A sum of money at compound interest amounts to thrice itself in 3years, in how money years will it be 81 times it self ?

Answer

To solve the problem step by step, we will use the information provided about the compound interest and the relationship between the time and the amount.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
We know that a sum of money amounts to thrice itself in 3 years. This means if the principal amount is \( P \), then after 3 years the amount \( A \) will be:
\[ A = 3P \]

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, \( r \) 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 (3 years):
\[ 3P = P(1 + r)^3 \]
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 3 = (1 + r)^3 \]

4. **Finding the Relationship Between Amounts:**
We want to find out how many years it will take for the amount to become 81 times itself. So we set up the equation for the second scenario:
\[ A = 81P \]
Using the compound interest formula again, we have:
\[ 81P = P(1 + r)^t \]
Dividing both sides by \( P \):
\[ 81 = (1 + r)^t \]

5. **Relating the Two Equations:**
We have two equations now:
- \( 3 = (1 + r)^3 \)
- \( 81 = (1 + r)^t \)

From the first equation, we can express \( (1 + r) \):
\[ (1 + r) = 3^{1/3} \]

6. **Substituting into the Second Equation:**
We can substitute \( (1 + r) \) into the second equation:
\[ 81 = (3^{1/3})^t \]
This simplifies to:
\[ 81 = 3^{t/3} \]

7. **Expressing 81 in Terms of Powers of 3:**
We know that \( 81 = 3^4 \). Therefore, we can equate the powers:
\[ 3^4 = 3^{t/3} \]

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

9. **Solving for \( t \):**
To find \( t \), multiply both sides by 3:
\[ t = 4 \times 3 = 12 \]

### Final Answer:
The amount will be 81 times itself in **12 years**.