A loan of `Rs. 12,300` at `5%` per annum compound interest, is to be repaid in two equal annual instalments at the end of every year. Find the amount of each Instalment.

Options

A. `Rs. 6,651`
B. `Rs. 6,615`
C. `Rs. 6,516 `
D. `Rs. 6,159 `

Answer

To solve the problem of finding the amount of each installment for a loan of Rs. 12,300 at 5% per annum compound interest, which is to be repaid in two equal annual installments, we can follow these steps:

### Step 1: Understand the Loan and Interest
We have a principal amount (P) of Rs. 12,300 and an interest rate (r) of 5% per annum. The loan is to be repaid in two equal installments at the end of each year.

### Step 2: Convert the Interest Rate
Convert the interest rate from percentage to decimal for calculations:
\[ r = \frac{5}{100} = 0.05 \]

### Step 3: Let the Installment Amount be 'A'
Let the amount of each installment be \( A \).

### Step 4: Calculate the Amount After One Year
At the end of the first year, the amount due will be the principal plus interest, minus the first installment:
\[ \text{Amount after 1 year} = P(1 + r) - A \]
Substituting the values:
\[ \text{Amount after 1 year} = 12300(1 + 0.05) - A = 12300 \times 1.05 - A = 12915 - A \]

### Step 5: Calculate the Amount After Two Years
At the end of the second year, the amount due will be the amount after the first year plus interest, minus the second installment:
\[ \text{Amount after 2 years} = (P(1 + r) - A)(1 + r) - A \]
Substituting the values:
\[ \text{Amount after 2 years} = (12915 - A)(1 + 0.05) - A = (12915 - A) \times 1.05 - A \]
Expanding this:
\[ = 13561.75 - 1.05A - A = 13561.75 - 2.05A \]

### Step 6: Set the Equation to Zero
Since the loan is fully paid off after the second installment, we set the amount after two years to zero:
\[ 13561.75 - 2.05A = 0 \]

### Step 7: Solve for A
Rearranging the equation gives:
\[ 2.05A = 13561.75 \]
\[ A = \frac{13561.75}{2.05} \approx 6600 \]

### Step 8: Conclusion
Thus, the amount of each installment is approximately Rs. 6,600.