A loan has to be returned in two equal yearly instalments each of Rs 44,100. If the rate of interest is 5% p.a., compounded annually, then the total interest paid is :

Options

A. Rs 5,840
B. Rs 6,000
C. Rs 6,200
D. Rs 6,280

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the problem
We need to find the total interest paid on a loan that is repaid in two equal yearly installments of Rs 44,100 each at a compound interest rate of 5% per annum.

### Step 2: Identify the variables
- Installment amount (X) = Rs 44,100
- Rate of interest (R) = 5% per annum
- Number of years (n) = 2

### Step 3: Use the formula for the present value of annuities
The formula for the present value (P) of an annuity (installments) is given by:
\[ P = \frac{X}{(1 + \frac{R}{100})} + \frac{X}{(1 + \frac{R}{100})^2} \]
Where:
- \(X\) = installment amount
- \(R\) = rate of interest
- \(n\) = number of installments

### Step 4: Substitute the values into the formula
Substituting the known values into the formula:
\[ P = \frac{44100}{(1 + \frac{5}{100})} + \frac{44100}{(1 + \frac{5}{100})^2} \]
Calculating \(1 + \frac{5}{100}\):
\[ 1 + \frac{5}{100} = 1.05 \]
Now substituting this back into the formula:
\[ P = \frac{44100}{1.05} + \frac{44100}{(1.05)^2} \]

### Step 5: Calculate each term
Calculating the first term:
\[ \frac{44100}{1.05} = 42000 \]
Calculating the second term:
\[ \frac{44100}{(1.05)^2} = \frac{44100}{1.1025} \approx 40000 \]

### Step 6: Add the present values to find the principal
Now, adding both present values:
\[ P = 42000 + 40000 = 82000 \]

### Step 7: Calculate the total amount paid
The total amount paid over the two years is:
\[ \text{Total Amount Paid} = 2 \times 44100 = 88200 \]

### Step 8: Calculate the total interest paid
The total interest paid is calculated by subtracting the principal from the total amount paid:
\[ \text{Total Interest} = \text{Total Amount Paid} - P = 88200 - 82000 = 6200 \]

### Final Answer
The total interest paid is Rs 6,200.

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