In how much time does Rs. 800 at 5% per annum compound interest amount to Rs. 882?

Answer

To solve the problem of how much time it takes for Rs. 800 at 5% per annum compound interest to amount to Rs. 882, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values**:
- Principal (P) = Rs. 800
- Rate of interest (R) = 5% per annum
- Amount (A) = Rs. 882
- Time (T) = ? (This is what we need to find)

2. **Use the compound interest formula**:
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Substituting the known values into the formula:
\[ 882 = 800 \left(1 + \frac{5}{100}\right)^T \]

3. **Simplify the equation**:
Calculate \(1 + \frac{5}{100}\):
\[ 1 + \frac{5}{100} = 1 + 0.05 = 1.05 \]
Now substitute this back into the equation:
\[ 882 = 800 \times (1.05)^T \]

4. **Divide both sides by 800**:
\[ \frac{882}{800} = (1.05)^T \]
Simplifying the left side:
\[ 1.1025 = (1.05)^T \]

5. **Take logarithm on both sides**:
To solve for T, we can take the logarithm of both sides:
\[ \log(1.1025) = T \cdot \log(1.05) \]

6. **Calculate the logarithms**:
Using a calculator:
\[ \log(1.1025) \approx 0.0414 \quad \text{and} \quad \log(1.05) \approx 0.0212 \]

7. **Solve for T**:
\[ T = \frac{\log(1.1025)}{\log(1.05)} \approx \frac{0.0414}{0.0212} \approx 1.95 \]
Rounding this gives us approximately 2 years.

### Final Answer:
Therefore, it takes approximately **2 years** for Rs. 800 at 5% per annum compound interest to amount to Rs. 882.

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