Kamal took Rs. 6800 as a loan which along with interest is to be repaid in two equal annual instalments. If the rate of interest is `12"" (1)/(2)`%, compounded annually, then the value of each instalment is

Options

A. Rs. 8100
B. Rs. 4150
C. Rs. 4050
D. Rs. 4000

Answer

To solve the problem of Kamal's loan repayment in two equal annual installments, we can follow these steps:

### Step 1: Understand the Problem
Kamal took a loan of Rs. 6800, which he needs to repay in two equal installments. The interest rate is 12.5% compounded annually. We need to find the value of each installment.

### Step 2: Define Variables
Let the annual installment be denoted as \( x \).

### Step 3: Calculate the Effective Interest Rate
The interest rate given is 12.5%, which can be expressed as a fraction:
\[ r = \frac{12.5}{100} = \frac{25}{200} = \frac{1}{8} = 0.125 \]

### Step 4: Calculate Present Values of Installments
The present value of the first installment (to be paid at the end of the first year) is:
\[ P_1 = \frac{x}{(1 + r)} = \frac{x}{1.125} \]

The present value of the second installment (to be paid at the end of the second year) is:
\[ P_2 = \frac{x}{(1 + r)^2} = \frac{x}{(1.125)^2} = \frac{x}{1.265625} \]

### Step 5: Set Up the Equation
The total present value of both installments must equal the loan amount:
\[ P_1 + P_2 = 6800 \]
Substituting the expressions for \( P_1 \) and \( P_2 \):
\[ \frac{x}{1.125} + \frac{x}{1.265625} = 6800 \]

### Step 6: Find a Common Denominator
To combine the fractions, we can find a common denominator, which is \( 1.125 \times 1.265625 \):
\[ \frac{x \cdot 1.265625 + x \cdot 1.125}{1.125 \times 1.265625} = 6800 \]
This simplifies to:
\[ \frac{x(1.125 + 1.265625)}{1.125 \times 1.265625} = 6800 \]

### Step 7: Calculate the Sum of the Numerator
Calculating \( 1.125 + 1.265625 \):
\[ 1.125 + 1.265625 = 2.390625 \]
Thus, we have:
\[ \frac{x \cdot 2.390625}{1.125 \times 1.265625} = 6800 \]

### Step 8: Calculate the Denominator
Calculating \( 1.125 \times 1.265625 \):
\[ 1.125 \times 1.265625 = 1.4267578125 \]

### Step 9: Solve for \( x \)
Now, substituting back into the equation:
\[ \frac{x \cdot 2.390625}{1.4267578125} = 6800 \]
Multiplying both sides by \( 1.4267578125 \):
\[ x \cdot 2.390625 = 6800 \cdot 1.4267578125 \]
Calculating the right-hand side:
\[ 6800 \cdot 1.4267578125 = 9688.5 \]
Now, divide both sides by \( 2.390625 \):
\[ x = \frac{9688.5}{2.390625} \approx 4050 \]

### Conclusion
Thus, the value of each installment is approximately Rs. 4050.