The compound Interest on a cerattain sum of money for 2 years at `5%` is `Rs.328`, then the sum is

Options

A. `Rs. 3000`
B. `Rs. 3600`
C. `Rs. 3200`
D. `Rs. 3400`

Answer

To find the principal sum (P) when the compound interest (CI) for 2 years at a rate of 5% is Rs. 328, we can follow these steps:

### Step 1: Understand the formula for Compound Interest
The formula for compound interest is given by:
\[ A = P(1 + r)^n \]
Where:
- \(A\) = Amount after n years
- \(P\) = Principal amount (initial sum)
- \(r\) = Rate of interest (in decimal)
- \(n\) = Number of years

### Step 2: Convert the percentage to decimal
The rate of interest is given as 5%. In decimal, this is:
\[ r = \frac{5}{100} = 0.05 \]

### Step 3: Set up the equation for 2 years
Since the interest is for 2 years, we can substitute \(n = 2\) into the formula:
\[ A = P(1 + 0.05)^2 \]
This simplifies to:
\[ A = P(1.05)^2 \]

### Step 4: Calculate \( (1.05)^2 \)
Calculating \( (1.05)^2 \):
\[ (1.05)^2 = 1.1025 \]
Thus, the equation becomes:
\[ A = P \times 1.1025 \]

### Step 5: Relate Amount to Compound Interest
The compound interest for 2 years is given as Rs. 328. The amount \(A\) can also be expressed as:
\[ A = P + \text{CI} \]
Thus:
\[ A = P + 328 \]

### Step 6: Set up the equation
Now we can set the two expressions for \(A\) equal to each other:
\[ P \times 1.1025 = P + 328 \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ P \times 1.1025 - P = 328 \]
Factoring out \(P\):
\[ P(1.1025 - 1) = 328 \]
This simplifies to:
\[ P(0.1025) = 328 \]

### Step 8: Solve for \(P\)
Now, divide both sides by 0.1025 to find \(P\):
\[ P = \frac{328}{0.1025} \]
Calculating this gives:
\[ P = 3200 \]

### Conclusion
The principal sum is Rs. 3200.

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