If some amount becomes 27 times of itself in 3 years at certain rate of interest then find the rate of compound interest?

Options

A. `200%`
B. `300%`
C. `100%`
D. `400%`

Answer

To solve the problem of finding the rate of compound interest when an amount becomes 27 times itself in 3 years, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We know that a principal amount (P) becomes 27 times itself in 3 years. Therefore, the final amount (A) can be expressed as:
\[ A = 27P \]

2. **Identify the Formula**:
The formula for compound interest is:
\[ A = P(1 + r)^n \]
where:
- \( A \) = final amount
- \( P \) = principal amount
- \( r \) = rate of interest (in decimal)
- \( n \) = number of years

3. **Substitute Known Values**:
From the problem, we know:
- \( A = 27P \)
- \( n = 3 \)
Therefore, substituting these values into the formula gives us:
\[ 27P = P(1 + r)^3 \]

4. **Simplify the Equation**:
We can divide both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 27 = (1 + r)^3 \]

5. **Take the Cube Root**:
To solve for \( r \), we take the cube root of both sides:
\[ 1 + r = \sqrt[3]{27} \]
Since \( \sqrt[3]{27} = 3 \):
\[ 1 + r = 3 \]

6. **Solve for \( r \)**:
Now, subtract 1 from both sides:
\[ r = 3 - 1 = 2 \]

7. **Convert to Percentage**:
To find the rate of interest as a percentage, we multiply by 100:
\[ \text{Rate of Interest} = r \times 100 = 2 \times 100 = 200\% \]

8. **Final Answer**:
The rate of compound interest is:
\[ \text{200\%} \]