In how many years Rs. 3000 amount to Rs. 5184 at 20% per annum compound interest ?

Answer

To solve the problem of how many years it takes for Rs. 3000 to amount to Rs. 5184 at a compound interest rate of 20% per annum, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the formula for compound interest**:
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
where:
- \( A \) = the amount of money accumulated after n years, including interest.
- \( P \) = the principal amount (the initial amount of money).
- \( r \) = the annual interest rate (in percentage).
- \( t \) = the number of years the money is invested or borrowed.

2. **Substitute the known values into the formula**:
Here, we have:
- \( A = 5184 \)
- \( P = 3000 \)
- \( r = 20 \)

Substituting these values into the formula gives us:
\[ 5184 = 3000 \left(1 + \frac{20}{100}\right)^t \]

3. **Simplify the equation**:
First, simplify \( 1 + \frac{20}{100} \):
\[ 1 + \frac{20}{100} = 1 + 0.2 = 1.2 \]
Now, the equation becomes:
\[ 5184 = 3000 \times (1.2)^t \]

4. **Isolate the term with \( t \)**:
Divide both sides by 3000:
\[ \frac{5184}{3000} = (1.2)^t \]
Simplifying the left side:
\[ \frac{5184}{3000} = 1.728 \]
So we have:
\[ 1.728 = (1.2)^t \]

5. **Convert to a more manageable form**:
We can express \( 1.728 \) as \( \left(\frac{6}{5}\right)^3 \) since \( 1.728 = 1.2^3 \):
\[ 1.728 = (1.2)^3 \]
Therefore, we can write:
\[ (1.2)^3 = (1.2)^t \]

6. **Equate the exponents**:
Since the bases are the same, we can set the exponents equal to each other:
\[ t = 3 \]

### Final Answer:
The time taken for Rs. 3000 to amount to Rs. 5184 at 20% per annum compound interest is **3 years**.