of \( (1 + r)^6 \):
\[ (1 + r)^t = (1 + r)^{6 \cdot \frac{t}{6}} = (3)^{\frac{t}{6}} \]

8. **Setting the Exponents Equal**:
Now we have:
\[ (3)^{\frac{t}{6}} = 3^4 \]
Since the bases are the same, we can set the exponents equal to each other:
\[ \frac{t}{6} = 4 \]

9. **Solving for \( t \)**:
Multiplying both sides by 6 gives:
\[ t = 24 \]

### Final Answer:
It will take **24 years** for the sum of money to become 81 times its original amount at compound interest.