A certain amount becomes 3 times in 4 years, when invested at compound interest. In how many years will it become 243 times?

Options

A. 25
B. 60
C. 12
D. 20

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the given information
We know that a certain amount becomes 3 times in 4 years when invested at compound interest. This means that if the principal amount is \( P \), then after 4 years, the amount \( A \) is given by:
\[ A = P \times 3 \]

### Step 2: Write the formula for compound interest
The formula for the amount \( A \) in compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where:
- \( A \) is the final amount,
- \( P \) is the principal amount,
- \( r \) is the rate of interest,
- \( n \) is the number of years.

### Step 3: Set up the equation for the first scenario
From the first scenario, we can set up the equation:
\[ P \left(1 + \frac{r}{100}\right)^4 = 3P \]
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ \left(1 + \frac{r}{100}\right)^4 = 3 \]

### Step 4: Solve for the rate of interest
To find \( r \), we take the fourth root of both sides:
\[ 1 + \frac{r}{100} = 3^{\frac{1}{4}} \]
Subtracting 1 from both sides:
\[ \frac{r}{100} = 3^{\frac{1}{4}} - 1 \]
Thus,
\[ r = 100 \left(3^{\frac{1}{4}} - 1\right) \]

### Step 5: Set up the equation for the second scenario
Now, we want to find out how many years it will take for the amount to become 243 times the principal. We can express this as:
\[ A = P \times 243 \]
Using the compound interest formula:
\[ P \left(1 + \frac{r}{100}\right)^n = 243P \]
Dividing both sides by \( P \):
\[ \left(1 + \frac{r}{100}\right)^n = 243 \]

### Step 6: Express 243 in terms of powers of 3
We know that:
\[ 243 = 3^5 \]
Thus, we can rewrite the equation as:
\[ \left(1 + \frac{r}{100}\right)^n = 3^5 \]

### Step 7: Relate the two scenarios
From the first scenario, we have:
\[ \left(1 + \frac{r}{100}\right)^4 = 3 \]
Taking both equations, we can relate them:
\[ \left(1 + \frac{r}{100}\right)^n = \left(\left(1 + \frac{r}{100}\right)^4\right)^{\frac{n}{4}} = 3^{\frac{n}{4}} \]
Setting this equal to \( 3^5 \):
\[ 3^{\frac{n}{4}} = 3^5 \]
Thus, we can equate the exponents:
\[ \frac{n}{4} = 5 \]

### Step 8: Solve for \( n \)
Multiplying both sides by 4 gives:
\[ n = 20 \]

### Final Answer
The amount will become 243 times the principal in **20 years**.

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