The problem asks for the value of each of the two equal annual installments Somaraj paid to clear a loan of Rs. 11,550 at a 10% annual compound interest rate.

Loan Installment Calculation Concept

When a loan is repaid in equal installments, the sum of the present values of all installments must equal the original loan amount (principal). The present value of a future payment is calculated by discounting it back to the present using the interest rate.

Installment Formula Derivation

Let:

• Principal Loan Amount (P) = Rs. 11,550
• Annual Interest Rate (R) = 10% or 0.10
• Number of Installments (n) = 2
• Value of each equal installment = Rs. x

The present value (PV) of the installments must equal the principal:

$ P = \frac{x}{(1+R)^1} + \frac{x}{(1+R)^2} $

Calculating the Installment Value

Substitute the known values into the formula:

$ 11550 = \frac{x}{(1+0.10)^1} + \frac{x}{(1+0.10)^2} $

$ 11550 = \frac{x}{1.1} + \frac{x}{(1.1)^2} $

$ 11550 = \frac{x}{1.1} + \frac{x}{1.21} $

To solve for x, find a common denominator (1.21):

$ 11550 = \frac{1.1x}{1.21} + \frac{x}{1.21} $

$ 11550 = \frac{1.1x + x}{1.21} $

$ 11550 = \frac{2.1x}{1.21} $

Now, rearrange the equation to isolate x:

$ x = \frac{11550 \times 1.21}{2.1} $

$ x = \frac{13975.5}{2.1} $

$ x = 6655 $

Result

The value of each annual installment is Rs. 6655.