If a certain sum becomes 3 times In 6 years at compound interest, then in how many years, it will become 81 times?

Options

A. 81
B. 162
C. 27
D. 24

Answer

To solve the problem step by step, we will use the concept of compound interest and the relationship between the time and the growth of the investment.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We know that a certain sum becomes 3 times in 6 years at compound interest. We need to find out how many years it will take for the same sum to become 81 times.

2. **Expressing the Growth**:
The sum becomes 3 times in 6 years. We can express this as:
\[ A = P(1 + r)^n \]
where \( A \) is the amount after time \( n \), \( P \) is the principal amount, \( r \) is the rate of interest, and \( n \) is the number of years.

In our case, after 6 years:
\[ 3P = P(1 + r)^6 \]
Dividing both sides by \( P \):
\[ 3 = (1 + r)^6 \]

3. **Finding the Relationship for 81 Times**:
We want to find out how many years \( x \) it will take for the sum to become 81 times:
\[ 81P = P(1 + r)^x \]
Dividing both sides by \( P \):
\[ 81 = (1 + r)^x \]

4. **Expressing 81 in Terms of 3**:
We can express 81 as:
\[ 81 = 3^4 \]
Therefore, we can rewrite the equation:
\[ 3^4 = (1 + r)^x \]

5. **Using the Previous Equation**:
From our earlier step, we know that:
\[ 3 = (1 + r)^6 \]
Now we can relate these two equations. We can express \( (1 + r)^x \) in terms of \( (1 + r)^6 \):
\[ (1 + r)^x = (3)^{\frac{x}{6}} \]

6. **Equating the Powers**:
Now we equate the two expressions:
\[ 3^4 = (3)^{\frac{x}{6}} \]
This implies:
\[ 4 = \frac{x}{6} \]

7. **Solving for \( x \)**:
To find \( x \), we multiply both sides by 6:
\[ x = 4 \times 6 = 24 \]

### Final Answer:
The sum will become 81 times in **24 years**.

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