If the compound interest of a certain sum of money for 2 years at `5%` p.a. is Rs. 328. What will be the simple interest on the same sum of money for the same time and at the same rate ?

Options

A. Rs.320
B. Rs.322
C. Rs.325
D. Rs.326

Answer

To solve the problem step by step, we will first find the principal amount using the given compound interest, and then we will calculate the simple interest using that principal.

### Step 1: Understand the Given Information
- Compound Interest (CI) for 2 years = Rs. 328
- Rate of Interest (R) = 5% per annum
- Time (T) = 2 years

### Step 2: Use the Formula for Compound Interest
The formula for Compound Interest after 2 years is:
\[ CI = P \left( \left(1 + \frac{R}{100}\right)^T - 1 \right) \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest
- \( T \) = Time in years

Substituting the known values:
\[ 328 = P \left( \left(1 + \frac{5}{100}\right)^2 - 1 \right) \]

### Step 3: Calculate \( \left(1 + \frac{5}{100}\right)^2 \)
First, calculate \( 1 + \frac{5}{100} \):
\[ 1 + \frac{5}{100} = 1 + 0.05 = 1.05 \]
Now, calculate \( (1.05)^2 \):
\[ (1.05)^2 = 1.1025 \]

### Step 4: Substitute Back into the CI Formula
Now substitute back into the CI formula:
\[ 328 = P \left( 1.1025 - 1 \right) \]
\[ 328 = P \times 0.1025 \]

### Step 5: Solve for Principal \( P \)
Now, solve for \( P \):
\[ P = \frac{328}{0.1025} \]
Calculating this gives:
\[ P = 3200 \]

### Step 6: Calculate Simple Interest (SI)
Now that we have the principal, we can calculate the Simple Interest using the formula:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the values:
\[ SI = \frac{3200 \times 5 \times 2}{100} \]
\[ SI = \frac{32000}{100} = 320 \]

### Final Answer
The Simple Interest on the same sum of money for the same time and at the same rate is Rs. 320.

---