The simple interest on a sum of money at 4% per annum for 2 years is Rs. 80 . The compound interest in the same sum for the same period is

Options

A. Rs 82.60
B. Rs 82.20
C. Rs 81.80
D. Rs. 81.60

Answer

To solve the problem, we need to find the compound interest on a sum of money given that the simple interest for 2 years at a rate of 4% per annum is Rs. 80.

### Step-by-Step Solution:

1. **Identify the formula for Simple Interest (SI)**:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time in years.

2. **Substitute the known values into the SI formula**:
Given \( SI = 80 \), \( R = 4\% \), and \( T = 2 \) years, we can set up the equation:
\[ 80 = \frac{P \times 4 \times 2}{100} \]

3. **Simplify the equation**:
\[ 80 = \frac{8P}{100} \]
Multiplying both sides by 100:
\[ 8000 = 8P \]

4. **Solve for the principal \( P \)**:
\[ P = \frac{8000}{8} = 1000 \]
So, the principal amount is Rs. 1000.

5. **Identify the formula for Compound Interest (CI)**:
The formula for compound interest is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]

6. **Substitute the known values into the CI formula**:
Using \( P = 1000 \), \( R = 4\% \), and \( T = 2 \):
\[ CI = 1000 \left(1 + \frac{4}{100}\right)^2 - 1000 \]

7. **Calculate \( \left(1 + \frac{4}{100}\right)^2 \)**:
\[ = 1000 \left(1 + 0.04\right)^2 - 1000 \]
\[ = 1000 \left(1.04\right)^2 - 1000 \]
\[ = 1000 \times 1.0816 - 1000 \]
\[ = 1081.6 - 1000 \]

8. **Calculate the Compound Interest**:
\[ CI = 1081.6 - 1000 = 81.6 \]

Thus, the compound interest for the same sum for the same period is **Rs. 81.60**.