Compound Interest: Calculate Borrowed Amount

This solution explains how to determine the original loan amount (principal) based on future payments and a given compound interest rate.

Problem Breakdown

Ramesh borrowed money and made two payments:

• Rs. 4,565 at the end of Year 1.
• Rs. 6,534 at the end of Year 2.

The annual compound interest rate is 10% ($r = 0.10$). We need to find the initial amount borrowed.

Calculating Principal Using Present Value

The principal amount borrowed is the sum of the present values (PV) of all future payments. The present value of a future payment is calculated using the formula:

$ PV = \frac{\text{Future Payment}}{(1 + r)^t} $

Where:

• $r$ = annual interest rate
• $t$ = number of years

Step 1: Find PV of the First Payment

The first payment of Rs. 4,565 is made after 1 year.

$ PV_1 = \frac{4565}{(1 + 0.10)^1} = \frac{4565}{1.1} $

$ PV_1 = 4150 $

Step 2: Find PV of the Second Payment

The second payment of Rs. 6,534 is made after 2 years.

$ PV_2 = \frac{6534}{(1 + 0.10)^2} = \frac{6534}{(1.1)^2} = \frac{6534}{1.21} $

$ PV_2 = 5400 $

Step 3: Sum the Present Values

The total amount borrowed is the sum of $PV_1$ and $PV_2$.

$ \text{Amount Borrowed} = PV_1 + PV_2 $

$ \text{Amount Borrowed} = 4150 + 5400 $

$ \text{Amount Borrowed} = 9550 $

Final Answer

The amount borrowed by Ramesh was Rs. 9,550.