What will be the principal, if the rate of 5% per annum can yield ? 410 as compound interest in 2 yr?

Options

A. ? 4000
B. ? 42000
C. ? 8000
D. ?21000

Answer

To find the principal amount (P) that yields a compound interest of ₹410 at a rate of 5% per annum over 2 years, we can follow these steps:

### Step 1: Understand the Formula for Compound Interest
The formula for compound interest (CI) is:
\[ CI = A - P \]
Where:
- \( A \) is the total amount after interest
- \( P \) is the principal amount

### Step 2: Express the Total Amount (A)
The total amount \( A \) can be calculated using the formula:
\[ A = P \left(1 + \frac{R}{100}\right)^N \]
Where:
- \( R \) is the rate of interest (5% in this case)
- \( N \) is the number of years (2 years in this case)

### Step 3: Substitute Values into the Formula
Substituting the values into the formula gives:
\[ A = P \left(1 + \frac{5}{100}\right)^2 \]
\[ A = P \left(1 + 0.05\right)^2 \]
\[ A = P \left(1.05\right)^2 \]
\[ A = P \times 1.1025 \]

### Step 4: Relate Compound Interest to Principal
Now, we know that:
\[ CI = A - P \]
Substituting \( A \) from the previous step:
\[ 410 = P \times 1.1025 - P \]
\[ 410 = P(1.1025 - 1) \]
\[ 410 = P(0.1025) \]

### Step 5: Solve for Principal (P)
Now, we can solve for \( P \):
\[ P = \frac{410}{0.1025} \]
Calculating this gives:
\[ P = 4000 \]

### Conclusion
The principal amount is ₹4000.

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